Handbook of Integral Equations 2008

Andrei D. Polyanin Alexander V. Manzhirov SECOND EDITION I,----. . - .- -- - . ., . . ,. . , - . . - . .. Chapm...

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Andrei D. Polyanin Alexander V. Manzhirov

SECOND EDITION I,----.

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Chapman & HalllCRC Taylor 6 Francis G m p

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HANDBOOK OF

INTEGRAL EQUATIONS SECOND EDITION

Handbooks of Mathematical Equations

Handbook of Linear Partial Differential Equations for Engineers and Scientists A. D. Polyanin, 2002 Handbook of First Order Partial Differential Equations A. D. Polyanin, V. F. Zaitsev, and A. Moussiaux, 2002 Handbook of Exact Solutions for Ordinary Differential Equations, 2nd Edition A. D. Polyanin and V. F. Zaitsev, 2003 Handbook of Nonlinear Partial Differential Equations A. D. Polyanin and V. F. Zaitsev, 2004 Handbook of Integral Equations, 2nd Edition A. D. Polyanin and A. V. Manzhirov, 2008 See also: Handbook of Mathematics for Engineers and Scientists A. D. Polyanin and A. V. Manzhirov, 2007

HANDBOOK OF

INTEGRAL EQUATIONS SECOND EDITION

Andrei D. Polyanin Alexander V. Manzhirov

Chapman & Hall/CRC Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 © 2008 by Taylor & Francis Group, LLC Chapman & Hall/CRC is an imprint of Taylor & Francis Group, an Informa business No claim to original U.S. Government works Printed in the United States of America on acid-free paper 10 9 8 7 6 5 4 3 2 1 International Standard Book Number-13: 978-1-58488-507-8 (Hardcover) This book contains information obtained from authentic and highly regarded sources. Reprinted material is quoted with permission, and sources are indicated. A wide variety of references are listed. Reasonable efforts have been made to publish reliable data and information, but the author and the publisher cannot assume responsibility for the validity of all materials or for the consequences of their use. Except as permitted under U.S. Copyright Law, no part of this book may be reprinted, reproduced, transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any information storage or retrieval system, without written permission from the publishers. For permission to photocopy or use material electronically from this work, please access www.copyright.com (http:// www.copyright.com/) or contact the Copyright Clearance Center, Inc. (CCC) 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400. CCC is a not-for-profit organization that provides licenses and registration for a variety of users. For organizations that have been granted a photocopy license by the CCC, a separate system of payment has been arranged. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe. Library of Congress Cataloging-in-Publication Data Polianin, A. D. (Andrei Dmitrievich) Handbook of integral equations / Andrei D. Polyanin and Alexander V. Manzhirov. -- 2nd ed. p. cm. Includes bibliographical references and index. ISBN-13: 978-1-58488-507-8 (hardcover : alk. paper) ISBN-10: 1-58488-507-6 (hardcover : alk. paper) 1. Integral equations--Handbooks, manuals, etc. I. Manzhirov, A. V. (Aleksandr Vladimirovich) II. Title. QA431.P65 2008 515’.45--dc22 Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com and the CRC Press Web site at http://www.crcpress.com

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CONTENTS Authors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxix Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxxi Some Remarks and Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .xxxiii

Part I. Exact Solutions of Integral Equations 1. Linear Equations of the First Kind with Variable Limit of Integration . . . . . . . . . . . .

3

1.1. Equations Whose Kernels Contain Power-Law Functions . . . . . . . . . . . . . . . . . . . . . . . . 1.1-1. Kernels Linear in the Arguments x and t . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1-2. Kernels Quadratic in the Arguments x and t . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1-3. Kernels Cubic in the Arguments x and t . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1-4. Kernels Containing Higher-Order Polynomials in x and t . . . . . . . . . . . . . . . . . . 1.1-5. Kernels Containing Rational Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1-6. Kernels Containing Square Roots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1-7. Kernels Containing Arbitrary Powers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1-8. Two-Dimensional Equation of the Abel Type . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4 4 4 5 6 7 9 12 15

1.2. Equations Whose Kernels Contain Exponential Functions . . . . . . . . . . . . . . . . . . . . . . . . 1.2-1. Kernels Containing Exponential Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2-2. Kernels Containing Power-Law and Exponential Functions . . . . . . . . . . . . . . . . .

15 15 19

1.3. Equations Whose Kernels Contain Hyperbolic Functions . . . . . . . . . . . . . . . . . . . . . . . . . 1.3-1. Kernels Containing Hyperbolic Cosine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3-2. Kernels Containing Hyperbolic Sine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3-3. Kernels Containing Hyperbolic Tangent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3-4. Kernels Containing Hyperbolic Cotangent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3-5. Kernels Containing Combinations of Hyperbolic Functions . . . . . . . . . . . . . . . . .

22 22 28 36 38 39

1.4. Equations Whose Kernels Contain Logarithmic Functions . . . . . . . . . . . . . . . . . . . . . . . . 1.4-1. Kernels Containing Logarithmic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4-2. Kernels Containing Power-Law and Logarithmic Functions . . . . . . . . . . . . . . . . .

42 42 45

1.5. Equations Whose Kernels Contain Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . 1.5-1. Kernels Containing Cosine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5-2. Kernels Containing Sine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5-3. Kernels Containing Tangent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5-4. Kernels Containing Cotangent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5-5. Kernels Containing Combinations of Trigonometric Functions . . . . . . . . . . . . . .

46 46 52 60 62 63

1.6. Equations Whose Kernels Contain Inverse Trigonometric Functions . . . . . . . . . . . . . . . . 1.6-1. Kernels Containing Arccosine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6-2. Kernels Containing Arcsine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6-3. Kernels Containing Arctangent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6-4. Kernels Containing Arccotangent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

66 66 68 70 71

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1.7. Equations Whose Kernels Contain Combinations of Elementary Functions . . . . . . . . . . 1.7-1. Kernels Containing Exponential and Hyperbolic Functions . . . . . . . . . . . . . . . . . 1.7-2. Kernels Containing Exponential and Logarithmic Functions . . . . . . . . . . . . . . . . 1.7-3. Kernels Containing Exponential and Trigonometric Functions . . . . . . . . . . . . . . . 1.7-4. Kernels Containing Hyperbolic and Logarithmic Functions . . . . . . . . . . . . . . . . . 1.7-5. Kernels Containing Hyperbolic and Trigonometric Functions . . . . . . . . . . . . . . . 1.7-6. Kernels Containing Logarithmic and Trigonometric Functions . . . . . . . . . . . . . .

73 73 77 78 83 84 85

1.8. Equations Whose Kernels Contain Special Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.8-1. Kernels Containing Error Function or Exponential Integral . . . . . . . . . . . . . . . . . 1.8-2. Kernels Containing Sine and Cosine Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.8-3. Kernels Containing Fresnel Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.8-4. Kernels Containing Incomplete Gamma Functions . . . . . . . . . . . . . . . . . . . . . . . . 1.8-5. Kernels Containing Bessel Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.8-6. Kernels Containing Modified Bessel Functions . . . . . . . . . . . . . . . . . . . . . . . . . . 1.8-7. Kernels Containing Legendre Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.8-8. Kernels Containing Associated Legendre Functions . . . . . . . . . . . . . . . . . . . . . . . 1.8-9. Kernels Containing Confluent Hypergeometric Functions . . . . . . . . . . . . . . . . . . 1.8-10. Kernels Containing Hermite Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.8-11. Kernels Containing Chebyshev Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.8-12. Kernels Containing Laguerre Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.8-13. Kernels Containing Jacobi Theta Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.8-14. Kernels Containing Other Special Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . .

86 86 87 87 88 88 97 105 107 107 108 109 110 110 111

1.9. Equations Whose Kernels Contain Arbitrary Functions . . . . . . . . . . . . . . . . . . . . . . . . . . 1.9-1. Equations with Degenerate Kernel: K(x, t) = g1 (x)h1 (t) + g2 (x)h2 (t) . . . . . . . . . 1.9-2. Equations with Difference Kernel: K(x, t) = K(x – t) . . . . . . . . . . . . . . . . . . . . . 1.9-3. Other Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

111 111 114 122

1.10. Some Formulas and Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 2. Linear Equations of the Second Kind with Variable Limit of Integration . . . . . . . . . . 127 2.1. Equations Whose Kernels Contain Power-Law Functions . . . . . . . . . . . . . . . . . . . . . . . . 2.1-1. Kernels Linear in the Arguments x and t . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1-2. Kernels Quadratic in the Arguments x and t . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1-3. Kernels Cubic in the Arguments x and t . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1-4. Kernels Containing Higher-Order Polynomials in x and t . . . . . . . . . . . . . . . . . . 2.1-5. Kernels Containing Rational Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1-6. Kernels Containing Square Roots and Fractional Powers . . . . . . . . . . . . . . . . . . . 2.1-7. Kernels Containing Arbitrary Powers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

127 127 129 132 133 136 138 139

2.2. Equations Whose Kernels Contain Exponential Functions . . . . . . . . . . . . . . . . . . . . . . . . 144 2.2-1. Kernels Containing Exponential Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144 2.2-2. Kernels Containing Power-Law and Exponential Functions . . . . . . . . . . . . . . . . . 151 2.3. Equations Whose Kernels Contain Hyperbolic Functions . . . . . . . . . . . . . . . . . . . . . . . . . 2.3-1. Kernels Containing Hyperbolic Cosine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3-2. Kernels Containing Hyperbolic Sine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3-3. Kernels Containing Hyperbolic Tangent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3-4. Kernels Containing Hyperbolic Cotangent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3-5. Kernels Containing Combinations of Hyperbolic Functions . . . . . . . . . . . . . . . . .

154 154 156 161 162 164

2.4. Equations Whose Kernels Contain Logarithmic Functions . . . . . . . . . . . . . . . . . . . . . . . . 164 2.4-1. Kernels Containing Logarithmic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164 2.4-2. Kernels Containing Power-Law and Logarithmic Functions . . . . . . . . . . . . . . . . . 165

CONTENTS

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2.5. Equations Whose Kernels Contain Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . 2.5-1. Kernels Containing Cosine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5-2. Kernels Containing Sine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5-3. Kernels Containing Tangent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5-4. Kernels Containing Cotangent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5-5. Kernels Containing Combinations of Trigonometric Functions . . . . . . . . . . . . . .

166 166 169 174 175 176

2.6. Equations Whose Kernels Contain Inverse Trigonometric Functions . . . . . . . . . . . . . . . . 2.6-1. Kernels Containing Arccosine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6-2. Kernels Containing Arcsine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6-3. Kernels Containing Arctangent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6-4. Kernels Containing Arccotangent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

176 176 177 178 178

2.7. Equations Whose Kernels Contain Combinations of Elementary Functions . . . . . . . . . . 2.7-1. Kernels Containing Exponential and Hyperbolic Functions . . . . . . . . . . . . . . . . . 2.7-2. Kernels Containing Exponential and Logarithmic Functions . . . . . . . . . . . . . . . . 2.7-3. Kernels Containing Exponential and Trigonometric Functions . . . . . . . . . . . . . . . 2.7-4. Kernels Containing Hyperbolic and Logarithmic Functions . . . . . . . . . . . . . . . . . 2.7-5. Kernels Containing Hyperbolic and Trigonometric Functions . . . . . . . . . . . . . . . 2.7-6. Kernels Containing Logarithmic and Trigonometric Functions . . . . . . . . . . . . . .

179 179 180 181 185 186 187

2.8. Equations Whose Kernels Contain Special Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187 2.8-1. Kernels Containing Bessel Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187 2.8-2. Kernels Containing Modified Bessel Functions . . . . . . . . . . . . . . . . . . . . . . . . . . 189 2.9. Equations Whose Kernels Contain Arbitrary Functions . . . . . . . . . . . . . . . . . . . . . . . . . . 2.9-1. Equations with Degenerate Kernel: K(x, t) = g1 (x)h1 (t) + · · · + gn (x)hn (t) . . . . 2.9-2. Equations with Difference Kernel: K(x, t) = K(x – t) . . . . . . . . . . . . . . . . . . . . . 2.9-3. Other Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

191 191 203 212

2.10. Some Formulas and Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215 3. Linear Equations of the First Kind with Constant Limits of Integration . . . . . . . . . . . 217 3.1. Equations Whose Kernels Contain Power-Law Functions . . . . . . . . . . . . . . . . . . . . . . . . 3.1-1. Kernels Linear in the Arguments x and t . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1-2. Kernels Quadratic in the Arguments x and t . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1-3. Kernels Containing Integer Powers of x and t or Rational Functions . . . . . . . . . . 3.1-4. Kernels Containing Square Roots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1-5. Kernels Containing Arbitrary Powers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1-6. Equations Containing the Unknown Function of a Complicated Argument . . . . . 3.1-7. Singular Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

217 217 219 220 222 223 227 228

3.2. Equations Whose Kernels Contain Exponential Functions . . . . . . . . . . . . . . . . . . . . . . . . 3.2-1. Kernels Containing Exponential Functions of the Form eλ|x–t| . . . . . . . . . . . . . . . 3.2-2. Kernels Containing Exponential Functions of the Forms eλx and eµt . . . . . . . . . 3.2-3. Kernels Containing Exponential Functions of the Form eλxt . . . . . . . . . . . . . . . . 3.2-4. Kernels Containing Power-Law and Exponential Functions . . . . . . . . . . . . . . . . . 2 3.2-5. Kernels Containing Exponential Functions of the Form eλ(x±t) . . . . . . . . . . . . . 3.2-6. Other Kernels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

231 231 234 234 236 236 237

3.3. Equations Whose Kernels Contain Hyperbolic Functions . . . . . . . . . . . . . . . . . . . . . . . . . 3.3-1. Kernels Containing Hyperbolic Cosine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3-2. Kernels Containing Hyperbolic Sine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3-3. Kernels Containing Hyperbolic Tangent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3-4. Kernels Containing Hyperbolic Cotangent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

238 238 238 241 242

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3.4. Equations Whose Kernels Contain Logarithmic Functions . . . . . . . . . . . . . . . . . . . . . . . . 3.4-1. Kernels Containing Logarithmic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4-2. Kernels Containing Power-Law and Logarithmic Functions . . . . . . . . . . . . . . . . . 3.4-3. Equation Containing the Unknown Function of a Complicated Argument . . . . . .

242 242 244 246

3.5. Equations Whose Kernels Contain Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . 3.5-1. Kernels Containing Cosine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5-2. Kernels Containing Sine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5-3. Kernels Containing Tangent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5-4. Kernels Containing Cotangent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5-5. Kernels Containing a Combination of Trigonometric Functions . . . . . . . . . . . . . . 3.5-6. Equations Containing the Unknown Function of a Complicated Argument . . . . . 3.5-7. Singular Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

246 246 247 251 252 252 254 255

3.6. Equations Whose Kernels Contain Combinations of Elementary Functions . . . . . . . . . . 3.6-1. Kernels Containing Hyperbolic and Logarithmic Functions . . . . . . . . . . . . . . . . . 3.6-2. Kernels Containing Logarithmic and Trigonometric Functions . . . . . . . . . . . . . . 3.6-3. Kernels Containing Combinations of Exponential and Other Elementary Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

255 255 256 257

3.7. Equations Whose Kernels Contain Special Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.7-1. Kernels Containing Error Function, Exponential Integral or Logarithmic Integral 3.7-2. Kernels Containing Sine Integrals, Cosine Integrals, or Fresnel Integrals . . . . . . 3.7-3. Kernels Containing Gamma Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.7-4. Kernels Containing Incomplete Gamma Functions . . . . . . . . . . . . . . . . . . . . . . . . 3.7-5. Kernels Containing Bessel Functions of the First Kind . . . . . . . . . . . . . . . . . . . . . 3.7-6. Kernels Containing Bessel Functions of the Second Kind . . . . . . . . . . . . . . . . . . 3.7-7. Kernels Containing Combinations of the Bessel Functions . . . . . . . . . . . . . . . . . 3.7-8. Kernels Containing Modified Bessel Functions of the First Kind . . . . . . . . . . . . . 3.7-9. Kernels Containing Modified Bessel Functions of the Second Kind . . . . . . . . . . 3.7-10. Kernels Containing a Combination of Bessel and Modified Bessel Functions . . 3.7-11. Kernels Containing Legendre Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.7-12. Kernels Containing Associated Legendre Functions . . . . . . . . . . . . . . . . . . . . . . 3.7-13. Kernels Containing Kummer Confluent Hypergeometric Functions . . . . . . . . . . 3.7-14. Kernels Containing Tricomi Confluent Hypergeometric Functions . . . . . . . . . . 3.7-15. Kernels Containing Whittaker Confluent Hypergeometric Functions . . . . . . . . . 3.7-16. Kernels Containing Gauss Hypergeometric Functions . . . . . . . . . . . . . . . . . . . . 3.7-17. Kernels Containing Parabolic Cylinder Functions . . . . . . . . . . . . . . . . . . . . . . . . 3.7-18. Kernels Containing Other Special Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . .

258 258 258 260 260 261 264 265 266 266 269 270 271 272 274 274 276 276 277

3.8. Equations Whose Kernels Contain Arbitrary Functions . . . . . . . . . . . . . . . . . . . . . . . . . . 3.8-1. Equations with Degenerate Kernel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.8-2. Equations Containing Modulus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.8-3. Equations with Difference Kernel: K(x, t) = K(x – t) . . . . . . . . . . . . . . . . . . . . . b 3.8-4. Other Equations of the Form a K(x, t)y(t) dt = F (x) . . . . . . . . . . . . . . . . . . . . . b 3.8-5. Equations of the Form a K(x, t)y(· · ·) dt = F (x) . . . . . . . . . . . . . . . . . . . . . . . .

278 278 279 284 285

3.9. Dual Integral Equations of the First Kind . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.9-1. Kernels Containing Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.9-2. Kernels Containing Bessel Functions of the First Kind . . . . . . . . . . . . . . . . . . . . . 3.9-3. Kernels Containing Bessel Functions of the Second Kind . . . . . . . . . . . . . . . . . . 3.9-4. Kernels Containing Legendre Spherical Functions of the First Kind, i2 = –1 . . .

295 295 297 299 299

289

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4. Linear Equations of the Second Kind with Constant Limits of Integration . . . . . . . . . 4.1. Equations Whose Kernels Contain Power-Law Functions . . . . . . . . . . . . . . . . . . . . . . . . 4.1-1. Kernels Linear in the Arguments x and t . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1-2. Kernels Quadratic in the Arguments x and t . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1-3. Kernels Cubic in the Arguments x and t . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1-4. Kernels Containing Higher-Order Polynomials in x and t . . . . . . . . . . . . . . . . . . 4.1-5. Kernels Containing Rational Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1-6. Kernels Containing Arbitrary Powers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1-7. Singular Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2. Equations Whose Kernels Contain Exponential Functions . . . . . . . . . . . . . . . . . . . . . . . . 4.2-1. Kernels Containing Exponential Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2-2. Kernels Containing Power-Law and Exponential Functions . . . . . . . . . . . . . . . . . 4.3. Equations Whose Kernels Contain Hyperbolic Functions . . . . . . . . . . . . . . . . . . . . . . . . . 4.3-1. Kernels Containing Hyperbolic Cosine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3-2. Kernels Containing Hyperbolic Sine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3-3. Kernels Containing Hyperbolic Tangent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3-4. Kernels Containing Hyperbolic Cotangent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3-5. Kernels Containing Combination of Hyperbolic Functions . . . . . . . . . . . . . . . . . 4.4. Equations Whose Kernels Contain Logarithmic Functions . . . . . . . . . . . . . . . . . . . . . . . . 4.4-1. Kernels Containing Logarithmic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4-2. Kernels Containing Power-Law and Logarithmic Functions . . . . . . . . . . . . . . . . . 4.5. Equations Whose Kernels Contain Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . 4.5-1. Kernels Containing Cosine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5-2. Kernels Containing Sine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5-3. Kernels Containing Tangent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5-4. Kernels Containing Cotangent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5-5. Kernels Containing Combinations of Trigonometric Functions . . . . . . . . . . . . . . 4.5-6. Singular Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6. Equations Whose Kernels Contain Inverse Trigonometric Functions . . . . . . . . . . . . . . . . 4.6-1. Kernels Containing Arccosine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6-2. Kernels Containing Arcsine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6-3. Kernels Containing Arctangent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6-4. Kernels Containing Arccotangent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.7. Equations Whose Kernels Contain Combinations of Elementary Functions . . . . . . . . . . 4.7-1. Kernels Containing Exponential and Hyperbolic Functions . . . . . . . . . . . . . . . . . 4.7-2. Kernels Containing Exponential and Logarithmic Functions . . . . . . . . . . . . . . . . 4.7-3. Kernels Containing Exponential and Trigonometric Functions . . . . . . . . . . . . . . . 4.7-4. Kernels Containing Hyperbolic and Logarithmic Functions . . . . . . . . . . . . . . . . . 4.7-5. Kernels Containing Hyperbolic and Trigonometric Functions . . . . . . . . . . . . . . . 4.7-6. Kernels Containing Logarithmic and Trigonometric Functions . . . . . . . . . . . . . . 4.8. Equations Whose Kernels Contain Special Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.8-1. Kernels Containing Bessel Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.8-2. Kernels Containing Modified Bessel Functions . . . . . . . . . . . . . . . . . . . . . . . . . . 4.9. Equations Whose Kernels Contain Arbitrary Functions . . . . . . . . . . . . . . . . . . . . . . . . . . 4.9-1. Equations with Degenerate Kernel: K(x, t) = g1 (x)h1 (t) + · · · + gn (x)hn (t) . . . . 4.9-2. Equations with Difference Kernel: K(x, t) = K(x – t) . . . . . . . . . . . . . . . . . . . . . b 4.9-3. Other Equations of the Form y(x) + a K(x, t)y(t) dt = F (x) . . . . . . . . . . . . . . . b 4.9-4. Equations of the Form y(x) + a K(x, t)y(· · ·) dt = F (x) . . . . . . . . . . . . . . . . . . . 4.10. Some Formulas and Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

ix 301 301 301 304 307 311 314 317 319 320 320 326 327 327 329 332 333 334 334 334 335 335 335 337 342 343 344 344 344 344 345 346 347 348 348 349 349 351 352 353 353 353 355 357 357 372 374 381 390

x

CONTENTS

5. Nonlinear Equations of the First Kind with Variable Limit of Integration . . . . . . . . . 5.1. Equations with Quadratic Nonlinearity That Contain Arbitrary Parameters . . . . . . . . . . . x 5.1-1. Equations of the Form 0 y(t)y(x – t) dt = f (x) . . . . . . . . . . . . . . . . . . . . . . . . . . x 5.1-2. Equations of the Form 0 K(x, t)y(t)y(x – t) dt = f (x) . . . . . . . . . . . . . . . . . . . . x 5.1-3. Equations of the Form 0 y(t)y(· · ·) dt = f (x) . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2. Equations with Quadratic Nonlinearity That Contain Arbitrary Functions . . . . . . . . . . . . x 5.2-1. Equations of the Form a K(x, t)[Ay(t) + By 2 (t)] dt = f (x) . . . . . . . . . . . . . . . . x 5.2-2. Equations of the Form a K(x, t)y(t)y(ax + bt) dt = f (x) . . . . . . . . . . . . . . . . . . 5.3. Equations with Nonlinearity ofGeneral Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x 5.3-1. Equations of the Form a K(x, t)f (t, y(t)) dt = g(x) . . . . . . . . . . . . . . . . . . . . . . 5.3-2. Other Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

393 393 393 395 396 397 397 398 399 399 401

6. Nonlinear Equations of the Second Kind with Variable Limit of Integration . . . . . . . 6.1. Equations with Quadratic Nonlinearity  xThat Contain Arbitrary Parameters . . . . . . . . . . . 6.1-1. Equations of the Form y(x) + a K(x, t)y 2(t) dt = F (x) . . . . . . . . . . . . . . . . . . . x 6.1-2. Equations of the Form y(x) + a K(x, t)y(t)y(x – t) dt = F (x) . . . . . . . . . . . . . . 6.2. Equations with Quadratic Nonlinearity  xThat Contain Arbitrary Functions . . . . . . . . . . . . 6.2-1. Equations of the Form y(x) + a K(x, t)y 2(t) dt = F (x) . . . . . . . . . . . . . . . . . . . 6.2-2. Other Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3. Equations with Power-Law Nonlinearity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3-1. Equations Containing Arbitrary Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3-2. Equations Containing Arbitrary Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4. Equations with Exponential Nonlinearity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4-1. Equations Containing Arbitrary Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4-2. Equations Containing Arbitrary Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5. Equations with Hyperbolic Nonlinearity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5-1. Integrands with Nonlinearity of the Form cosh[βy(t)] . . . . . . . . . . . . . . . . . . . . . 6.5-2. Integrands with Nonlinearity of the Form sinh[βy(t)] . . . . . . . . . . . . . . . . . . . . . 6.5-3. Integrands with Nonlinearity of the Form tanh[βy(t)] . . . . . . . . . . . . . . . . . . . . . 6.5-4. Integrands with Nonlinearity of the Form coth[βy(t)] . . . . . . . . . . . . . . . . . . . . . 6.6. Equations with Logarithmic Nonlinearity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.6-1. Integrands Containing Power-Law Functions of x and t . . . . . . . . . . . . . . . . . . . . 6.6-2. Integrands Containing Exponential Functions of x and t . . . . . . . . . . . . . . . . . . . 6.6-3. Other Integrands . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.7. Equations with Trigonometric Nonlinearity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.7-1. Integrands with Nonlinearity of the Form cos[βy(t)] . . . . . . . . . . . . . . . . . . . . . . 6.7-2. Integrands with Nonlinearity of the Form sin[βy(t)] . . . . . . . . . . . . . . . . . . . . . . 6.7-3. Integrands with Nonlinearity of the Form tan[βy(t)] . . . . . . . . . . . . . . . . . . . . . . 6.7-4. Integrands with Nonlinearity of the Form cot[βy(t)] . . . . . . . . . . . . . . . . . . . . . . 6.8. Equations with Nonlinearity of General  xForm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.8-1. Equations of the Form y(x) + a K(x, t)G y(t) dt = F (x) . . . . . . . . . . . . . . . . .   x 6.8-2. Equations of the Form y(x) + a K(x – t)G t, y(t) dt = F (x) . . . . . . . . . . . . . . 6.8-3. Other Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

403 403 403 406 406 406 407 408 408 410 411 411 413 414 414 415 416 418 419 419 419 420 420 420 422 423 424 425 425 428 431

7. Nonlinear Equations of the First Kind with Constant Limits of Integration . . . . . . . . 7.1. Equations with Quadratic Nonlinearity That Contain Arbitrary Parameters . . . . . . . . . . . b 7.1-1. Equations of the Form a K(t)y(x)y(t) dt = F (x) . . . . . . . . . . . . . . . . . . . . . . . . b 7.1-2. Equations of the Form a K(t)y(t)y(xt) dt = F (x) . . . . . . . . . . . . . . . . . . . . . . . 7.1-3. Other Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

433 433 433 435 436

CONTENTS

7.2. Equations with Quadratic Nonlinearity That Contain Arbitrary Functions . . . . . . . . . . . . b 7.2-1. Equations of the Form a K(t)y(t)y(· · ·) dt = F (x) . . . . . . . . . . . . . . . . . . . . . . . b 7.2-2. Equations of the Form a [K(x, t)y(t) + M (x, t)y 2 (t)] dt = F (x) . . . . . . . . . . . . . 7.3. Equations with Power-Law Nonlinearity That Contain Arbitrary Functions . . . . . . . . . . b 7.3-1. Equations of the Form a K(t)y µ (x)y γ (t) dt = F (x) . . . . . . . . . . . . . . . . . . . . . . b 7.3-2. Equations of the Form a K(t)y γ (t)y(xt) dt = F (x) . . . . . . . . . . . . . . . . . . . . . . b 7.3-3. Equations of the Form a K(t)y γ (t)y(x + βt) dt = F (x) . . . . . . . . . . . . . . . . . . . b 7.3-4. Equations of the Form a [K(x, t)y(t) + M (x, t)y γ (t)] dt = f (x) . . . . . . . . . . . . . 7.3-5. Other Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4. Equations with Nonlinearity of General Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . b     7.4-1. Equations of the Form a ϕ y(x) K t, y(t) dt = F (x) . . . . . . . . . . . . . . . . . . . .   b 7.4-2. Equations of the Form a y(xt)K t, y(t) dt = F (x) . . . . . . . . . . . . . . . . . . . . . .   b 7.4-3. Equations of the Form a y(x + βt)K t, y(t) dt = F (x) . . . . . . . . . . . . . . . . . . . b 7.4-4. Equations of the Form a [K(x, t)y(t) + ϕ(x)Ψ(t, y(t))] dt = F (x) . . . . . . . . . . . 7.4-5. Other Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8. Nonlinear Equations of the Second Kind with Constant Limits of Integration . . . . . . 8.1. Equations with Quadratic Nonlinearity That Contain Arbitrary Parameters . . . . . . . . . . . b 8.1-1. Equations of the Form y(x) + a K(x, t)y 2 (t) dt = F (x) . . . . . . . . . . . . . . . . . . . b 8.1-2. Equations of the Form y(x) + a K(x, t)y(x)y(t) dt = F (x) . . . . . . . . . . . . . . . . . b 8.1-3. Equations of the Form y(x) + a K(t)y(t)y(· · ·) dt = F (x) . . . . . . . . . . . . . . . . . 8.2. Equations with Quadratic Nonlinearity That Contain Arbitrary Functions . . . . . . . . . . . . b 8.2-1. Equations of the Form y(x) + a K(x, t)y 2 (t) dt = F (x) . . . . . . . . . . . . . . . . . . . b Knm (x, t)y n (x)y m (t) dt = F (x), n + m ≤ 2 8.2-2. Equations of the Form y(x) + a b 8.2-3. Equations of the Form y(x) + a K(t)y(t)y(· · ·) dt = F (x) . . . . . . . . . . . . . . . . . 8.3. Equations with Power-Law Nonlinearity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . b 8.3-1. Equations of the Form y(x) + a K(x, t)y β (t) dt = F (x) . . . . . . . . . . . . . . . . . . . 8.3-2. Other Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4. Equations with Exponential Nonlinearity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4-1. Integrands with Nonlinearity of the Form exp[βy(t)] . . . . . . . . . . . . . . . . . . . . . . 8.4-2. Other Integrands . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.5. Equations with Hyperbolic Nonlinearity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.5-1. Integrands with Nonlinearity of the Form cosh[βy(t)] . . . . . . . . . . . . . . . . . . . . . 8.5-2. Integrands with Nonlinearity of the Form sinh[βy(t)] . . . . . . . . . . . . . . . . . . . . . 8.5-3. Integrands with Nonlinearity of the Form tanh[βy(t)] . . . . . . . . . . . . . . . . . . . . . 8.5-4. Integrands with Nonlinearity of the Form coth[βy(t)] . . . . . . . . . . . . . . . . . . . . . 8.5-5. Other Integrands . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.6. Equations with Logarithmic Nonlinearity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.6-1. Integrands with Nonlinearity of the Form ln[βy(t)] . . . . . . . . . . . . . . . . . . . . . . . 8.6-2. Other Integrands . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.7. Equations with Trigonometric Nonlinearity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.7-1. Integrands with Nonlinearity of the Form cos[βy(t)] . . . . . . . . . . . . . . . . . . . . . . 8.7-2. Integrands with Nonlinearity of the Form sin[βy(t)] . . . . . . . . . . . . . . . . . . . . . . 8.7-3. Integrands with Nonlinearity of the Form tan[βy(t)] . . . . . . . . . . . . . . . . . . . . . . 8.7-4. Integrands with Nonlinearity of the Form cot[βy(t)] . . . . . . . . . . . . . . . . . . . . . . 8.7-5. Other Integrands . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xi 437 437 443 444 444 444 445 446 446 447 447 447 449 450 451 453 453 453 454 455 456 456 457 460 464 464 465 467 467 468 468 468 469 469 470 471 472 472 473 473 473 474 475 475 476

xii

CONTENTS

8.8. Equations with Nonlinearity of General Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .   b 8.8-1. Equations of the Form y(x) + a K(|x – t|)G y(t) dt = F (x) . . . . . . . . . . . . . . .   b 8.8-2. Equations of the Form y(x) + a K(x, t)G t, y(t) dt = F (x) . . . . . . . . . . . . . . .  b  8.8-3. Equations of the Form y(x) + a G x, t, y(t) dt = F (x) . . . . . . . . . . . . . . . . . . .   b 8.8-4. Equations of the Form y(x) + a y(xt)G t, y(t) dt = F (x) . . . . . . . . . . . . . . . . .   b 8.8-5. Equations of the Form y(x) + a y(x + βt)G t, y(t) dt = F (x) . . . . . . . . . . . . . . 8.8-6. Other Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

477 477 479 483 485 487 494

Part II. Methods for Solving Integral Equations 9. Main Definitions and Formulas. Integral Transforms . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1. Some Definitions, Remarks, and Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1-1. Some Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1-2. Structure of Solutions to Linear Integral Equations . . . . . . . . . . . . . . . . . . . . . . . 9.1-3. Integral Transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1-4. Residues. Calculation Formulas. Cauchy’s Residue Theorem . . . . . . . . . . . . . . . 9.1-5. Jordan Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2. Laplace Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2-1. Definition. Inversion Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2-2. Inverse Transforms of Rational Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2-3. Inversion of Functions with Finitely Many Singular Points . . . . . . . . . . . . . . . . . 9.2-4. Convolution Theorem. Main Properties of the Laplace Transform . . . . . . . . . . . . 9.2-5. Limit Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2-6. Representation of Inverse Transforms as Convergent Series . . . . . . . . . . . . . . . . . 9.2-7. Representation of Inverse Transforms as Asymptotic Expansions as x → ∞ . . . 9.2-8. Post–Widder Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3. Mellin Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3-1. Definition. Inversion Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3-2. Main Properties of the Mellin Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3-3. Relation Among the Mellin, Laplace, and Fourier Transforms . . . . . . . . . . . . . . . 9.4. Fourier Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4-1. Definition. Inversion Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4-2. Asymmetric Form of the Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4-3. Alternative Fourier Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4-4. Convolution Theorem. Main Properties of the Fourier Transforms . . . . . . . . . . . 9.5. Fourier Cosine and Sine Transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.5-1. Fourier Cosine Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.5-2. Fourier Sine Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.6. Other Integral Transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.6-1. Hankel Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.6-2. Meijer Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.6-3. Kontorovich–Lebedev Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.6-4. Y -transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.6-5. Summary Table of Integral Transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x 10. Methods for Solving Linear Equations of the Form a K(x, t)y(t) dt = f (x) . . . . . 10.1. Volterra Equations of the First Kind . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.1-1. Equations of the First Kind. Function and Kernel Classes . . . . . . . . . . . . . . . . 10.1-2. Existence and Uniqueness of a Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.1-3. Some Problems Leading to Volterra Integral Equations of the First Kind . . . .

501 501 501 502 503 504 505 505 505 506 507 507 507 509 509 510 510 510 511 511 512 512 512 512 513 514 514 514 515 515 516 516 516 517 519 519 519 520 520

CONTENTS

xiii

10.2. Equations with Degenerate Kernel: K(x, t) = g1 (x)h1 (t) + · · · + gn (x)hn (t) . . . . . . . . . 522 10.2-1. Equations with Kernel of the Form K(x, t) = g1 (x)h1 (t) + g2 (x)h2 (t) . . . . . . . 522 10.2-2. Equations with General Degenerate Kernel . . . . . . . . . . . . . . . . . . . . . . . . . . . . 523 10.3. Reduction of Volterra Equations of the First Kind to Volterra Equations of the Second Kind . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 524 10.3-1. First Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 524 10.3-2. Second Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 524 10.4. Equations with Difference Kernel: K(x, t) = K(x – t) . . . . . . . . . . . . . . . . . . . . . . . . . . 10.4-1. Solution Method Based on the Laplace Transform . . . . . . . . . . . . . . . . . . . . . . 10.4-2. Case in Which the Transform of the Solution is a Rational Function . . . . . . . . 10.4-3. Convolution Representation of a Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.4-4. Application of an Auxiliary Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.4-5. Reduction to Ordinary Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . 10.4-6. Reduction of a Volterra Equation to a Wiener–Hopf Equation . . . . . . . . . . . . .

524 524 525 526 527 527 528

10.5. Method of Fractional Differentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.5-1. Definition of Fractional Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.5-2. Definition of Fractional Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.5-3. Main Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.5-4. Solution of the Generalized Abel Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.5-5. Erd´elyi–Kober Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

529 529 529 530 531 532

10.6. Equations with Weakly Singular Kernel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 532 10.6-1. Method of Transformation of the Kernel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 532 10.6-2. Kernel with Logarithmic Singularity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 533 10.7. Method of Quadratures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.7-1. Quadrature Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.7-2. General Scheme of the Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.7-3. Algorithm Based on the Trapezoidal Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.7-4. Algorithm for an Equation with Degenerate Kernel . . . . . . . . . . . . . . . . . . . . .

534 534 535 536 536

10.8. Equations with Infinite Integration Limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.8-1. Equation of the First Kind with Variable Lower Limit of Integration . . . . . . . . 10.8-2. Reduction to a Wiener–Hopf Equation of the First Kind . . . . . . . . . . . . . . . . . x 11. Methods for Solving Linear Equations of the Form y(x) – a K(x, t)y(t) dt = f (x)

537 537 538 539

11.1. Volterra Integral Equations of the Second Kind . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 539 11.1-1. Preliminary Remarks. Equations for the Resolvent . . . . . . . . . . . . . . . . . . . . . 539 11.1-2. Relationship Between Solutions of Some Integral Equations . . . . . . . . . . . . . . 540 11.2. Equations with Degenerate Kernel: K(x, t) = g1 (x)h1 (t) + · · · + gn (x)hn (t) . . . . . . . . . 11.2-1. Equations with Kernel of the Form K(x, t) = ϕ(x) + ψ(x)(x – t) . . . . . . . . . . . 11.2-2. Equations with Kernel of the Form K(x, t) =  ϕ(t) + ψ(t)(t – x) . . . . . . . . . . . . 11.2-3. Equations with Kernel of the Form K(x, t) = nm=1 ϕm (x)(x – t)m–1 . . . . . . . n 11.2-4. Equations with Kernel of the Form K(x, t) = m=1 ϕm (t)(t – x)m–1 . . . . . . . 11.2-5. Equations with Degenerate Kernel of the General Form . . . . . . . . . . . . . . . . . .

540 540 541 542 543 543

11.3. Equations with Difference Kernel: K(x, t) = K(x – t) . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3-1. Solution Method Based on the Laplace Transform . . . . . . . . . . . . . . . . . . . . . . 11.3-2. Method Based on the Solution of an Auxiliary Equation . . . . . . . . . . . . . . . . . 11.3-3. Reduction to Ordinary Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3-4. Reduction to a Wiener–Hopf Equation of the Second Kind . . . . . . . . . . . . . . . 11.3-5. Method of Fractional Integration for the Generalized Abel Equation . . . . . . . . 11.3-6. Systems of Volterra Integral Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

544 544 546 547 547 548 549

xiv

CONTENTS

11.4. Operator Methods for Solving Linear Integral Equations . . . . . . . . . . . . . . . . . . . . . . . . 11.4-1. Application of a Solution of a “Truncated” Equation of the First Kind . . . . . . 11.4-2. Application of the Auxiliary Equation of the Second Kind . . . . . . . . . . . . . . . . 11.4-3. Method for Solving “Quadratic” Operator Equations . . . . . . . . . . . . . . . . . . . . 11.4-4. Solution of Operator Equations of Polynomial Form . . . . . . . . . . . . . . . . . . . . 11.4-5. Some Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.5. Construction of Solutions of Integral Equations with Special Right-Hand Side . . . . . . . 11.5-1. General Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.5-2. Generating Function of Exponential Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.5-3. Power-Law Generating Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.5-4. Generating Function Containing Sines and Cosines . . . . . . . . . . . . . . . . . . . . . 11.6. Method of Model Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.6-1. Preliminary Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.6-2. Description of the Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.6-3. Model Solution in the Case of an Exponential Right-Hand Side . . . . . . . . . . . 11.6-4. Model Solution in the Case of a Power-Law Right-Hand Side . . . . . . . . . . . . . 11.6-5. Model Solution in the Case of a Sine-Shaped Right-Hand Side . . . . . . . . . . . . 11.6-6. Model Solution in the Case of a Cosine-Shaped Right-Hand Side . . . . . . . . . . 11.6-7. Some Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.7. Method of Differentiation for Integral Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.7-1. Equations with Kernel Containing a Sum of Exponential Functions . . . . . . . . 11.7-2. Equations with Kernel Containing a Sum of Hyperbolic Functions . . . . . . . . . 11.7-3. Equations with Kernel Containing a Sum of Trigonometric Functions . . . . . . . 11.7-4. Equations Whose Kernels Contain Combinations of Various Functions . . . . . . 11.8. Reduction of Volterra Equations of the Second Kind to Volterra Equations of the First Kind . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.8-1. First Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.8-2. Second Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.9. Successive Approximation Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.9-1. General Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.9-2. Formula for the Resolvent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.10. Method of Quadratures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.10-1. General Scheme of the Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.10-2. Application of the Trapezoidal Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.10-3. Case of a Degenerate Kernel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.11. Equations with Infinite Integration Limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.11-1. Equation of the Second Kind with Variable Lower Integration Limit . . . . . . 11.11-2. Reduction to a Wiener–Hopf Equation of the Second Kind . . . . . . . . . . . . . b 12. Methods for Solving Linear Equations of the Form a K(x, t)y(t) dt = f (x) . . . . . 12.1. Some Definition and Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.1-1. Fredholm Integral Equations of the First Kind . . . . . . . . . . . . . . . . . . . . . . . . . 12.1-2. Integral Equations of the First Kind with Weak Singularity . . . . . . . . . . . . . . . 12.1-3. Integral Equations of Convolution Type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.1-4. Dual Integral Equations of the First Kind . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.1-5. Some Problems Leading to Integral Equations of the First Kind . . . . . . . . . . . 12.2. Integral Equations of the First Kind with Symmetric Kernel . . . . . . . . . . . . . . . . . . . . . 12.2-1. Solution of an Integral Equation in Terms of Series in Eigenfunctions of Its Kernel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2-2. Method of Successive Approximations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

549 549 551 552 553 554 555 555 555 557 558 559 559 560 561 562 562 563 563 564 564 564 564 565 565 565 566 566 566 567 568 568 568 569 569 570 571 573 573 573 574 574 575 575 577 577 579

CONTENTS

12.3. Integral Equations of the First Kind with Nonsymmetric Kernel . . . . . . . . . . . . . . . . . . 12.3-1. Representation of a Solution in the Form of Series. General Description . . . . 12.3-2. Special Case of a Kernel That is a Generating Function . . . . . . . . . . . . . . . . . . 12.3-3. Special Case of the Right-Hand Side Represented in Terms of Orthogonal Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.3-4. General Case. Galerkin’s Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.3-5. Utilization of the Schmidt Kernels for the Construction of Solutions of Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xv 580 580 580 582 582 582

12.4. Method of Differentiation for Integral Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 583 12.4-1. Equations with Modulus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 583 12.4-2. Other Equations. Some Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 585 12.5. Method of Integral Transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.5-1. Equation with Difference Kernel on the Entire Axis . . . . . . . . . . . . . . . . . . . . . 12.5-2. Equations with Kernel K(x, t) = K(x/t) on the Semiaxis . . . . . . . . . . . . . . . . 12.5-3. Equation with Kernel K(x, t) = K(xt) and Some Generalizations . . . . . . . . . .

586 586 587 587

12.6. Krein’s Method and Some Other Exact Methods for Integral Equations of Special Types 12.6-1. Krein’s Method for an Equation with Difference Kernel with a Weak Singularity 12.6-2. Kernel is the Sum of a Nondegenerate Kernel and an Arbitrary Degenerate Kernel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.6-3. Reduction of Integral Equations of the First Kind to Equations of the Second Kind . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

588 588

591

12.7. Riemann Problem for the Real Axis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.7-1. Relationships Between the Fourier Integral and the Cauchy Type Integral . . . . 12.7-2. One-Sided Fourier Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.7-3. Analytic Continuation Theorem and the Generalized Liouville Theorem . . . . 12.7-4. Riemann Boundary Value Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.7-5. Problems with Rational Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.7-6. Exceptional Cases. The Homogeneous Problem . . . . . . . . . . . . . . . . . . . . . . . . 12.7-7. Exceptional Cases. The Nonhomogeneous Problem . . . . . . . . . . . . . . . . . . . . .

592 592 593 595 595 601 602 604

589

12.8. Carleman Method for Equations of the Convolution Type of the First Kind . . . . . . . . . 606 12.8-1. Wiener–Hopf Equation of the First Kind . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 606 12.8-2. Integral Equations of the First Kind with Two Kernels . . . . . . . . . . . . . . . . . . . 607 12.9. Dual Integral Equations of the First Kind . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.9-1. Carleman Method for Equations with Difference Kernels . . . . . . . . . . . . . . . . 12.9-2. General Scheme of Finding Solutions of Dual Integral Equations . . . . . . . . . . 12.9-3. Exact Solutions of Some Dual Equations of the First Kind . . . . . . . . . . . . . . . . 12.9-4. Reduction of Dual Equations to a Fredholm Equation . . . . . . . . . . . . . . . . . . .

610 610 611 613 615

12.10. Asymptotic Methods for Solving Equations with Logarithmic Singularity . . . . . . . . . 12.10-1. Preliminary Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.10-2. Solution for Large λ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.10-3. Solution for Small λ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.10-4. Integral Equation of Elasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

618 618 619 620 621

12.11. Regularization Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 621 12.11-1. Lavrentiev Regularization Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 621 12.11-2. Tikhonov Regularization Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 622 12.12. Fredholm Integral Equation of the First Kind as an Ill-Posed Problem . . . . . . . . . . . . 623 12.12-1. General Notions of Well-Posed and Ill-Posed Problems . . . . . . . . . . . . . . . . 623 12.12-2. Integral Equation of the First Kind is an Ill-Posed Problem . . . . . . . . . . . . . 624

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13. Methods for Solving Linear Equations of the Form y(x) –

b a

K(x, t)y(t) dt = f (x) 625

13.1. Some Definition and Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.1-1. Fredholm Equations and Equations with Weak Singularity of the Second Kind 13.1-2. Structure of the Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.1-3. Integral Equations of Convolution Type of the Second Kind . . . . . . . . . . . . . . 13.1-4. Dual Integral Equations of the Second Kind . . . . . . . . . . . . . . . . . . . . . . . . . . .

625 625 626 626 627

13.2. Fredholm Equations of the Second Kind with Degenerate Kernel. Some Generalizations 13.2-1. Simplest Degenerate Kernel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.2-2. Degenerate Kernel in the General Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.2-3. Kernel is the Sum of a Nondegenerate Kernel and an Arbitrary Degenerate Kernel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

627 627 628

13.3. Solution as a Power Series in the Parameter. Method of Successive Approximations . . 13.3-1. Iterated Kernels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.3-2. Method of Successive Approximations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.3-3. Construction of the Resolvent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.3-4. Orthogonal Kernels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

632 632 633 633 634

631

13.4. Method of Fredholm Determinants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 635 13.4-1. Formula for the Resolvent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 635 13.4-2. Recurrent Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 636 13.5. Fredholm Theorems and the Fredholm Alternative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 637 13.5-1. Fredholm Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 637 13.5-2. Fredholm Alternative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 638 13.6. Fredholm Integral Equations of the Second Kind with Symmetric Kernel . . . . . . . . . . . 13.6-1. Characteristic Values and Eigenfunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.6-2. Bilinear Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.6-3. Hilbert–Schmidt Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.6-4. Bilinear Series of Iterated Kernels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.6-5. Solution of the Nonhomogeneous Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.6-6. Fredholm Alternative for Symmetric Equations . . . . . . . . . . . . . . . . . . . . . . . . 13.6-7. Resolvent of a Symmetric Kernel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.6-8. Extremal Properties of Characteristic Values and Eigenfunctions . . . . . . . . . . 13.6-9. Kellog’s Method for Finding Characteristic Values in the Case of Symmetric Kernel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.6-10. Trace Method for the Approximation of Characteristic Values . . . . . . . . . . . . 13.6-11. Integral Equations Reducible to Symmetric Equations . . . . . . . . . . . . . . . . . . 13.6-12. Skew-Symmetric Integral Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.6-13. Remark on Nonsymmetric Kernels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

639 639 640 641 642 642 643 644 644 645 646 647 647 647

13.7. Integral Equations with Nonnegative Kernels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.7-1. Positive Principal Eigenvalues. Generalized Jentzch Theorem . . . . . . . . . . . . . 13.7-2. Positive Solutions of a Nonhomogeneous Integral Equation . . . . . . . . . . . . . . . 13.7-3. Estimates for the Spectral Radius . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.7-4. Basic Definition and Theorems for Oscillating Kernels . . . . . . . . . . . . . . . . . . 13.7-5. Stochastic Kernels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

648 648 649 649 651 654

13.8. Operator Method for Solving Integral Equations of the Second Kind . . . . . . . . . . . . . . 655 13.8-1. Simplest Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 655 13.8-2. Solution of Equations of the Second Kind on the Semiaxis . . . . . . . . . . . . . . . 655

CONTENTS

xvii

13.9. Methods of Integral Transforms and Model Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . 13.9-1. Equation with Difference Kernel on the Entire Axis . . . . . . . . . . . . . . . . . . . . . 13.9-2. Equation with the Kernel K(x, t) = t–1 Q(x/t) on the Semiaxis . . . . . . . . . . . . 13.9-3. Equation with the Kernel K(x, t) = tβ Q(xt) on the Semiaxis . . . . . . . . . . . . . 13.9-4. Method of Model Solutions for Equations on the Entire Axis . . . . . . . . . . . . .

656 656 657 658 659

13.10. Carleman Method for Integral Equations of Convolution Type of the Second Kind . . 13.10-1. Wiener–Hopf Equation of the Second Kind . . . . . . . . . . . . . . . . . . . . . . . . . 13.10-2. Integral Equation of the Second Kind with Two Kernels . . . . . . . . . . . . . . . 13.10-3. Equations of Convolution Type with Variable Integration Limit . . . . . . . . . . 13.10-4. Dual Equation of Convolution Type of the Second Kind . . . . . . . . . . . . . . .

660 660 664 668 670

13.11. Wiener–Hopf Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.11-1. Some Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.11-2. Homogeneous Wiener–Hopf Equation of the Second Kind . . . . . . . . . . . . . 13.11-3. General Scheme of the Method. The Factorization Problem . . . . . . . . . . . . 13.11-4. Nonhomogeneous Wiener–Hopf Equation of the Second Kind . . . . . . . . . . 13.11-5. Exceptional Case of a Wiener–Hopf Equation of the Second Kind . . . . . . .

671 671 673 676 677 678

13.12. Krein’s Method for Wiener–Hopf Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.12-1. Some Remarks. The Factorization Problem . . . . . . . . . . . . . . . . . . . . . . . . . 13.12-2. Solution of the Wiener–Hopf Equations of the Second Kind . . . . . . . . . . . . 13.12-3. Hopf–Fock Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

679 679 681 683

13.13. Methods for Solving Equations with Difference Kernels on a Finite Interval . . . . . . . 13.13-1. Krein’s Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.13-2. Kernels with Rational Fourier Transforms . . . . . . . . . . . . . . . . . . . . . . . . . . 13.13-3. Reduction to Ordinary Differential Equations . . . . . . . . . . . . . . . . . . . . . . . .

683 683 685 686

13.14. Method of Approximating a Kernel by a Degenerate One . . . . . . . . . . . . . . . . . . . . . . 687 13.14-1. Approximation of the Kernel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 687 13.14-2. Approximate Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 688 13.15. Bateman Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 689 13.15-1. General Scheme of the Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 689 13.15-2. Some Special Cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 690 13.16. Collocation Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.16-1. General Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.16-2. Approximate Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.16-3. Eigenfunctions of the Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

692 692 693 694

13.17. Method of Least Squares . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 695 13.17-1. Description of the Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 695 13.17-2. Construction of Eigenfunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 696 13.18. Bubnov–Galerkin Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 697 13.18-1. Description of the Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 697 13.18-2. Characteristic Values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 697 13.19. Quadrature Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.19-1. General Scheme for Fredholm Equations of the Second Kind . . . . . . . . . . . 13.19-2. Construction of the Eigenfunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.19-3. Specific Features of the Application of Quadrature Formulas . . . . . . . . . . . .

698 698 699 700

13.20. Systems of Fredholm Integral Equations of the Second Kind . . . . . . . . . . . . . . . . . . . . 701 13.20-1. Some Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 701 13.20-2. Method of Reducing a System of Equations to a Single Equation . . . . . . . . 701

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13.21. Regularization Method for Equations with Infinite Limits of Integration . . . . . . . . . . . 13.21-1. Basic Equation and Fredholm Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.21-2. Regularizing Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.21-3. Regularization Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

702 702 703 704

14. Methods for Solving Singular Integral Equations of the First Kind . . . . . . . . . . . . . . 707 14.1. Some Definitions and Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 707 14.1-1. Integral Equations of the First Kind with Cauchy Kernel . . . . . . . . . . . . . . . . . 707 14.1-2. Integral Equations of the First Kind with Hilbert Kernel . . . . . . . . . . . . . . . . . 707 14.2. Cauchy Type Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.2-1. Definition of the Cauchy Type Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.2-2. H¨older Condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.2-3. Principal Value of a Singular Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.2-4. Multivalued Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.2-5. Principal Value of a Singular Curvilinear Integral . . . . . . . . . . . . . . . . . . . . . . . 14.2-6. Poincar´e–Bertrand Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

708 708 709 709 711 712 714

14.3. Riemann Boundary Value Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.3-1. Principle of Argument. The Generalized Liouville Theorem . . . . . . . . . . . . . . 14.3-2. Hermite Interpolation Polynomial . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.3-3. Notion of the Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.3-4. Statement of the Riemann Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.3-5. Solution of the Homogeneous Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.3-6. Solution of the Nonhomogeneous Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.3-7. Riemann Problem with Rational Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . 14.3-8. Riemann Problem for a Half-Plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.3-9. Exceptional Cases of the Riemann Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.3-10. Riemann Problem for a Multiply Connected Domain . . . . . . . . . . . . . . . . . . . 14.3-11. Riemann Problem for Open Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.3-12. Riemann Problem with a Discontinuous Coefficient . . . . . . . . . . . . . . . . . . . . 14.3-13. Riemann Problem in the General Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.3-14. Hilbert Boundary Value Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

714 714 716 716 718 720 721 723 725 727 731 734 739 741 742

14.4. Singular Integral Equations of the First Kind . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.4-1. Simplest Equation with Cauchy Kernel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.4-2. Equation with Cauchy Kernel on the Real Axis . . . . . . . . . . . . . . . . . . . . . . . . 14.4-3. Equation of the First Kind on a Finite Interval . . . . . . . . . . . . . . . . . . . . . . . . . 14.4-4. General Equation of the First Kind with Cauchy Kernel . . . . . . . . . . . . . . . . . . 14.4-5. Equations of the First Kind with Hilbert Kernel . . . . . . . . . . . . . . . . . . . . . . . .

743 743 743 744 745 746

14.5. Multhopp–Kalandiya Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.5-1. Solution That is Unbounded at the Endpoints of the Interval . . . . . . . . . . . . . . 14.5-2. Solution Bounded at One Endpoint of the Interval . . . . . . . . . . . . . . . . . . . . . . 14.5-3. Solution Bounded at Both Endpoints of the Interval . . . . . . . . . . . . . . . . . . . . .

747 747 749 750

14.6. Hypersingular Integral Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.6-1. Hypersingular Integral Equations with Cauchy- and Hilbert-Type Kernels . . . 14.6-2. Definition of Hypersingular Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.6-3. Exact Solution of the Simplest Hypersingular Equation with Cauchy-Type Kernel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.6-4. Exact Solution of the Simplest Hypersingular Equation with Hilbert-Type Kernel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.6-5. Numerical Methods for Hypersingular Equations . . . . . . . . . . . . . . . . . . . . . . .

751 751 751 753 754 754

CONTENTS

xix

15. Methods for Solving Complete Singular Integral Equations . . . . . . . . . . . . . . . . . . . . 757 15.1. Some Definitions and Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.1-1. Integral Equations with Cauchy Kernel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.1-2. Integral Equations with Hilbert Kernel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.1-3. Fredholm Equations of the Second Kind on a Contour . . . . . . . . . . . . . . . . . . .

757 757 759 759

15.2. Carleman Method for Characteristic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.2-1. Characteristic Equation with Cauchy Kernel . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.2-2. Transposed Equation of a Characteristic Equation . . . . . . . . . . . . . . . . . . . . . . 15.2-3. Characteristic Equation on the Real Axis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.2-4. Exceptional Case of a Characteristic Equation . . . . . . . . . . . . . . . . . . . . . . . . . 15.2-5. Characteristic Equation with Hilbert Kernel . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.2-6. Tricomi Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

761 761 764 765 767 769 769

15.3. Complete Singular Integral Equations Solvable in a Closed Form . . . . . . . . . . . . . . . . . 770 15.3-1. Closed-Form Solutions in the Case of Constant Coefficients . . . . . . . . . . . . . . 770 15.3-2. Closed-Form Solutions in the General Case . . . . . . . . . . . . . . . . . . . . . . . . . . . 771 15.4. Regularization Method for Complete Singular Integral Equations . . . . . . . . . . . . . . . . . 15.4-1. Certain Properties of Singular Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.4-2. Regularizer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.4-3. Methods of Left and Right Regularization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.4-4. Problem of Equivalent Regularization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.4-5. Fredholm Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.4-6. Carleman–Vekua Approach to the Regularization . . . . . . . . . . . . . . . . . . . . . . . 15.4-7. Regularization in Exceptional Cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.4-8. Complete Equation with Hilbert Kernel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

772 772 774 775 776 777 778 779 780

15.5. Analysis of Solutions Singularities for Complete Integral Equations with Generalized Cauchy Kernels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.5-1. Statement of the Problem and Preliminary Remarks . . . . . . . . . . . . . . . . . . . . . 15.5-2. Auxiliary Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.5-3. Equations for the Exponents of Singularity of a Solution . . . . . . . . . . . . . . . . . 15.5-4. Analysis of Equations for Singularity Exponents . . . . . . . . . . . . . . . . . . . . . . . 15.5-5. Application to an Equation Arising in Fracture Mechanics . . . . . . . . . . . . . . . .

783 783 784 787 789 791

15.6. Direct Numerical Solution of Singular Integral Equations with Generalized Kernels . . 15.6-1. Preliminary Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.6-2. Quadrature Formulas for Integrals with the Jacobi Weight Function . . . . . . . . 15.6-3. Approximation of Solutions in Terms of a System of Orthogonal Polynomials 15.6-4. Some Special Functions and Their Calculations . . . . . . . . . . . . . . . . . . . . . . . . 15.6-5. Numerical Solution of Singular Integral Equations . . . . . . . . . . . . . . . . . . . . . . 15.6-6. Numerical Solutions of Singular Integral Equations of Bueckner Type . . . . . .

792 792 793 795 797 799 801

16. Methods for Solving Nonlinear Integral Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 805 16.1. Some Definitions and Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.1-1. Nonlinear Equations with Variable Limit of Integration (Volterra Equations) . 16.1-2. Nonlinear Equations with Constant Integration Limits (Urysohn Equations) . . 16.1-3. Some Special Features of Nonlinear Integral Equations . . . . . . . . . . . . . . . . . .

805 805 806 807

16.2. Exact Methods for Nonlinear Equations with Variable Limit of Integration . . . . . . . . . . 809 16.2-1. Method of Integral Transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 809 16.2-2. Method of Differentiation for Nonlinear Equations with Degenerate Kernel . . 810

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16.3. Approximate and Numerical Methods for Nonlinear Equations with Variable Limit of Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.3-1. Successive Approximation Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.3-2. Newton–Kantorovich Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.3-3. Collocation Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.3-4. Quadrature Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

811 811 813 815 816

16.4. Exact Methods for Nonlinear Equations with Constant Integration Limits . . . . . . . . . . 16.4-1. Nonlinear Equations with Degenerate Kernels . . . . . . . . . . . . . . . . . . . . . . . . . 16.4-2. Method of Integral Transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.4-3. Method of Differentiating for Integral Equations . . . . . . . . . . . . . . . . . . . . . . . 16.4-4. Method for Special Urysohn Equations of the First Kind . . . . . . . . . . . . . . . . . 16.4-5. Method for Special Urysohn Equations of the Second Kind . . . . . . . . . . . . . . . 16.4-6. Some Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

817 817 819 820 821 822 824

16.5. Approximate and Numerical Methods for Nonlinear Equations with Constant Integration Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.5-1. Successive Approximation Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.5-2. Newton–Kantorovich Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.5-3. Quadrature Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.5-4. Tikhonov Regularization Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

826 826 827 829 829

16.6 Existence and Uniqueness Theorems for Nonlinear Equations . . . . . . . . . . . . . . . . . . . . 830 16.6-1. Hammerstein Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 830 16.6-2. Urysohn Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 832 16.7. Nonlinear Equations with a Parameter: Eigenfunctions, Eigenvalues, Bifurcation Points 16.7-1. Eigenfunctions and Eigenvalues of Nonlinear Integral Equations . . . . . . . . . . . 16.7-2. Local Solutions of a Nonlinear Integral Equation with a Parameter . . . . . . . . . 16.7-3. Bifurcation Points of Nonlinear Integral Equations . . . . . . . . . . . . . . . . . . . . . .

834 834 835 835

17. Methods for Solving Multidimensional Mixed Integral Equations . . . . . . . . . . . . . . . 839 17.1. Some Definition and Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.1-1. Basic Classes of Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.1-2. Mixed Equations on a Finite Interval . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.1-3. Mixed Equation on a Ring-Shaped (Circular) Domain . . . . . . . . . . . . . . . . . . . 17.1-4. Mixed Equations on a Closed Bounded Set . . . . . . . . . . . . . . . . . . . . . . . . . . . .

839 839 840 841 842

17.2. Methods of Solution of Mixed Integral Equations on a Finite Interval . . . . . . . . . . . . . . 17.2-1. Equation with a Hilbert–Schmidt Kernel and a Given Right-Hand Side . . . . . . 17.2-2. Equation with Hilbert–Schmidt Kernel and Auxiliary Conditions . . . . . . . . . . 17.2-3. Equation with a Schmidt Kernel and a Given Right-Hand Side on an Interval . 17.2-4. Equation with a Schmidt Kernel and Auxiliary Conditions . . . . . . . . . . . . . . .

843 843 845 848 851

17.3. Methods of Solving Mixed Integral Equations on a Ring-Shaped Domain . . . . . . . . . . 17.3-1. Equation with a Hilbert–Schmidt Kernel and a Given Right-Hand Side . . . . . . 17.3-2. Equation with a Hilbert–Schmidt Kernel and Auxiliary Conditions . . . . . . . . . 17.3-3. Equation with a Schmidt Kernel and a Given Right-Hand Side . . . . . . . . . . . . 17.3-4. Equation with a Schmidt Kernel and Auxiliary Conditions on Ring-Shaped Domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

855 855 856 859 862

17.4. Projection Method for Solving Mixed Equations on a Bounded Set . . . . . . . . . . . . . . . . 17.4-1. Mixed Operator Equation with a Given Right-Hand Side . . . . . . . . . . . . . . . . . 17.4-2. Mixed Operator Equations with Auxiliary Conditions . . . . . . . . . . . . . . . . . . . 17.4-3. General Projection Problem for Operator Equation . . . . . . . . . . . . . . . . . . . . . .

866 866 869 873

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18. Application of Integral Equations for the Investigation of Differential Equations . . 875 18.1. Reduction of the Cauchy Problem for ODEs to Integral Equations . . . . . . . . . . . . . . . . 18.1-1. Cauchy Problem for First-Order ODEs. Uniqueness and Existence Theorems 18.1-2. Cauchy Problem for First-Order ODEs. Method of Successive Approximations 18.1-3. Cauchy Problem for Second-Order ODEs. Method of Successive Approximations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.1-4. Cauchy Problem for a Special n-Order Linear ODE . . . . . . . . . . . . . . . . . . . . .

875 875 876 876 876

18.2. Reduction of Boundary Value Problems for ODEs to Volterra Integral Equations. Calculation of Eigenvalues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 877 18.2-1. Reduction of Differential Equations to Volterra Integral Equations . . . . . . . . . 877 18.2-2. Application of Volterra Equations to the Calculation of Eigenvalues . . . . . . . . 879 18.3. Reduction of Boundary Value Problems for ODEs to Fredholm Integral Equations with the Help of the Green’s Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.3-1. Linear Ordinary Differential Equations. Fundamental Solutions . . . . . . . . . . . 18.3-2. Boundary Value Problems for nth Order Differential Equations. Green’s Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.3-3. Boundary Value Problems for Second-Order Differential Equations. Green’s Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.3-4. Nonlinear Problem of Nonisothermal Flow in Plane Channel . . . . . . . . . . . . .

881 881 882 883 884

18.4. Reduction of PDEs with Boundary Conditions of the Third Kind to Integral Equations 18.4-1. Usage of Particular Solutions of PDEs for the Construction of Other Solutions 18.4-2. Mass Transfer to a Particle in Fluid Flow Complicated by a Surface Reaction 18.4-3. Integral Equations for Surface Concentration and Diffusion Flux . . . . . . . . . . 18.4-4. Method of Numerical Integration of the Equation for Surface Concentration .

887 887 888 890 891

18.5. Representation of Linear Boundary Value Problems in Terms of Potentials . . . . . . . . . . 18.5-1. Basic Types of Potentials for the Laplace Equation and Their Properties . . . . . 18.5-2. Integral Identities. Green’s Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.5-3. Reduction of Interior Dirichlet and Neumann Problems to Integral Equations . 18.5-4. Reduction of Exterior Dirichlet and Neumann Problems to Integral Equations

892 892 895 895 896

18.6. Representation of Solutions of Nonlinear PDEs in Terms of Solutions of Linear Integral Equations (Inverse Scattering) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 898 18.6-1. Description of the Zakharov–Shabat Method . . . . . . . . . . . . . . . . . . . . . . . . . . 898 18.6-2. Korteweg–de Vries Equation and Other Nonlinear Equations . . . . . . . . . . . . . 899

Supplements Supplement 1. Elementary Functions and Their Properties . . . . . . . . . . . . . . . . . . . . . . . 905 1.1. Power, Exponential, and Logarithmic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1-1. Properties of the Power Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1-2. Properties of the Exponential Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1-3. Properties of the Logarithmic Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

905 905 905 906

1.2. Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2-1. Simplest Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2-2. Reduction Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2-3. Relations Between Trigonometric Functions of Single Argument . . . . . . . . . . . . 1.2-4. Addition and Subtraction of Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . 1.2-5. Products of Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2-6. Powers of Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2-7. Addition Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

907 907 907 908 908 908 908 909

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1.2-8. Trigonometric Functions of Multiple Arguments . . . . . . . . . . . . . . . . . . . . . . . . . 1.2-9. Trigonometric Functions of Half Argument . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2-10. Differentiation Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2-11. Integration Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2-12. Expansion in Power Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2-13. Representation in the Form of Infinite Products . . . . . . . . . . . . . . . . . . . . . . . . . 1.2-14. Euler and de Moivre Formulas. Relationship with Hyperbolic Functions . . . . .

909 909 910 910 910 910 911

1.3. Inverse Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3-1. Definitions of Inverse Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3-2. Simplest Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3-3. Some Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3-4. Relations Between Inverse Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . 1.3-5. Addition and Subtraction of Inverse Trigonometric Functions . . . . . . . . . . . . . . . 1.3-6. Differentiation Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3-7. Integration Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3-8. Expansion in Power Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

911 911 912 912 912 912 913 913 913

1.4. Hyperbolic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4-1. Definitions of Hyperbolic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4-2. Simplest Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4-3. Relations Between Hyperbolic Functions of Single Argument (x ≥ 0) . . . . . . . . 1.4-4. Addition and Subtraction of Hyperbolic Functions . . . . . . . . . . . . . . . . . . . . . . . . 1.4-5. Products of Hyperbolic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4-6. Powers of Hyperbolic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4-7. Addition Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4-8. Hyperbolic Functions of Multiple Argument . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4-9. Hyperbolic Functions of Half Argument . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4-10. Differentiation Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4-11. Integration Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4-12. Expansion in Power Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4-13. Representation in the Form of Infinite Products . . . . . . . . . . . . . . . . . . . . . . . . . 1.4-14. Relationship with Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . .

913 913 913 914 914 914 914 915 915 915 916 916 916 916 916

1.5. Inverse Hyperbolic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5-1. Definitions of Inverse Hyperbolic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5-2. Simplest Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5-3. Relations Between Inverse Hyperbolic Functions . . . . . . . . . . . . . . . . . . . . . . . . . 1.5-4. Addition and Subtraction of Inverse Hyperbolic Functions . . . . . . . . . . . . . . . . . 1.5-5. Differentiation Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5-6. Integration Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5-7. Expansion in Power Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

917 917 917 917 917 917 918 918

Supplement 2. Finite Sums and Infinite Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 919 2.1. Finite Numerical Sums . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1-1. Progressions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  2.1-2. Sums of Powers of Natural Numbers Having the Form k m . . . . . . . . . . . . . . .  2.1-3. Alternating Sums of Powers of Natural Numbers, (–1)k k m . . . . . . . . . . . . . . . 2.1-4. Other Sums Containing Integers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1-5. Sums Containing Binomial Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1-6. Other Numerical Sums . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

919 919 919 920 920 920 921

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xxiii

2.2. Finite Functional Sums . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2-1. Sums Involving Hyperbolic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2-2. Sums Involving Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3. Infinite Numerical Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3-1. Progressions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3-2. Other Numerical Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4. Infinite Functional Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4-1. Power Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4-2. Trigonometric Series in One Variable Involving Sine . . . . . . . . . . . . . . . . . . . . . . 2.4-3. Trigonometric Series in One Variable Involving Cosine . . . . . . . . . . . . . . . . . . . . 2.4-4. Trigonometric Series in Two Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

922 922 922 924 924 924 925 925 927 928 930

Supplement 3. Tables of Indefinite Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1. Integrals Involving Rational Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1-1. Integrals Involving a + bx . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1-2. Integrals Involving a + x and b + x . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1-3. Integrals Involving a2 + x2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1-4. Integrals Involving a2 – x2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1-5. Integrals Involving a3 + x3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1-6. Integrals Involving a3 – x3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1-7. Integrals Involving a4 ± x4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2. Integrals Involving Irrational Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2-1. Integrals Involving x1/2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2-2. Integrals Involving (a + bx)p/2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2-3. Integrals Involving (x2 + a2 )1/2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2-4. Integrals Involving (x2 – a2 )1/2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2-5. Integrals Involving (a2 – x2 )1/2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2-6. Integrals Involving Arbitrary Powers. Reduction Formulas . . . . . . . . . . . . . . . . . 3.3. Integrals Involving Exponential Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4. Integrals Involving Hyperbolic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4-1. Integrals Involving cosh x . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4-2. Integrals Involving sinh x . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4-3. Integrals Involving tanh x or coth x . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5. Integrals Involving Logarithmic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6. Integrals Involving Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6-1. Integrals Involving cos x (n = 1, 2, . . . ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6-2. Integrals Involving sin x (n = 1, 2, . . . ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6-3. Integrals Involving sin x and cos x . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6-4. Reduction Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6-5. Integrals Involving tan x and cot x . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.7. Integrals Involving Inverse Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . .

933 933 933 933 934 935 936 936 937 937 937 938 938 938 939 939 940 940 940 941 942 943 944 944 945 947 947 947 948

Supplement 4. Tables of Definite Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1. Integrals Involving Power-Law Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1-1. Integrals Over a Finite Interval . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1-2. Integrals Over an Infinite Interval . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2. Integrals Involving Exponential Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3. Integrals Involving Hyperbolic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4. Integrals Involving Logarithmic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

951 951 951 952 954 955 955

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4.5. Integrals Involving Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 956 4.5-1. Integrals Over a Finite Interval . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 956 4.5-2. Integrals Over an Infinite Interval . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 957 4.6. Integrals Involving Bessel Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 958 4.6-1. Integrals Over an Infinite Interval . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 958 4.6-2. Other Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 959 Supplement 5. Tables of Laplace Transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 961 5.1. General Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 961 5.2. Expressions with Power-Law Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 963 5.3. Expressions with Exponential Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 963 5.4. Expressions with Hyperbolic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 964 5.5. Expressions with Logarithmic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 965 5.6. Expressions with Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 966 5.7. Expressions with Special Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 967 Supplement 6. Tables of Inverse Laplace Transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 969 6.1. General Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 969 6.2. Expressions with Rational Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 971 6.3. Expressions with Square Roots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 975 6.4. Expressions with Arbitrary Powers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 977 6.5. Expressions with Exponential Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 978 6.6. Expressions with Hyperbolic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 979 6.7. Expressions with Logarithmic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 980 6.8. Expressions with Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 981 6.9. Expressions with Special Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 981 Supplement 7. Tables of Fourier Cosine Transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 983 7.1. General Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 983 7.2. Expressions with Power-Law Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 983 7.3. Expressions with Exponential Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 984 7.4. Expressions with Hyperbolic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 985 7.5. Expressions with Logarithmic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 985 7.6. Expressions with Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 986 7.7. Expressions with Special Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 987 Supplement 8. Tables of Fourier Sine Transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 989 8.1. General Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 989 8.2. Expressions with Power-Law Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 989 8.3. Expressions with Exponential Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 990 8.4. Expressions with Hyperbolic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 991 8.5. Expressions with Logarithmic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 992 8.6. Expressions with Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 992 8.7. Expressions with Special Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 993

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Supplement 9. Tables of Mellin Transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 997 9.1. General Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 997 9.2. Expressions with Power-Law Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 998 9.3. Expressions with Exponential Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 998 9.4. Expressions with Logarithmic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 999 9.5. Expressions with Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 999 9.6. Expressions with Special Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1000 Supplement 10. Tables of Inverse Mellin Transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1001 10.1. Expressions with Power-Law Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1001 10.2. Expressions with Exponential and Logarithmic Functions . . . . . . . . . . . . . . . . . . . . . . . 1002 10.3. Expressions with Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1003 10.4. Expressions with Special Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1004 Supplement 11. Special Functions and Their Properties . . . . . . . . . . . . . . . . . . . . . . . . . . 1007 11.1. Some Coefficients, Symbols, and Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1007 11.1-1. Binomial Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1007 11.1-2. Pochhammer Symbol . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1007 11.1-3. Bernoulli Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1008 11.1-4. Euler Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1008 11.2. Error Functions. Exponential and Logarithmic Integrals . . . . . . . . . . . . . . . . . . . . . . . . 1009 11.2-1. Error Function and Complementary Error Function . . . . . . . . . . . . . . . . . . . . . 1009 11.2-2. Exponential Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1010 11.2-3. Logarithmic Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1010 11.3. Sine Integral and Cosine Integral. Fresnel Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1011 11.3-1. Sine Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1011 11.3-2. Cosine Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1011 11.3-3. Fresnel Integrals and Generalized Fresnel Integrals . . . . . . . . . . . . . . . . . . . . . 1012 11.4. Gamma Function, Psi Function, and Beta Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1012 11.4-1. Gamma Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1012 11.4-2. Psi Function (Digamma Function) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1013 11.4-3. Beta Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1014 11.5. Incomplete Gamma and Beta Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1014 11.5-1. Incomplete Gamma Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1014 11.5-2. Incomplete Beta Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1015 11.6. Bessel Functions (Cylindrical Functions) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1016 11.6-1. Definitions and Basic Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1016 11.6-2. Integral Representations and Asymptotic Expansions . . . . . . . . . . . . . . . . . . . . 1017 11.6-3. Zeros of Bessel Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1019 11.6-4. Orthogonality Properties of Bessel Functions . . . . . . . . . . . . . . . . . . . . . . . . . . 1019 11.6-5. Hankel Functions (Bessel Functions of the Third Kind) . . . . . . . . . . . . . . . . . . 1020 11.7. Modified Bessel Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1021 11.7-1. Definitions. Basic Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1021 11.7-2. Integral Representations and Asymptotic Expansions . . . . . . . . . . . . . . . . . . . . 1022 11.8. Airy Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1023 11.8-1. Definition and Basic Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1023 11.8-2. Power Series and Asymptotic Expansions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1023

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11.9. Confluent Hypergeometric Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1024 11.9-1. Kummer and Tricomi Confluent Hypergeometric Functions . . . . . . . . . . . . . . 1024 11.9-2. Integral Representations and Asymptotic Expansions . . . . . . . . . . . . . . . . . . . . 1027 11.9-3. Whittaker Confluent Hypergeometric Functions . . . . . . . . . . . . . . . . . . . . . . . . 1027 11.10. Gauss Hypergeometric Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1028 11.10-1. Various Representations of the Gauss Hypergeometric Function . . . . . . . . . 1028 11.10-2. Basic Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1028 11.11. Legendre Polynomials, Legendre Functions, and Associated Legendre Functions . . . 1030 11.11-1. Legendre Polynomials and Legendre Functions . . . . . . . . . . . . . . . . . . . . . . 1030 11.11-2. Associated Legendre Functions with Integer Indices and Real Argument . . 1031 11.11-3. Associated Legendre Functions. General Case . . . . . . . . . . . . . . . . . . . . . . . 1032 11.12. Parabolic Cylinder Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1034 11.12-1. Definitions. Basic Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1034 11.12-2. Integral Representations, Asymptotic Expansions, and Linear Relations . . . 1035 11.13. Elliptic Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1035 11.13-1. Complete Elliptic Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1035 11.13-2. Incomplete Elliptic Integrals (Elliptic Integrals) . . . . . . . . . . . . . . . . . . . . . . 1037 11.14. Elliptic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1038 11.14-1. Jacobi Elliptic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1039 11.14-2. Weierstrass Elliptic Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1042 11.15. Jacobi Theta Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1043 11.15-1. Series Representation of the Jacobi Theta Functions. Simplest Properties . . 1043 11.15-2. Various Relations and Formulas. Connection with Jacobi Elliptic Functions 1044 11.16. Mathieu Functions and Modified Mathieu Functions . . . . . . . . . . . . . . . . . . . . . . . . . . 1045 11.16-1. Mathieu Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1045 11.16-2. Modified Mathieu Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1046 11.17. Orthogonal Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1047 11.17-1. Laguerre Polynomials and Generalized Laguerre Polynomials . . . . . . . . . . . 1047 11.17-2. Chebyshev Polynomials and Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1048 11.17-3. Hermite Polynomials and Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1050 11.17-4. Jacobi Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1051 11.17-5. Gegenbauer Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1051 11.18. Nonorthogonal Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1052 11.18-1. Bernoulli Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1052 11.18-2. Euler Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1053 Supplement 12. Some Notions of Functional Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1055 12.1. Functions of Bounded Variation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1055 12.1-1. Definition of a Function of Bounded Variation . . . . . . . . . . . . . . . . . . . . . . . . . 1055 12.1-2. Classes of Functions of Bounded Variation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1056 12.1-3. Properties of Functions of Bounded Variation . . . . . . . . . . . . . . . . . . . . . . . . . . 1056 12.1-4. Criteria for Functions to Have Bounded Variation . . . . . . . . . . . . . . . . . . . . . . 1057 12.1-5. Properties of Continuous Functions of Bounded Variation . . . . . . . . . . . . . . . . 1057 12.2. Stieltjes Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1057 12.2-1. Basic Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1057 12.2-2. Properties of the Stieltjes Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1058 12.2-3. Existence Theorems for the Stieltjes Integral . . . . . . . . . . . . . . . . . . . . . . . . . . 1058

CONTENTS

xxvii

12.3. Lebesgue Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1059 12.3-1. Riemann Integral and the Lebesgue Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . 1059 12.3-2. Sets of Zero Measure. Notion of “Almost Everywhere” . . . . . . . . . . . . . . . . . . 1060 12.3-3. Step Functions and Measurable Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1060 12.3-4. Definition and Properties of the Lebesgue Integral . . . . . . . . . . . . . . . . . . . . . . 1061 12.3-5. Measurable Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1062 12.3-6. Integration Over Measurable Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1063 12.3-7. Case of an Infinite Interval . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1063 12.3-8. Case of Several Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1064 12.3-9. Spaces Lp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1064 12.4. Linear Normed Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1065 12.4-1. Linear Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1065 12.4-2. Linear Normed Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1065 12.4-3. Space of Continuous Functions C(a, b) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1066 12.4-4. Lebesgue Space Lp (a, b) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1066 12.4-5. H¨older Space Cα (0, 1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1066 12.4-6. Space of Functions of Bounded Variation V (0, 1) . . . . . . . . . . . . . . . . . . . . . . . 1066 12.5. Euclidean and Hilbert Spaces. Linear Operators in Hilbert Spaces . . . . . . . . . . . . . . . . 1067 12.5-1. Preliminary Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1067 12.5-2. Euclidean and Hilbert Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1067 12.5-3. Linear Operators in Hilbert Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1068 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1071 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1081

AUTHORS

Andrei D. Polyanin, D.Sc., Ph.D., is a well-known scientist of broad interests and is active in various areas of mathematics, mechanics, and chemical engineering sciences. He is one of the most prominent authors in the field of reference literature on mathematics and physics. Professor Polyanin graduated with honors from the Department of Mechanics and Mathematics of Moscow State University in 1974. He received his Ph.D. degree in 1981 and D.Sc. degree in 1986 at the Institute for Problems in Mechanics of the Russian (former USSR) Academy of Sciences. Since 1975, Professor Polyanin has been working at the Institute for Problems in Mechanics of the Russian Academy of Sciences; he is also Professor of Mathematics at Bauman Moscow State Technical University. He is a member of the Russian National Committee on Theoretical and Applied Mechanics and of the Mathematics and Mechanics Expert Council of the Higher Certification Committee of the Russian Federation. Professor Polyanin has made important contributions to exact and approximate analytical methods in the theory of differential equations, mathematical physics, integral equations, engineering mathematics, theory of heat and mass transfer, and chemical hydrodynamics. He obtained exact solutions for several thousand ordinary differential, partial differential, and integral equations. Professor Polyanin is an author of more than 30 books in English, Russian, German, and Bulgarian as well as over 120 research papers and three patents. He has written a number of fundamental handbooks, including A. D. Polyanin and V. F. Zaitsev, Handbook of Exact Solutions for Ordinary Differential Equations, CRC Press, 1995 and 2003; A. D. Polyanin and A. V. Manzhirov, Handbook of Integral Equations, CRC Press, 1998; A. D. Polyanin, Handbook of Linear Partial Differential Equations for Engineers and Scientists, Chapman & Hall/CRC Press, 2002; A. D. Polyanin, V. F. Zaitsev, and A. Moussiaux, Handbook of First Order Partial Differential Equations, Taylor & Francis, 2002; A. D. Polyanin and V. F. Zaitsev, Handbook of Nonlinear Partial Differential Equations, Chapman & Hall/CRC Press, 2004, and A. D. Polyanin and A. V. Manzhirov, Handbook of Mathematics for Engineers and Scientists, Chapman & Hall/CRC Press, 2007. Professor Polyanin is editor of the book series Differential and Integral Equations and Their Applications, Chapman & Hall/CRC Press, London/Boca Raton, and Physical and Mathematical Reference Literature, Fizmatlit, Moscow. He is also Editor-in-Chief of the international scientificeducational Website EqWorld—The World of Mathematical Equations (http://eqworld.ipmnet.ru), which is visited by over 1700 users a day worldwide. Professor Polyanin is a member of the Editorial Board of the journal Theoretical Foundations of Chemical Engineering. In 1991, Professor Polyanin was awarded a Chaplygin Prize of the Russian Academy of Sciences for his research in mechanics. In 2001, he received an award from the Ministry of Education of the Russian Federation. Address: Institute for Problems in Mechanics, Vernadsky Ave. 101 Bldg 1, 119526 Moscow, Russia Home page: http://eqworld.ipmnet.ru/polyanin-ew.htm

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Alexander V. Manzhirov, D.Sc., Ph.D., is a noted scientist in the fields of mechanics and applied mathematics, integral equations, and their applications. After graduating with honors from the Department of Mechanics and Mathematics of Rostov State University in 1979, Alexander Manzhirov attended postgraduate courses at Moscow Institute of Civil Engineering. He received his Ph.D. degree in 1983 at Moscow Institute of Electronic Engineering Industry and D.Sc. degree in 1993 at the Institute for Problems in Mechanics of the Russian (former USSR) Academy of Sciences. Since 1983, Alexander Manzhirov has been working at the Institute for Problems in Mechanics of the Russian Academy of Sciences. Currently, he is head of the Laboratory for Modeling in Solid Mechanics at the same institute. Professor Manzhirov is also head of a branch of the Department of Applied Mathematics at Bauman Moscow State Technical University, professor of mathematics at Moscow State University of Engineering and Computer Science, vice-chairman of Mathematics and Mechanics Expert Council of the Higher Certification Committee of the Russian Federation, executive secretary of Solid Mechanics Scientific Council of the Russian Academy of Sciences, and expert in mathematics, mechanics, and computer science of the Russian Foundation for Basic Research. He is a member of the Russian National Committee on Theoretical and Applied Mechanics and the European Mechanics Society (EUROMECH), and member of the editorial board of the journal Mechanics of Solids and the international scientific-educational Website EqWorld—The World of Mathematical Equations (http://eqworld.ipmnet.ru). Professor Manzhirov has made important contributions to new mathematical methods for solving problems in the fields of integral equations and their applications, mechanics of growing solids, contact mechanics, tribology, viscoelasticity, and creep theory. He is an author of more than ten books (including Contact Problems in Mechanics of Growing Solids [in Russian], Nauka, Moscow, 1991; Handbook of Integral Equations, CRC Press, Boca Raton, 1998; Handbuch der Integralgleichungen: Exacte L¨osungen, Spektrum Akad. Verlag, Heidelberg, 1999; Contact Problems in the Theory of Creep [in Russian], National Academy of Sciences of Armenia, Erevan, 1999; A. D. Polyanin and A. V. Manzhirov, Handbook of Mathematics for Engineers and Scientists, Chapman & Hall/CRC Press, Boca Raton, 2007), more than 70 research papers, and two patents. Professor Manzhirov is a winner of the First Competition of the Science Support Foundation 2001, Moscow. Address: Institute for Problems in Mechanics, Vernadsky Ave. 101 Bldg 1, 119526 Moscow, Russia. Home page: http://eqworld.ipmnet.ru/en/board/manzhirov.htm.

PREFACE TO THE NEW EDITION Handbook of Integral Equations, Second Edition, a unique reference for engineers and scientists, contains over 2,500 integral equations with solutions, as well as analytical and numerical methods for solving linear and nonlinear equations. It considers Volterra, Fredholm, Wiener–Hopf, Hammerstein, Urysohn, and other equations, which arise in mathematics, physics, engineering sciences, economics, etc. In total, the number of equations described is an order of magnitude greater than in any other book available. The second edition has been substantially updated, revised, and extended. It includes new chapters on mixed multidimensional equations, methods of integral equations for ODEs and PDEs, and about 400 new equations with exact solutions. It presents a considerable amount of new material on Volterra, Fredholm, singular, hypersingular, dual, and nonlinear integral equations, integral transforms, and special functions. Many examples were added for illustrative purposes. The new edition has been increased by a total of over 300 pages. Note that the first part of the book can be used as a database of test problems for numerical and approximate methods for solving linear and nonlinear integral equations. We would like to express our deep gratitude to Alexei Zhurov and Vasilii Silvestrov for fruitful discussions. We also appreciate the help of Grigory Yosifian in translating new sections of this book and valuable remarks. The authors hope that the handbook will prove helpful for a wide audience of researchers, college and university teachers, engineers, and students in various fields of applied mathematics, mechanics, physics, chemistry, biology, economics, and engineering sciences. A. D. Polyanin A. V. Manzhirov

PREFACE TO THE FIRST EDITION Integral equations are encountered in various fields of science and numerous applications (in elasticity, plasticity, heat and mass transfer, oscillation theory, fluid dynamics, filtration theory, electrostatics, electrodynamics, biomechanics, game theory, control, queuing theory, electrical engineering, economics, medicine, etc.). Exact (closed-form) solutions of integral equations play an important role in the proper understanding of qualitative features of many phenomena and processes in various areas of natural science. Lots of equations of physics, chemistry, and biology contain functions or parameters which are obtained from experiments and hence are not strictly fixed. Therefore, it is expedient to choose the structure of these functions so that it would be easier to analyze and solve the equation. As a possible selection criterion, one may adopt the requirement that the model integral equation admits a solution in a closed form. Exact solutions can be used to verify the consistency and estimate errors of various numerical, asymptotic, and approximate methods. More than 2,100 integral equations and their solutions are given in the first part of the book (Chapters 1–6). A lot of new exact solutions to linear and nonlinear equations are included. Special attention is paid to equations of general form, which depend on arbitrary functions. The other equations contain one or more free parameters (the book actually deals with families of integral xxxi

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PREFACE

equations); it is the reader’s option to fix these parameters. In total, the number of equations described in this handbook is an order of magnitude greater than in any other book currently available. The second part of the book (Chapters 7–14) presents exact, approximate analytical, and numerical methods for solving linear and nonlinear integral equations. Apart from the classical methods, some new methods are also described. When selecting the material, the authors have given a pronounced preference to practical aspects of the matter; that is, to methods that allow effectively “constructing” the solution. For the reader’s better understanding of the methods, each section is supplied with examples of specific equations. Some sections may be used by lecturers of colleges and universities as a basis for courses on integral equations and mathematical physics equations for graduate and postgraduate students. For the convenience of a wide audience with different mathematical backgrounds, the authors tried to do their best, wherever possible, to avoid special terminology. Therefore, some of the methods are outlined in a schematic and somewhat simplified manner, with necessary references made to books where these methods are considered in more detail. For some nonlinear equations, only solutions of the simplest form are given. The book does not cover two-, three-, and multidimensional integral equations. The handbook consists of chapters, sections, and subsections. Equations and formulas are numbered separately in each section. The equations within a section are arranged in increasing order of complexity. The extensive table of contents provides rapid access to the desired equations. For the reader’s convenience, the main material is followed by a number of supplements, where some properties of elementary and special functions are described, tables of indefinite and definite integrals are given, as well as tables of Laplace, Mellin, and other transforms, which are used in the book. The first and second parts of the book, just as many sections, were written so that they could be read independently from each other. This allows the reader to quickly get to the heart of the matter. We would like to express our deep gratitude to Rolf Sulanke and Alexei Zhurov for fruitful discussions and valuable remarks. We also appreciate the help of Vladimir Nazaikinskii and Alexander Shtern in translating the second part of this book, and are thankful to Inna Shingareva for her assistance in preparing the camera-ready copy of the book. The authors hope that the handbook will prove helpful for a wide audience of researchers, college and university teachers, engineers, and students in various fields of mathematics, mechanics, physics, chemistry, biology, economics, and engineering sciences. A. D. Polyanin A. V. Manzhirov

SOME REMARKS AND NOTATION 1. In Chapters 1–11, 14, and 18 in the original integral equations, the independent variable is denoted by x, the integration variable by t, and the unknown function by y = y(x). 2. For a function of one variable f = f (x), we use the following notation for the derivatives: fx =

df , dx

 fxx =

d2 f , dx2

 fxxx =

d3 f , dx3

 fxxxx =

d4 f , dx4

and fx(n) =

dn f for n ≥ 5. dxn

Occasionally, we use the similar notation for partial derivatives of a function of two variables, ∂ for example, Kx (x, t) = K(x, t). ∂x  d n g(x), which is defined recursively by 3. In some cases, we use the operator notation f (x) dx 

d f (x) dx

n

d g(x) = f (x) dx



d f (x) dx

n–1

g(x) .

4. It is indicated in the beginning of Chapters 1–8 that f = f (x), g = g(x), K = K(x), etc. are arbitrary functions, and A, B, etc. are free parameters. This means that: (a) f = f (x), g = g(x), K = K(x), etc. are assumed to be continuous real-valued functions of real arguments;* (b) if the solution contains derivatives of these functions, then the functions are assumed to be sufficiently differentiable;** (c) if the solution contains integrals with these functions (in combination with other functions), then the integrals are supposed to converge; (d) the free parameters A, B, etc. may assume any real values for which the expressions occurring A in the equation and the solution make sense (for example, if a solution contains a factor , 1–A then it is implied that A ≠ 1; as a rule, this is not specified in the text). 5. The notations Re z and Im z stand, respectively, for the real and the imaginary part of a complex quantity z. 6. In the first part of the book (Chapters 1–8) when referencing a particular equation, we use a notation like 2.3.15, which implies equation 15 from Section 2.3. 7. To highlight portions of the text, the following symbols are used in the book:  indicates important information pertaining to a group of equations (Chapters 1–8); indicates the literature used in the preparation of the text in specific equations (Chapters 1–8) or sections (Chapters 9–18). * Less severe restrictions on these functions are presented in the second part of the book. ** Restrictions (b) and (c) imposed on f = f (x), g = g(x), K = K(x), etc. are not mentioned in the text.

xxxiii

Part I

Exact Solutions of Integral Equations

Chapter 1

Linear Equations of the First Kind with Variable Limit of Integration  Notation: f = f (x), g = g(x), h = h(x), K = K(x), and M = M (x) are arbitrary functions (these may be composite functions of the argument depending on two variables x and t); A, B, C, D, E, a, b, c, α, β, γ, λ, and µ are free parameters; and m and n are nonnegative integers.  Preliminary remarks. For equations of the form x K(x, t)y(t) dt = f (x),

a ≤ x ≤ b,

a

where the functions K(x, t) and f (x) are continuous, the right-hand side must satisfy the following conditions: 1◦ . If K(a, a) ≠ 0, then we must have f (a) = 0 (for example, the right-hand sides of equations 1.1.1 and 1.2.1 must satisfy this condition). 2◦ . If K(a, a) = Kx (a, a) = · · · = Kx(n–1) (a, a) = 0, 0 < Kx(n) (a, a) < ∞, then the right-hand side of the equation must satisfy the conditions f (a) = fx (a) = · · · = fx(n) (a) = 0. For example, with n = 1, these are constraints for the right-hand side of equation 1.1.2. 3◦ . If K(a, a) = Kx (a, a) = · · · = Kx(n–1) (a, a) = 0, Kx(n) (a, a) = ∞, then the right-hand side of the equation must satisfy the conditions f (a) = fx (a) = · · · = fx(n–1) (a) = 0. For example, with n = 1, this is a constraint for the right-hand side of equation 1.1.30. 4◦ . For unbounded K(x, t) with integrable power-law or logarithmic singularity at x = t and continuous f (x), no additional conditions are imposed on the right-hand side of the integral equation (e.g., see Abel’s equation 1.1.36). In the case of a difference kernel, K(x, t) = K(x – t), that can be represented as x → t in the form   K(x – t) = A(x – t)λ + o (x – t)λ (0 < |A| < ∞), the right-hand side of the integral equation, for λ ≥ 0, must satisfy the conditions f (a) = fx (a) = · · · = fx([λ]) (a) = 0, where [λ] is the integer part of λ. For –1 < λ < 0, there are no additional conditions imposed on the function f (x). In Chapter 1, conditions 1◦ –3◦ are as a rule not specified. 3

4

LINEAR EQUATIONS OF THE FIRST KIND WITH VARIABLE LIMIT OF INTEGRATION

1.1. Equations Whose Kernels Contain Power-Law Functions 1.1-1. Kernels Linear in the Arguments x and t.

x

y(t) dt = f (x).

1. a

2.

Solution: y(x) = fx (x). x (x – t)y(t) dt = f (x).

3.

 Solution: y(x) = fxx (x). x (Ax + Bt + C)y(t) dt = f (x).

a

a

This is a special case of equation 1.9.5 with g(x) = x. 1◦ . Solution with B ≠ –A:

x – A

– B  d A+B A+B (A + B)x + C y(x) = (A + B)t + C ft (t) dt . dx a 2◦ . Solution with B = –A:

   A  A  x 1 d  exp – x t ft (t) dt . y(x) = exp C dx C C a

1.1-2. Kernels Quadratic in the Arguments x and t.

x

4.

(x – t)2 y(t) dt = f (x),

a

5.

 (x). Solution: y(x) = 12 fxxx x (x2 – t2 )y(t) dt = f (x), a

 f (a) = fx (a) = fxx (a) = 0.

f (a) = fx (a) = 0.

6.

This is a special case of equation 1.9.2 with g(x) = x2 .  1  Solution: y(x) = xfxx (x) – fx (x) . 2x2 x   Ax2 + Bt2 y(t) dt = f (x).

7.

This is a special case of equation1.9.4 with g(x) = x2 . For B= –A, see equation 1.1.5. x 2A 2B d 1 x– A+B t– A+B ft (t) dt . Solution: y(x) = A + B dx a x   Ax2 + Bt2 + C y(t) dt = f (x).

a

a

This is a special case of equation 1.9.5 with g(x) = x2 . Solution:

x A B d y(x) = sign ϕ(x) |ϕ(x)|– A+B |ϕ(t)|– A+B ft (t) dt , dx a

ϕ(x) = (A + B)x2 + C.

1.1. EQUATIONS WHOSE KERNELS CONTAIN POWER-LAW FUNCTIONS



x

8.

 Ax2 + (B – A)xt – Bt2 y(t) dt = f (x),

a

5

f (a) = fx (a) = 0.

Differentiating with respect to x yields an equation of the form 1.1.3:

x

[2Ax + (B – A)t]y(t) dt = fx (x).

a

Solution: y(x) =

x

9.

  x 2A A–B d 1 x– A+B t A+B ftt (t) dt . A + B dx a

  Ax2 + Bt2 + Cx + Dt + E y(t) dt = f (x).

a

This is a special case of equation 1.9.6 with g(x) = Ax2 + Cx and h(t) = Bt2 + Dt + E.

x

10.

  Axt + Bt2 + Cx + Dt + E y(t) dt = f (x).

a

This is a special case of equation 1.9.15 with g1 (x) = x, h1 (t) = At + C, g2 (x) = 1, and h2 (t) = Bt2 + Dt + E.

x

11.

  Ax2 + Bxt + Cx + Dt + E y(t) dt = f (x).

a

This is a special case of equation 1.9.15 with g1 (x) = Bx + D, h1 (t) = t, g2 (x) = Ax2 + Cx + E, and h2 (t) = 1. 1.1-3. Kernels Cubic in the Arguments x and t.

x

12.

(x – t)3 y(t) dt = f (x),

a

  f (a) = fx (a) = fxx (a) = fxxx (a) = 0.

 Solution: y(x) = 16 fxxxx (x).



x

13. a

14.

(x3 – t3 )y(t) dt = f (x),

f (a) = fx (a) = 0.

This is a special case of equation 1.9.2 with g(x) = x3 .  1  Solution: y(x) = xfxxx (x) – 2fx (x) . 3x3 x   Ax3 + Bt3 y(t) dt = f (x). a

This is a special case of equation 1.9.4 with g(x)= x3 . For B = –A, see equation  1.1.13. x 3A 3B d 1 x– A+B t– A+B ft (t) dt . Solution with 0 ≤ a ≤ x: y(x) = A + B dx a

x

15.

  Ax3 + Bt3 + C y(t) dt = f (x).

a

This is a special case of equation 1.9.5 with g(x) = x3 .

6

LINEAR EQUATIONS OF THE FIRST KIND WITH VARIABLE LIMIT OF INTEGRATION



x

16.

(x2 t – xt2 )y(t) dt = f (x),

a

f (a) = fx (a) = 0.

17.

2 This is a special case of equation  with g(x) = x and h(x) = x.  1.9.11 2 1 1 d f (x) . Solution: y(x) = 2 x dx x x (Ax2 t + Bxt2 )y(t) dt = f (x).

18.

This is a special case of equation 1.9.12 with g(x) = x2 and h(x) = x. For B = –A, see equation 1.1.16. Solution:  

x B A 1 d 1 d y(x) = x– A+B f (t) dt . t– A+B (A + B)x dx dt t a x (Ax3 + Bxt2 )y(t) dt = f (x).

a

a

19.

This is a special case of equation 1.9.15 with g1 (x) = Ax3 , h1 (t) = 1, g2 (x) = Bx, and h2 (t) = t2 . x (Ax3 + Bx2 t)y(t) dt = f (x).

20.

This is a special case of equation 1.9.15 with g1 (x) = Ax3 , h1 (t) = 1, g2 (x) = Bx2 , and h2 (t) = t. x (Ax2 t + Bt3 )y(t) dt = f (x).

a

a

21.

This is a special case of equation 1.9.15 with g1 (x) = Ax2 , h1 (t) = t, g2 (x) = B, and h2 (t) = t3 . x (Axt2 + Bt3 )y(t) dt = f (x). a

22.

This is a special case of equation 1.9.15 with g1 (x) = Ax, h1 (t) = t2 , g2 (x) = B, and h2 (t) = t3 . x   A3 x3 + B3 t3 + A2 x2 + B2 t2 + A1 x + B1 t + C y(t) dt = f (x). a

This is a special case of equation 1.9.6 with g(x) = A3 x3 + A2 x2 + A1 x + C and h(t) = B3 t3 + B2 t2 + B1 t. 1.1-4. Kernels Containing Higher-Order Polynomials in x and t.

x

23.

(x – t)n y(t) dt = f (x),

n = 1, 2, . . .

a

It is assumed that the right-hand of the equation satisfies the conditions f (a) = fx (a) = · · · = fx(n) (a) = 0. 1 (n+1) f Solution: y(x) = (x). n! x Example. For f (x) = Axm , where m is a positive integer, m > n, the solution has the form y(x) =

Am! xm–n–1 . n! (m – n – 1)!

1.1. EQUATIONS WHOSE KERNELS CONTAIN POWER-LAW FUNCTIONS



x

24.

(xn – tn )y(t) dt = f (x),

a

Solution: y(x) =

x

25.

f (a) = fx (a) = 0,

7

n = 1, 2, . . .

  1 d fx (x) . n dx xn–1

 n n+1  t x – xn tn+1 y(t) dt = f (x),

n = 2, 3, . . .

a n+1 This is a special case of equation and h(x) = xn .  1.9.11  with g(x) = x 2 f (x) 1 d . Solution: y(x) = n 2 x dx xn

1.1-5. Kernels Containing Rational Functions.

x

26. 0

y(t) dt x+t

= f (x).

1◦ . For a polynomial right-hand side, f (x) =

N 

An xn , the solution has the form

n=0

y(x) =

N  An n x , Bn n=0

2◦ . For f (x) = xλ

N 

  n  (–1)k . Bn = (–1)n ln 2 + k k=1

An xn , where λ is an arbitrary number (λ > –1), the solution has the

n=0

form y(x) = xλ



N  An n x , Bn

1

Bn = 0

n=0

tλ+n dt . 1+t

  N An xn , the solution has the form 3◦ . For f (x) = ln x n=0 N N  An n  An In n x + x , Bn Bn2 n=0 n=0    2   n n  (–1)k (–1)k n n π , In = (–1) . Bn = (–1) ln 2 + + k 12 k2

y(x) = ln x

k=1

4◦ . For f (x) =

N 

k=1

 An ln x)n , the solution of the equation has the form

n=0

y(x) =

N 

An Yn (x),

n=0

where the functions Yn = Yn (x) are given by Yn (x) =

 λ 

x dn , n dλ I(λ) λ=0

I(λ) = 0

1

z λ dz . 1+z

8

LINEAR EQUATIONS OF THE FIRST KIND WITH VARIABLE LIMIT OF INTEGRATION

5◦ . For f (x) =

N 

N 

An cos(λn ln x) +

n=1

Bn sin(λn ln x), the solution of the equation has the

n=1

form y(x) =

N 

Cn cos(λn ln x) +

n=1

N 

Dn sin(λn ln x),

n=1

where the constants Cn and Dn are found by the method of undetermined coefficients. 6◦ . For arbitrary f (x), the transformation x = 12 e2z ,

t = 12 e2τ ,

y(t) = e–τ w(τ ),

f (x) = e–z g(z)

leads to an integral equation with difference kernel of the form 1.9.27:

z

–∞



x

27.

y(t) dt ax + bt

0

= f (x),

a > 0,

w(τ ) dτ = g(z). cosh(z – τ )

a + b > 0.

1◦ . For a polynomial right-hand side, f (x) =

N 

An xn , the solution has the form

n=0



N  An n x , y(x) = Bn n=0

2◦ . For f (x) = xλ

N 

1

tn dt . a + bt

Bn = 0

An xn , where λ is an arbitrary number (λ > –1), the solution has the

n=0

form



N  An n x , y(x) = x Bn λ

1

Bn = 0

n=0

tλ+n dt . a + bt

  N 3◦ . For f (x) = ln x An xn , the solution has the form n=0 N N  An n  An Cn n x – x , y(x) = ln x Bn Bn2 n=0 n=0



1

Bn = 0

tn dt , a + bt

Cn = 0

1

tn ln t dt. a + bt

4◦ . For some other special forms of the right-hand side (see items 4 and 5, equation 1.1.26), the solution may be found by the method of undetermined coefficients. 28. 0

x

y(t) dt ax2 + bt2

= f (x),

a > 0,

a + b > 0.

1◦ . For a polynomial right-hand side, f (x) =

N 

An xn , the solution has the form

n=0

y(x) =

N  An n+1 x , Bn n=0

Bn = 0

1

tn+1 dt . a + bt2

9

1.1. EQUATIONS WHOSE KERNELS CONTAIN POWER-LAW FUNCTIONS Example. For a = b = 1 and f (x) = Ax2 + Bx + C, the solution of the integral equation is: y(x) =

2◦ . For f (x) = xλ

N 

2A 4B 2 2C x3 + x + x. 1 – ln 2 4–π ln 2

An xn , where λ is an arbitrary number (λ > –1), the solution has the

n=0

form



N  An n+1 x , y(x) = x Bn n=0 λ

1

Bn = 0

tλ+n+1 dt . a + bt2

  N An xn , the solution has the form 3◦ . For f (x) = ln x n=0

y(x) = ln x

x

y(t) dt = f (x), axm + btm

29. 0



N N  An n+1  An Cn n+1 x – x , Bn Bn2 n=0 n=0

a > 0,

tn+1 dt , a + bt2

0

a + b > 0,

1◦ . For a polynomial right-hand side, f (x) =

1

Bn =

N 

Cn = 0

1

tn+1 ln t dt. a + bt2

m = 1, 2, . . .

An xn , the solution has the form

n=0



N  An m+n–1 x , Bn

y(x) =

0

n=0

2◦ . For f (x) = xλ

N 

1

Bn =

tm+n–1 dt . a + btm

An xn , where λ is an arbitrary number (λ > –1), the solution has the

n=0

form y(x) = xλ



N  An m+n–1 x , Bn

1

Bn = 0

n=0

tλ+m+n–1 dt . a + btm

  N An xn , the solution has the form 3◦ . For f (x) = ln x n=0 N N  An m+n–1  An Cn m+n–1 x – x , Bn Bn2 n=0 n=0 1 m+n–1 tm+n–1 dt t ln t , Cn = dt. m a + btm a + bt 0

y(x) = ln x Bn = 0

1

1.1-6. Kernels Containing Square Roots.

x

30.



x – t y(t) dt = f (x).

a

Differentiating with respect to x, we arrive at Abel’s equation 1.1.36: x y(t) dt √ = 2fx (x). x–t a Solution: y(x) =

2 d2 π dx2



x a

f (t) dt √ . x–t

10

LINEAR EQUATIONS OF THE FIRST KIND WITH VARIABLE LIMIT OF INTEGRATION



x

31.

√

x–

√  t y(t) dt = f (x).

a

32.

This is a special case of equation 1.1.45 with µ = 12 . d √   x fx (x) . Solution: y(x) = 2 dx x √   √ A x + B t y(t) dt = f (x). a

This is a special case of equation 1.1.46 with µ = 12 .

x

33.

√   1 + b x – t y(t) dt = f (x).

a

Differentiating with respect to x, we arrive at Abel’s equation of the second kind 2.1.46: b y(x) + 2

x

34.

a

x

y(t) dt √ = fx (x). x–t

√   √ t x – x t y(t) dt = f (x).

a

This is a special case of equation 1.9.11 with g(x) =

x

35.

This is a special case of equation 1.9.12 with g(x) = x

36. a

x and h(x) = x.

√   √ At x + Bx t y(t) dt = f (x).

a







x and h(t) = t.

y(t) dt = f (x). √ x–t

Abel’s equation. Solution: 1 d y(x) = π dx



x

a

f (t) dt 1 f (a) √ + = √ π π x – a x–t



x a

ft (t) dt √ . x–t

Reference: E. T. Whittaker and G. N. Watson (1958).



x

37. a

  1 b+ √ y(t) dt = f (x). x–t

Let us rewrite the equation in the form

x a

y(t) dt √ = f (x) – b x–t



x

y(t) dt. a

Assuming the right-hand side to be known, we solve this equation as Abel’s equation 1.1.36. After some manipulations, we arrive at Abel’s equation of the second kind 2.1.46: y(x) +

b π

a

x

y(t) dt √ = F (x), x–t

where F (x) =

1 d π dx

a

x

f (t) dt √ . x–t

11

1.1. EQUATIONS WHOSE KERNELS CONTAIN POWER-LAW FUNCTIONS



x

38. a

 1 1 y(t) dt = f (x). √ – √ x t

This is a special case of equation 1.1.45 with µ = – 21 .

 Solution: y(x) = –2 x3/2 fx (x) x , a > 0.

x



39. a

 A B √ + √ y(t) dt = f (x). x t

This is a special case of equation 1.1.46 with µ = – 21 . 40.

x



x–t

x+t –x Solution:

y(t) dt = f (x).

  |x| |x| f (t) – f (–t) t[f (t) – f (–t)] sign x d 1 d √ √ y(x) = dt – dt . 2π dx 0 x dx 0 x2 – t2 x2 – t2

Reference: A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992).



41.

x

y(t) dt = f (x). √ x2 – t2 a x tf (t) dt 2 d √ Solution: y = . π dx a x2 – t2 Reference: P. P. Zabreyko, A. I. Koshelev, et al. (1975).



x

42. 0



y(t) dt ax2 + bt2

= f (x),

a > 0,

a + b > 0.

1◦ . For a polynomial right-hand side, f (x) =

N 

An xn , the solution has the form

n=0



N  An n x , y(x) = Bn

0

n=0

2◦ . For f (x) = xλ

N 

1

Bn =

tn dt √ . a + bt2

An xn , where λ is an arbitrary number (λ > –1), the solution has the

n=0

form y(x) = xλ



N  An n x , Bn

1

Bn = 0

n=0

tλ+n dt √ . a + bt2

  N An xn , the solution has the form 3◦ . For f (x) = ln x n=0 N N  An n  An Cn n x – x , y(x) = ln x Bn Bn2 n=0

4◦ . For f (x) =

Bn = 0

n=0

N 

1

tn dt √ , a + bt2

Cn =

 An ln x)n , the solution of the equation has the form

n=0

y(x) =

N  n=0

An Yn (x),

0

1

tn ln t √ dt. a + bt2

12

LINEAR EQUATIONS OF THE FIRST KIND WITH VARIABLE LIMIT OF INTEGRATION

where the functions Yn = Yn (x) are given by Yn (x) =

5◦ . For f (x) =

N 

 λ 

x dn , dλn I(λ) λ=0

An cos(λn ln x) +

n=1

N 

I(λ) = 0

1

z λ dz √ . a + bz 2

Bn sin(λn ln x), the solution of the equation has the

n=1

form y(x) =

N 

Cn cos(λn ln x) +

n=1

N 

Dn sin(λn ln x),

n=1

where the constants Cn and Dn are found by the method of undetermined coefficients. 1.1-7. Kernels Containing Arbitrary Powers.

x

43.

(x – t)λ y(t) dt = f (x),

f (a) = 0,

0 < λ < 1.

a

Differentiating with respect to x, we arrive at the generalized Abel equation 1.1.47:

x a

Solution:

d2 y(x) = k 2 dx

y(t) dt 1 = fx (x). (x – t)1–λ λ a

x

f (t) dt , (x – t)λ

k=

sin(πλ) . πλ

Reference: F. D. Gakhov (1977).



x

44.

(x – t)µ y(t) dt = f (x).

a

For µ = 0, 1, 2, . . . , see equations 1.1.1, 1.1.2, 1.1.4, 1.1.12, and 1.1.23. For 0 < µ < 1, see equation 1.1.43. Set µ = n – λ, where n = 1, 2, . . . and 0 ≤ λ < 1, and f (a) = fx (a) = · · · = fx(n–1) (a) = 0. On differentiating the equation n times, we arrive at an equation of the form 1.1.47: a

x

y(t) dτ Γ(µ – n + 1) (n) f (x), = (x – t)λ Γ(µ + 1) x

where Γ(µ) is the gamma function. Example. Set f (x) = Axβ , where β ≥ 0, and let µ > –1 and µ – β ≠ 0, 1, 2, . . . In this case, the solution has A Γ(β + 1) the form y(x) = xβ–µ–1 . Γ(µ + 1) Γ(β – µ) Reference: M. L. Krasnov, A. I. Kisilev, and G. I. Makarenko (1971).



x

45.

(xµ – tµ )y(t) dt = f (x).

a

This is a special case of equation 1.9.2 with g(x) = xµ . 1 1–µ   x fx (x) x . Solution: y(x) = µ

13

1.1. EQUATIONS WHOSE KERNELS CONTAIN POWER-LAW FUNCTIONS



x

46.

  Axµ + Btµ y(t) dt = f (x).

a

This is a special case of equation1.9.4 with g(x) = xµ . For B= –A, see equation 1.1.44. x Aµ Bµ d 1 x– A+B t– A+B ft (t) dt . Solution: y(x) = A + B dx a

x

47.

y(t) dt (x – t)λ

a

= f (x),

0 < λ < 1.

The generalized Abel equation. Solution: y(x) =

sin(πλ) d π dx



  x  f (t) dt ft (t) dt f (a) sin(πλ) . = + 1–λ (x – t)1–λ π (x – a)1–λ a (x – t)

x

a

Reference: E. T. Whittaker and G. N. Watson (1958).



x

 b+

48. a

1

 y(t) dt = f (x),

(x – t)λ

0 < λ < 1.

Rewrite the equation in the form a

x

y(t) dt = f (x) – b (x – t)λ



x

y(t) dt. a

Assuming the right-hand side to be known, we solve this equation as the generalized Abel equation 1.1.47. After some manipulations, we arrive at Abel’s equation of the second kind 2.1.60: y(x) +

x

49.

√

b sin(πλ) π x–

a

x

y(t) dt = F (x), (x – t)1–λ

√ λ t y(t) dt = f (x),

where F (x) =

sin(πλ) d π dx

a

0 < λ < 1.

a

Solution:  2 x f (t) dt k √ d √ x y(x) = √ √ √ λ , x dx a t x– t

x

50. a

y(t) dt √ λ = f (x), √ x– t



x

51.

sin(πλ) . πλ

0 < λ < 1.

Solution: y(x) =

k=

sin(πλ) d 2π dx

a

x

f (t) dt √ √ √ 1–λ . t x– t

  Axλ + Btµ y(t) dt = f (x).

a

This is a special case of equation 1.9.6 with g(x) = Axλ and h(t) = Btµ .

x

f (t) dt . (x – t)1–λ

14

LINEAR EQUATIONS OF THE FIRST KIND WITH VARIABLE LIMIT OF INTEGRATION



x

52.

 1 + A(xλ tµ – xλ+µ ) y(t) dt = f (x).

a

This is a special case of equation 1.9.13 with g(x) = Axµ and h(x) = xλ . Solution: d y(x) = dx

x

53.



xλ Φ(x)

a

x

t f (t) t Φ(t) dt ,

  Aµ xµ+λ . Φ(x) = exp – µ+λ



–λ

  Axβ tγ + Bxδ tλ y(t) dt = f (x).

a

This is a special case of equation 1.9.15 with g1 (x) = Axβ , h1 (t) = tγ , g2 (x) = Bxδ , and h2 (t) = tλ .

x

54.

 Axλ (tµ – xµ ) + Bxβ (tγ – xγ ) y(t) dt = f (x).

a

This is a special case of equation 1.9.47 with g1 (x) = Axλ , h1 (x) = xµ , g2 (x) = Bxβ , and h2 (x) = xγ .

x

55.

 Axλ tµ + Bxλ+β tµ–β – (A + B)xλ+γ tµ–γ y(t) dt = f (x).

a

This is a special case of equation 1.9.49 with g(x) = x.

x

56.

tσ (xµ – tµ )λ y(t) dt = f (x),

σ > –1,

µ > 0,

λ > –1.

a

The transformation τ = tµ , z = xµ , w(τ ) = tσ–µ+1 y(t) leads to an equation of the form 1.1.43:

z

(z – τ )λ w(τ ) dτ = F (z), A

where A = aµ and F (z) = µf (z 1/µ ). Solution with –1 < λ < 0: µ sin(πλ) d y(x) = – πxσ dx 57. 0

x



x

t

µ–1

µ

µ –1–λ

(x – t )

 f (t) dt .

a

y(t) dt = f (x). (x + t)µ

This is a special case of equation 1.1.58 with λ = 1 and a = b = 1. The transformation x = 12 e2z ,

t = 12 e2τ ,

y(t) = e(µ–2)τ w(τ ),

f (x) = e–µz g(z)

leads to an equation with difference kernel of the form 1.9.27:

z

–∞

w(τ ) dτ = g(z). coshµ (z – τ )

15

1.2. EQUATIONS WHOSE KERNELS CONTAIN EXPONENTIAL FUNCTIONS



58.

x

y(t) dt = f (x), a > 0, a + b > 0. λ + btλ )µ (ax 0 1◦ . The substitution t = xz leads to a special case of equation 3.8.45: 1 y(xz) dz = xλµ–1 f (x). λ )µ (a + bz 0 n 

2◦ . For a polynomial right-hand side, f (x) =

(1)

Am xm , the solution has the form

m=0

y(x) = xλµ–1

n  Am m x , Im m=0

Im = 0

1

z m+λµ–1 dz . (a + bz λ )µ

The integrals Im are supposed to be convergent. 3◦ . The solution structure for some other right-hand sides of the integral equation may be obtained using (1) and the results presented for the more general equation 3.8.53 (see also equations 3.8.34–3.8.40).

59.

4◦ . For a = b, the equation can be reduced, just as equation 1.1.57, to an integral equation with difference kernel of the form 1.9.27. √  2λ  √ √ 2λ x √ x+ x–t + x– x–t y(t) dt = f (x). √ 2tλ x – t a The equation can be rewritten in terms of the Gaussian hypergeometric functions in the form x  x y(t) dt = f (x), where γ = 12 . (x – t)γ–1 F λ, –λ, γ; 1 – t a See 1.8.135 for the solution of this equation.

1.1-8. Two-Dimensional Equation of the Abel Type.

u(x, y) dx dy  = f (x0 , y0 ). (y0 – y)2 – (x0 – x)2

60. ∆

Here ∆ is an isosceles right triangle with apex at the point (x0 , y0 ) and base on the x-axis. Solution:  2  f (x, y) dx dy ∂ g ∂ 2g 1  u(x0 , y0 ) = , g(x0 , y0 ) = – . 2 2 2 2π ∂x0 ∂y0 (y0 – y)2 – (x0 – x)2 ∆

Reference: P. P. Zabreyko, A. I. Koshelev, et al. (1975).

1.2. Equations Whose Kernels Contain Exponential Functions 1.2-1. Kernels Containing Exponential Functions.

x

1.

eλ(x–t) y(t) dt = f (x).

a

Solution: y(x) = fx (x) – λf (x). Example. In the special case a = 0 and f (x) = Ax, the solution has the form y(x) = A(1 – λx).

16

LINEAR EQUATIONS OF THE FIRST KIND WITH VARIABLE LIMIT OF INTEGRATION



x

2.

eλx+βt y(t) dt = f (x).

a

 Solution: y(x) = e–(λ+β)x fx (x) – λf (x) . Example. In the special case a = 0 and f (x) = A sin(γx), the solution has the form y(x) = Ae–(λ+β)x × [γ cos(γx) – λ sin(γx)].



x

3.

 eλ(x–t) – 1 y(t) dt = f (x),

a

Solution: y(x) =

x

4.

1  λ fxx (x)

f (a) = fx (a) = 0.

– fx (x).

 eλ(x–t) + b y(t) dt = f (x).

a

For b = –1, see equation 1.2.3. Differentiating with respect to x yields an equation of the form 2.2.1: x f  (x) λ eλ(x–t) y(t) dt = x . y(x) + b+1 a b+1 Solution: y(x) =

x

5.

λ fx (x) – b + 1 (b + 1)2

a

x

  λb (x – t) ft (t) dt. exp b+1

 λx+βt  e + b y(t) dt = f (x).

a

This is a special case of equation 1.9.15 with g1 (x) = eλx , h1 (t) = eβt , g2 (x) = 1, and h2 (t) = b. For β = –λ, see equation 1.2.4.

x

6.

 λx  e – eλt y(t) dt = f (x),

a

f (a) = fx (a) = 0.

λx This is a special case of equation 1.9.2 with g(x)   =e . 1  f (x) – fx (x) . Solution: y(x) = e–λx λ xx



x

7.

 λx  e – eλt + b y(t) dt = f (x).

a

This is a special case of equation 1.9.3 with g(x) = eλx . For b = 0, see equation 1.2.6. Solution:  λt λx  1  λ λx x e –e y(x) = fx (x) – 2 e ft (t) dt. exp b b b a

x

8.

 λx  Ae + Beλt y(t) dt = f (x).

a

This is a special case of equation1.9.4 with g(x) = eλx . For B = –A, see equation  1.2.6.  Aλ  x  Bλ  d 1  exp – x t ft (t) dt . exp – Solution: y(x) = A + B dx A+B A+B a

x

9.

 λx  Ae + Beλt + C y(t) dt = f (x).

a

This is a special case of equation 1.9.5 with g(x) = eλx .

1.2. EQUATIONS WHOSE KERNELS CONTAIN EXPONENTIAL FUNCTIONS



x

10.

17

 λx  Ae + Beµt y(t) dt = f (x).

a

11.

This is a special case of equation 1.9.6 with g(x) = Aeλx and h(t) = Beµt . For λ = µ, see equation 1.2.8. x

λ(x–t)  e f (a) = fx (a) = 0. – eµ(x–t) y(t) dt = f (x), a

Solution: y(x) =

x

12.

 1  fxx – (λ + µ)fx + λµf , λ–µ

f = f (x).

 Aeλ(x–t) + Beµ(x–t) y(t) dt = f (x).

a

This is a special case of equation 1.9.15 with g1 (x) = Aeλx , h1 (t) = e–λt , g2 (x) = Beµx , and h2 (t) = e–µt . For B = –A, see equation 1.2.11. Solution:

   x eλx d dt B(λ – µ) f (t) y(x) = e(µ–λ)x Φ(x) , Φ(x) = exp x . A + B dx eµt t Φ(t) A+B a

x

13.

 Aeλ(x–t) + Beµ(x–t) + C y(t) dt = f (x).

a

14.

This is a special case of equation 1.2.14 with β = 0. x

λ(x–t)  Ae + Beµ(x–t) + Ceβ(x–t) y(t) dt = f (x). a

Differentiating the equation with respect to x yields x

 Aλeλ(x–t) + Bµeµ(x–t) + Cβeβ(x–t) y(t) dt = fx (x). (A + B + C)y(x) + a β(x–t)

Eliminating the term with e with the aid of the original equation, we arrive at an equation of the form 2.2.10: x

 (A + B + C)y(x) + A(λ – β)eλ(x–t) + B(µ – β)eµ(x–t) y(t) dt = fx (x) – βf (x). a

15.

In the special case A + B + C = 0, this is an equation of the form 1.2.12. x

λ(x–t)  Ae + Beµ(x–t) + Ceβ(x–t) – A – B – C y(t) dt = f (x), f (a) = fx (a) = 0. a

Differentiating with respect to x, we arrive at an equation of the form 1.2.14: x

 Aλeλ(x–t) + Bµeµ(x–t) + Cβeβ(x–t) y(t) dt = fx (x). a



x

16. a

 λx+µt  e – eµx+λt y(t) dt = f (x),

f (a) = fx (a) = 0.

This is a special case of equation 1.9.11 with g(x) = eλx and h(t) = eµt . Solution: f  – (λ + µ)fx (x) + λµf (x) . y(x) = xx (λ – µ) exp[(λ + µ)x]

18

LINEAR EQUATIONS OF THE FIRST KIND WITH VARIABLE LIMIT OF INTEGRATION



x

17.

 λx+µt  Ae + Beµx+λt y(t) dt = f (x).

a

This is a special case of equation 1.9.12 with g(x) = eλx and h(t) = eµt . For B = –A, see equation 1.2.16. Solution:  

x  µ–λ  d d f (t) 1 A B Φ x . dt , Φ(x) = exp (x) Φ (t) y(x) = µx µt (A + B)e dx dt e A+B a

x

18.

 λx+µt  Ae + Beβx+γt y(t) dt = f (x).

a

This is a special case of equation 1.9.15 with g1 (x) = Aeλx , h1 (t) = eµt , g2 (x) = Beβx , and h2 (t) = eγt .

x

19.

 2λx  Ae + Be2βt + Ceλx + Deβt + E y(t) dt = f (x).

a

This is a special case of equation 1.9.6 with g(x) = Ae2λx +Ceλx and h(t) = Be2βt +Deβt +E.

x

20.

 λx+βt  Ae + Be2βt + Ceλx + Deβt + E y(t) dt = f (x).

a

This is a special case of equation 1.9.15 with g1 (x) = eλx , h1 (t) = Aeβt + C, and g2 (x) = 1, h2 (t) = Be2βt + Deβt + E.

x

21.

 2λx  Ae + Beλx+βt + Ceλx + Deβt + E y(t) dt = f (x).

a

This is a special case of equation 1.9.15 with g1 (x) = Beλx + D, h1 (t) = eβt , and g2 (x) = Ae2λx + Ceλx + E, h2 (t) = 1.

x

22.

 1 + Aeλx (eµt – eµx ) y(t) dt = f (x).

a

This is a special case of equation 1.9.13 with g(x) = eµx and h(x) = Aeλx . Solution:

 x dt f (t) d λx e Φ(x) , y(x) = dx eλt t Φ(t) a

x

23.

  Aµ (λ+µ)x Φ(x) = exp e . λ+µ

 Aeλx (eµx – eµt ) + Beβx (eγx – eγt ) y(t) dt = f (x).

a

This is a special case of equation 1.9.47 with g1 (x) = Aeλx , h1 (t) = –eµt , g2 (x) = Beβx , and h2 (t) = –eγt.

x

24. a



A exp(λx + µt) + B exp[(λ + β)x + (µ – β)t]  – (A + B) exp[(λ + γ)x + (µ – γ)t] y(t) dt = f (x).

This is a special case of equation 1.9.49 with g1 (x) = ex .

1.2. EQUATIONS WHOSE KERNELS CONTAIN EXPONENTIAL FUNCTIONS



x

25.

 λx n e – eλt y(t) dt = f (x),

n = 1, 2, . . .

a

Solution: y(x) =

x

26.



1 λx  1 d n+1 e f (x). λn n! eλx dx

eλx – eλt y(t) dt = f (x),

λ > 0.

a

Solution:

27.

x



 d 2 2 y(x) = eλx e–λx π dx

y(t) dt

eλx – eλt Solution: a

= f (x),



x

x

a

eλt f (t) dt √ . eλx – eλt

λ > 0.

y(x) =

28.



(eλx – eλt )µ y(t) dt = f (x),

λ d π dx



x

eλt f (t) dt √ . eλx – eλt

a

λ > 0,

0 < µ < 1.

a

Solution: y(x) = ke 29.

x

y(t) dt

(eλx – eλt )µ Solution:

λx

 e

–λx

= f (x),

d 2 dx

λ > 0,



x

a

eλt f (t) dt , (eλx – eλt )µ

k=

sin(πµ) . πµ

0 < µ < 1.

a

y(x) =

λ sin(πµ) d π dx

a

x

eλt f (t) dt . (eλx – eλt )1–µ

1.2-2. Kernels Containing Power-Law and Exponential Functions.

x

30.

 A(x – t) + Beλ(x–t) y(t) dt = f (x).

a

Differentiating with respect to x, we arrive at an equation of the form 2.2.4: x

 By(x) + A + Bλeλ(x–t) y(t) dt = fx (x). a



x

31.

(x – t)eλ(x–t) y(t) dt = f (x),

a

32.

f (a) = fx (a) = 0.

 (x) – 2λfx (x) + λ2 f (x). Solution: y(x) = fxx x (Ax + Bt + C)eλ(x–t) y(t) dt = f (x). a

The substitution u(x) = e–λx y(x) leads to an equation of the form 1.1.3: x (Ax + Bt + C)u(t) dt = e–λx f (x). a

19

20

LINEAR EQUATIONS OF THE FIRST KIND WITH VARIABLE LIMIT OF INTEGRATION



x

33.

(Axeλt + Bteµx )y(t) dt = f (x).

a

This is a special case of equation 1.9.15 with g1 (x) = Ax, h1 (t) = eλt , and g2 (x) = Beµx , h2 (t) = t.

x

34.

 Axeλ(x–t) + Bteµ(x–t) y(t) dt = f (x).

a

This is a special case of equation 1.9.15 with g1 (x) = Axeλx , h1 (t) = e–λt , g2 (x) = Beµx , and h2 (t) = te–µt .

x

35.

(x – t)2 eλ(x–t) y(t) dt = f (x),

a

Solution: y(x) =

x

36.

1 2

 f (a) = fx (a) = fxx (a) = 0.

   fxxx (x) – 3λfxx (x) + 3λ2 fx (x) – λ3 f (x) .

(x – t)n eλ(x–t) y(t) dt = f (x),

n = 1, 2, . . .

a

It is assumed that f (a) = fx (a) = · · · = fx(n) (a) = 0.  1 λx dn+1 –λx e Solution: y(x) = e f (x) . n+1 n! dx

x

37.

(Axβ + Beλt )y(t) dt = f (x).

a

This is a special case of equation 1.9.6 with g(x) = Axβ and h(t) = Beλt .

x

38.

(Aeλx + Btβ )y(t) dt = f (x).

a

This is a special case of equation 1.9.6 with g(x) = Aeλx and h(t) = Btβ .

x

39.

(Axβ eλt + Btγ eµx )y(t) dt = f (x).

a

This is a special case of equation 1.9.15 with g1 (x) = Axβ , h1 (t) = eλt , g2 (x) = Beµx , and h2 (t) = tγ .

x

40.

eλ(x–t)



x – t y(t) dt = f (x).

a

Solution: y(x) =

x

41. a

2 λx d2 e π dx2



x

e–λt f (t) dt √ . x–t

x

e–λt f (t) dt √ . x–t

a

eλ(x–t) y(t) dt = f (x). √ x–t

Solution: y(x) =

1 λx d e π dx

a

1.2. EQUATIONS WHOSE KERNELS CONTAIN EXPONENTIAL FUNCTIONS



x

42.

(x – t)λ eµ(x–t) y(t) dt = f (x),

0 < λ < 1.

a

Solution: y(x) = keµx

43.

a

x

e–µt f (t) dt , (x – t)λ

k=

sin(πλ) . πλ

x

eλ(x–t) y(t) dt = f (x), µ a (x – t) Solution: y(x) =

d2 dx2

x

44.

√

x–

0 < µ < 1. sin(πµ) λx d e π dx

√ λ µ(x–t) t e y(t) dt = f (x),

a

x

e–λt f (t) dt. (x – t)1–µ

0 < λ < 1.

a

The substitution u(x) = e–µx y(x) leads to an equation of the form 1.1.49: x √ λ √ x – t u(t) dt = e–µx f (x). a



x

45. a

eµ(x–t) y(t) dt √ √  λ = f (x), x– t

0 < λ < 1.

The substitution u(x) = e–µx y(x) leads to an equation of the form 1.1.50: x u(t) dt √ = e–µx f (x). √ λ ( x – t) a

47.

eλ(x–t) y(t) dt = f (x). √ x2 – t2 a x te–λt 2 λx d √ f (t) dt. Solution: y = e π dx a x2 – t2 x exp[λ(x2 – t2 )]y(t) dt = f (x).

48.

Solution: y(x) = fx (x) – 2λxf (x). x [exp(λx2 ) – exp(λt2 )]y(t) dt = f (x).

46.

x

a

a

This is a special case of equation with g(x) = exp(λx2 ).  1.9.2  fx (x) 1 d . Solution: y(x) = 2λ dx x exp(λx2 )

x

49.

 A exp(λx2 ) + B exp(λt2 ) + C y(t) dt = f (x).

a

50.

This is a special case of equation 1.9.5 with g(x) = exp(λx2 ). x

 A exp(λx2 ) + B exp(µt2 ) y(t) dt = f (x). a

This is a special case of equation 1.9.6 with g(x) = A exp(λx2 ) and h(t) = B exp(µt2 ).

21

22

LINEAR EQUATIONS OF THE FIRST KIND WITH VARIABLE LIMIT OF INTEGRATION



x

51.



x – t exp[λ(x2 – t2 )]y(t) dt = f (x).

a

Solution:

d2 2 exp(λx2 ) 2 π dx

y(x) =

52.

53.

a

x

exp(–λt2 ) √ f (t) dt. x–t

x

exp[λ(x2 – t2 )] y(t) dt = f (x). √ x–t a Solution: x exp(–λt2 ) 1 2 d √ f (t) dt. y(x) = exp(λx ) π dx a x–t x (x – t)λ exp[µ(x2 – t2 )]y(t) dt = f (x), 0 < λ < 1. a

Solution: y(x) = k exp(µx2 )

x

54.

d2 dx2



x a

exp(–µt2 ) f (t) dt, (x – t)λ

k=

sin(πλ) . πλ

exp[λ(xβ – tβ )]y(t) dt = f (x).

a

Solution: y(x) = fx (x) – λβxβ–1 f (x).

x

55.

f (0) = fx (0) = 0.

(–1)[(x–t)/b] y(t) dt = f (x),

0

Here b = const and [A] stands for the integer part of the number A. Solution:     x–t 1 x + 1 ftt (t) dt. 2 y(x) = 2 0 b References: H. M. Srivastava and R. G. Buschman (1977), A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 434).

1.3. Equations Whose Kernels Contain Hyperbolic Functions 1.3-1. Kernels Containing Hyperbolic Cosine.

x

cosh[λ(x – t)]y(t) dt = f (x).

1. a

Solution: y(x) = fx (x) – λ2



x

f (x) dx. a

Reference: A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 435).



x

2.



 cosh[λ(x – t)] – 1 y(t) dt = f (x),

a

Solution: y(x) =

1  f (x) – fx (x). λ2 xxx

 f (a) = fx (a) = fxx (x) = 0.

23

1.3. EQUATIONS WHOSE KERNELS CONTAIN HYPERBOLIC FUNCTIONS



x

3.



 cosh[λ(x – t)] + b y(t) dt = f (x).

a

For b = 0, see equation 1.3.1. For b = –1, see equation 1.3.2. For λ = 0, see equation 1.1.1. Differentiating the equation with respect to x, we arrive at an equation of the form 2.3.16: x λ f  (x) y(x) + sinh[λ(x – t)]y(t) dt = x . b+1 a b+1 1◦ . Solution with b(b + 1) < 0: λ2 f  (x) – y(x) = x b + 1 k(b + 1)2





x

sin[k(x –

t)]ft (t) dt,

where k = λ

a

–b . b+1



2 . Solution with b(b + 1) > 0: f  (x) λ2 y(x) = x – b + 1 k(b + 1)2





x

sinh[k(x –

t)]ft (t) dt,

where k = λ

a

b . b+1

x

cosh(λx + βt)y(t) dt = f (x).

4. a

For β = –λ, see equation 1.3.1. Differentiating the equation with respect to x twice, we obtain x cosh[(λ + β)x]y(x) + λ sinh(λx + βt)y(t) dt = fx (x), (1) a x    cosh[(λ + β)x]y(x) x + λ sinh[(λ + β)x]y(x) + λ2 cosh(λx + βt)y(t) dt = fxx (x). (2) a

Eliminating the integral term from (2) with the aid of the original equation, we arrive at the first-order linear ordinary differential equation  wx + λ tanh[(λ + β)x]w = fxx (x) – λ2 f (x),

w = cosh[(λ + β)x]y(x).

(3)

Setting x = a in (1) yields the initial condition w(a) = fx (a). On solving equation (3) with this condition, after some manipulations we obtain the solution of the original integral equation in the form y(x) =



1 λ sinh[(λ + β)x] f  (x) – f (x) cosh[(λ + β)x] x cosh2 [(λ + β)x] x λβ + f (t) coshk–2 [(λ + β)t] dt, coshk+1 [(λ + β)x] a

k=

λ . λ+β

x

[cosh(λx) – cosh(λt)]y(t) dt = f (x).

5. a

6.

This is a special case of equation   1.9.2 with g(x) = cosh(λx). fx (x) 1 d . Solution: y(x) = λ dx sinh(λx) x [A cosh(λx) + B cosh(λt)]y(t) dt = f (x). a

This is a special case of equation 1.9.4 with g(x) = cosh(λx). For B = –A, see equation 1.3.5.

x – A

– B  1 d A+B A+B cosh(λx) cosh(λt) Solution: y(x) = ft (t) dt . A + B dx a

24

LINEAR EQUATIONS OF THE FIRST KIND WITH VARIABLE LIMIT OF INTEGRATION



x

7.

 A cosh(λx) + B cosh(µt) + C y(t) dt = f (x).

a

8.

This is a special case of equation 1.9.6 with g(x) = A cosh(λx) and h(t) = B cosh(µt) + C. x   A1 cosh[λ1 (x – t)] + A2 cosh[λ2 (x – t)] y(t) dt = f (x).

9.

The equation is equivalent to the equation x   B1 sinh[λ1 (x – t)] + B2 sinh[λ2 (x – t)] y(t) dt = F (x), a x A1 A2 B1 = , B2 = , F (x) = f (t) dt, λ1 λ2 a of the form 1.3.49. (Differentiating this equation yields the original equation.) x cosh2 [λ(x – t)]y(t) dt = f (x).

a

a

Differentiation yields an equation of the form 2.3.16: x sinh[2λ(x – t)]y(t) dt = fx (x). y(x) + λ a

Solution: y(x) = fx (x) –

10.

11.

x

2λ2 k



x

sinh[k(x – t)]ft (t) dt,

√ where k = λ 2.

a

 cosh2 (λx) – cosh2 (λt) y(t) dt = f (x), f (a) = fx (a) = 0. a   1 d fx (x) . Solution: y(x) = λ dx sinh(2λx) x

 A cosh2 (λx) + B cosh2 (λt) y(t) dt = f (x). a

12.

This is a special case of equation 1.9.4 with g(x) = cosh2 (λx). For B = –A, see equation 1.3.10. Solution:

x – 2A

– 2B  1 d A+B A+B y(x) = cosh(λx) cosh(λt) ft (t) dt . A + B dx a x

 A cosh2 (λx) + B cosh2 (µt) + C y(t) dt = f (x). a

13.

This is a special case of equation 1.9.6 with g(x) = A cosh2 (λx), and h(t) = B cosh2 (µt) + C. x cosh[λ(x – t)] cosh[λ(x + t)]y(t) dt = f (x). a

Using the formula cosh(α – β) cosh(α + β) = 12 [cosh(2α) + cosh(2β)],

α = λx,

β = λt,

we transform the original equation to an equation of the form 1.3.6 with A = B = 1: x [cosh(2λx) + cosh(2λt)]y(t) dt = 2f (x). a

Solution: y(x) =

  x f  (t) dt 1 d √ √ t . dx cosh(2λx) a cosh(2λt)

1.3. EQUATIONS WHOSE KERNELS CONTAIN HYPERBOLIC FUNCTIONS



25

x

[cosh(λx) cosh(µt) + cosh(βx) cosh(γt)]y(t) dt = f (x).

14. a

This is a special case of equation 1.9.15 with g1 (x) = cosh(λx), h1 (t) = cosh(µt), g2 (x) = cosh(βx), and h2 (t) = cosh(γt).

x

15.

cosh3 [λ(x – t)]y(t) dt = f (x).

a

Using the formula cosh3 β = a



x

16.

x

1 4

1 4

cosh 3β +

3 4

cosh[3λ(x – t)] +

cosh β, we arrive at an equation of the form 1.3.8: 3 4

 cosh[λ(x – t)] y(t) dt = f (x).

 cosh3 (λx) – cosh3 (λt) y(t) dt = f (x),

a

f (a) = fx (a) = 0.

  fx (x) 1 d Solution: y(x) = . 3λ dx sinh(λx) cosh2 (λx)

x

17.

 A cosh3 (λx) + B cosh3 (λt) y(t) dt = f (x).

a

This is a special case of equation 1.9.4 with g(x) = cosh3 (λx). For B = –A, see equation 1.3.16. Solution:

x – 3A

– 3B 1 d y(x) = cosh(λx) A+B cosh(λt) A+B ft (t) dt . A + B dx a

x

18.

 A cosh2 (λx) cosh(µt) + B cosh(βx) cosh2 (γt) y(t) dt = f (x).

a

This is a special case of equation 1.9.15 with g1 (x) = A cosh2 (λx), h1 (t) = cosh(µt), g2 (x) = B cosh(βx), and h2 (t) = cosh2 (γt).

x

19.

cosh4 [λ(x – t)]y(t) dt = f (x).

a

Let us transform the kernel of the integral equation using the formula cosh4 β =

1 8

cosh 4β +

1 2

cosh 2β + 38 ,

where β = λ(x – t),

and differentiate the resulting equation with respect to x. Then we obtain an equation of the form 2.3.18: x 1   y(x) + λ 2 sinh[4λ(x – t)] + sinh[2λ(x – t)] y(t) dt = fx (x). a



x

20.

[cosh(λx) – cosh(λt)]n y(t) dt = f (x),

n = 1, 2, . . .

a

The right-hand side of the equation is assumed to satisfy the conditions f (a) = fx (a) = · · · = fx(n) (a) = 0.  n+1 sinh(λx) 1 d f (x). Solution: y(x) = λn n! sinh(λx) dx

26

LINEAR EQUATIONS OF THE FIRST KIND WITH VARIABLE LIMIT OF INTEGRATION



x

21.



cosh x – cosh t y(t) dt = f (x).

a

Solution: y(x) = 22.

x



y(t) dt

cosh x – cosh t Solution: a

 1 d 2 x sinh t f (t) dt 2 √ sinh x . π sinh x dx cosh x – cosh t a

= f (x).

y(x) =

x

23.



1 d π dx

x

a

(cosh x – cosh t)λ y(t) dt = f (x),

sinh t f (t) dt √ . cosh x – cosh t

0 < λ < 1.

a

Solution:  y(x) = k sinh x

x

24.

1 d 2 sinh x dx

a

x

sinh t f (t) dt , (cosh x – cosh t)λ

k=

sin(πλ) . πλ

(coshµ x – coshµ t)y(t) dt = f (x).

a

25.

µ This is a special case of equation 1.9.2 with g(x)   = cosh x.  1 d fx (x) . Solution: y(x) = µ dx sinh x coshµ–1 x x   A coshµ x + B coshµ t y(t) dt = f (x). a

This is a special case of equation 1.9.4 with g(x) = coshµ x. For B = –A, see equation 1.3.24. Solution:

x – Aµ

– Bµ  d 1 A+B A+B cosh(λx) y(x) = cosh(λt) ft (t) dt . A + B dx a 26.

x

y(t) dt

λ a (cosh x – cosh t) Solution:

= f (x),

y(x) =

0 < λ < 1.

sin(πλ) d π dx

x

(x – t) cosh[λ(x – t)]y(t) dt = f (x),

27. a



x

sinh t f (t) dt . (cosh x – cosh t)1–λ

a

f (a) = fx (a) = 0.

Differentiating the equation twice yields x x  y(x) + 2λ sinh[λ(x – t)]y(t) dt + λ2 (x – t) cosh[λ(x – t)]y(t) dt = fxx (x). a

a

Eliminating the third term on the right-hand side with the aid of the original equation, we arrive at an equation of the form 2.3.16: x  y(x) + 2λ sinh[λ(x – t)]y(t) dt = fxx (x) – λ2 f (x). a

1.3. EQUATIONS WHOSE KERNELS CONTAIN HYPERBOLIC FUNCTIONS



x

cosh[λ(x – t)] √ y(t) dt = f (x), x–t

28. a

Solution: y(x) =

27

f (a) = fx (a) = 0.

2 x cosh[λ(x – t)]  √ [ftt (t) – λ2 f (t)] dt. πλ a x–t

Reference: A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 436).



x

29.



 √  x – t cosh λ x – t y(t) dt = f (x).

a

Solution: y(x) =

 √  1 x cos λ x – t  √ ft (t) dt. π a x–t

References: A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 437), S. G. Samko, A. A. Kilbas, and O. I. Marichev (1993).



x

30. a

 √  cosh λ x – t √ y(t) dt = f (x). x–t

Solution: y(x) =



1 d π dx

x a

 √  cos λ x – t √ f (t) dt. x–t

References: A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 437), S. G. Samko, A. A. Kilbas, and O. I. Marichev (1993).





31. x

 √  cosh λ t – x √ y(t) dt = f (x). t–x

Solution: y(x) = –

 √  1 d ∞ cos λ t – x √ f (t) dt. π dx x t–x

References: H. M. Srivastava and R. G. Buschman (1977), A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 439), S. G. Samko, A. A. Kilbas, and O. I. Marichev (1993).



x

32. 0

  √ cosh λ x2 – t2 √ y(t) dt = f (x). x2 – t2

Solution: 2 d y(x) = π dx



33. x



x

0

 √  cos λ x2 – t2 √ t f (t) dt. x2 – t2

  √ cosh λ t2 – x2 y(t) dt = f (x). √ t2 – x2

Solution: 2 d y(x) = – π dx





x

 √  cos λ t2 – x2 √ t f (t) dt. t2 – x2

28

LINEAR EQUATIONS OF THE FIRST KIND WITH VARIABLE LIMIT OF INTEGRATION

  √ cosh λ xt – t2 y(t) dt = f (x). √ x–t 0 Solution:  √  x cos λ x2 – xt 1 √ [f (t)/2 + tft (t)] dt. y(x) = πx 0 x–t

34.

x

References: A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 438), S. G. Samko, A. A. Kilbas, and O. I. Marichev (1993).

 √  cosh λ x2 – xt √ y(t) dt = f (x). x–t 0 Solution:   √ √ x  cos λ xt – t2 x d √ √ f (t) dt . x y(x) = π dx x–t 0

35.

x

References: A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 438), S. G. Samko, A. A. Kilbas, and O. I. Marichev (1993).



x

36. a

√  cosh λ (x – t)(x – t + γ) y(t) dt = f (x). √ x–t

  (a) = fxxx (a) = 0. It is assumed that f (a) = fx (a) = fxx Solution:

√  t x 2  d2 sinh λ (x – t)(x – t – γ) 2 2 √ y(x) = sinh[λ(t – s)] – λ f (s) ds dt. πλ2 a x–t–γ ds 2 a

References: A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 438), S. G. Samko, A. A. Kilbas, and O. I. Marichev (1993).



x

37.

Axβ + B coshγ (λt) + C]y(t) dt = f (x).

a

This is a special case of equation 1.9.6 with g(x) = Axβ and h(t) = B coshγ (λt) + C.

x

38.

A coshγ (λx) + Btβ + C]y(t) dt = f (x).

a

This is a special case of equation 1.9.6 with g(x) = A coshγ (λx) and h(t) = Btβ + C.

x

39.

  Axλ coshµ t + Btβ coshγ x y(t) dt = f (x).

a

This is a special case of equation 1.9.15 with g1 (x) = Axλ , h1 (t) = coshµ t, g2 (x) = B coshγ x, and h2 (t) = tβ . 1.3-2. Kernels Containing Hyperbolic Sine.

x

sinh[λ(x – t)]y(t) dt = f (x),

40. a

Solution: y(x) =

f (a) = fx (a) = 0.

1  f (x) – λf (x). λ xx

Reference: A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 435).

29

1.3. EQUATIONS WHOSE KERNELS CONTAIN HYPERBOLIC FUNCTIONS

41.

x

sinh[λ(x – t)] √ y(t) dt = f (x), f (a) = fx (a) = 0. x–t a Solution: x sinh[λ(x – t)]  2 √ [ftt (t) – λ2 f (t)] dt. y(x) = πλ a x–t Reference: A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 436).

42.

x

sinh[λ(x – t)]

(x – t)3/2 Solution:

y(t) dt = f (x),

a

2 y(x) = πλ

a

x

f (a) = fx (a) = 0.

  sinh[λ(x – t)]  f  (t) 2 √ dt. ftt (t) – λ f (t) + x–t x–t

Reference: A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 437).



x

43.



 sinh[λ(x – t)] + b y(t) dt = f (x).

a

For b = 0, see equation 1.3.40. Assume that b ≠ 0. Differentiating the equation with respect to x, we arrive at an equation of the form 2.3.3: λ x 1 y(x) + cosh[λ(x – t)]y(t) dt = fx (x). b a b Solution:

x 1  fx (x) + R(x – t)ft (t) dt, b a √    λ λ 1 + 4b2 λ λx sinh(kx) – cosh(kx) , k = . R(x) = 2 exp – b 2b 2bk 2b y(x) =



x

sinh(λx + βt)y(t) dt = f (x).

44. a

For β = –λ, see equation 1.3.40. Assume that β ≠ –λ. Differentiating the equation with respect to x twice yields x sinh[(λ + β)x]y(x) + λ cosh(λx + βt)y(t) dt = fx (x), a x    sinh[(λ + β)x]y(x) x + λ cosh[(λ + β)x]y(x) + λ2 sinh(λx + βt)y(t) dt = fxx (x).

(1) (2)

a

Eliminating the integral term from (2) with the aid of the original equation, we arrive at the first-order linear ordinary differential equation  wx + λ coth[(λ + β)x]w = fxx (x) – λ2 f (x),

w = sinh[(λ + β)x]y(x).

(3)

Setting x = a in (1) yields the initial condition w(a) = fx (a). On solving equation (3) with this condition, after some manipulations we obtain the solution of the original integral equation in the form y(x) =

1 λ cosh[(λ + β)x] fx (x) – f (x) sinh[(λ + β)x] sinh2 [(λ + β)x] x λβ – f (t) sinhk–2 [(λ + β)t] dt, sinhk+1 [(λ + β)x] a

k=

λ . λ+β

30

LINEAR EQUATIONS OF THE FIRST KIND WITH VARIABLE LIMIT OF INTEGRATION



x

[sinh(λx) – sinh(λt)]y(t) dt = f (x),

45. a

f (a) = fx (a) = 0.

This is a special case of equation 1.9.2  with g(x) = sinh(λx). fx (x) 1 d . Solution: y(x) = λ dx cosh(λx)

x

[A sinh(λx) + B sinh(λt)]y(t) dt = f (x).

46. a

47.

This is a special case of equation 1.9.4 For B = –A, see equation 1.3.45. with g(x) =A sinh(λx).

x –

– B  d 1 sinh(λx) A+B sinh(λt) A+B ft (t) dt . Solution: y(x) = A + B dx a x [A sinh(λx) + B sinh(µt)]y(t) dt = f (x).

48.

This is a special case of equation 1.9.6 with g(x) = A sinh(λx) and h(t) = B sinh(µt). x   µ sinh[λ(x – t)] – λ sinh[µ(x – t)] y(t) dt = f (x).

49.

  (a) = fxxx (a) = 0. It is assumed that f (a) = fx (a) = fxx Solution:  f  – (λ2 + µ2 )fxx + λ2 µ2 f y(x) = xxxx , f = f (x). µλ3 – λµ3 x   A1 sinh[λ1 (x – t)] + A2 sinh[λ2 (x – t)] y(t) dt = f (x), f (a) = fx (a) = 0.

a

a

a

1◦ . Introduce the notation x x sinh[λ1 (x – t)]y(t) dt, I2 = sinh[λ2 (x – t)]y(t) dt, I1 = ax ax J1 = cosh[λ1 (x – t)]y(t) dt, J2 = cosh[λ2 (x – t)]y(t) dt. a

a

Let us successively differentiate the integral equation four times. As a result, we have (the first line is the original equation): A1 I1 + A2 I2 = f ,

f = f (x),

(1)

A1 λ1 J1 + A2 λ2 J2 = fx ,  , (A1 λ1 + A2 λ2 )y + A1 λ21 I1 + A2 λ22 I2 = fxx

(2) (3)

 , (A1 λ1 + A2 λ2 )yx + A1 λ31 J1 + A2 λ32 J2 = fxxx  3 3 4  . (A1 λ1 + A2 λ2 )yxx + (A1 λ1 + A2 λ2 )y + A1 λ1 I1 + A2 λ42 I2 = fxxxx

(4) (5)

Eliminating I1 and I2 from (1), (3), and (5), we arrive at the following second-order linear ordinary differential equation with constant coefficients:    – λ1 λ2 (A1 λ2 + A2 λ1 )y = fxxxx – (λ21 + λ22 )fxx + λ21 λ22 f . (A1 λ1 + A2 λ2 )yxx

(6)

The initial conditions can be obtained by substituting x = a into (3) and (4):  (a), (A1 λ1 + A2 λ2 )y(a) = fxx

 (A1 λ1 + A2 λ2 )yx (a) = fxxx (a).

(7)

Solving the differential equation (6) under conditions (7) allows us to find the solution of the integral equation.

1.3. EQUATIONS WHOSE KERNELS CONTAIN HYPERBOLIC FUNCTIONS

2◦ . Denote ∆ = λ1 λ2

A1 λ2 + A2 λ1 . A1 λ1 + A2 λ2

2.1. Solution for ∆ > 0:  (x) + Bf (x) + C (A1 λ1 + A2 λ2 )y(x) = fxx

k=

√ ∆,



B = ∆ – λ21 – λ22 ,

sinh[k(x – t)]f (t) dt,

 1 C = √ ∆2 – (λ21 + λ22 )∆ + λ21 λ22 . ∆

 (A1 λ1 + A2 λ2 )y(x) = fxx (x) + Bf (x) + C



x

a

2.2. Solution for ∆ < 0:

k=

31



x

sin[k(x – t)]f (t) dt, a

–∆,

B = ∆ – λ21 – λ22 ,

 1 2 C= √ ∆ – (λ21 + λ22 )∆ + λ21 λ22 . –∆

2.3. Solution for ∆ = 0: (A1 λ1 + A2 λ2 )y(x) =

 fxx (x)



(λ21

+

λ22 )f (x) +

λ21 λ22

x

(x – t)f (t) dt. a

2.4. Solution for ∆ = ∞: y(x) =

50.

  – (λ21 + λ22 )fxx + λ21 λ22 f fxxxx , A1 λ31 + A2 λ32

f = f (x).

In the last case, the relation A1 λ1 + A2 λ2 = 0 is valid, and the right-hand side of the   integral equation is assumed to satisfy the conditions f (a) = fx (a) = fxx (a) = fxxx (a) = 0. x   A sinh[λ(x – t)] + B sinh[µ(x – t)] + C sinh[β(x – t)] y(t) dt = f (x). a

It assumed that f (a) = fx (a) = 0. Differentiating the integral equation twice yields x  2  Aλ sinh[λ(x – t)] + Bµ2 sinh[µ(x – t)] y(t) dt (Aλ + Bµ + Cβ)y(x) + a x 2  + Cβ sinh[β(x – t)]y(t) dt = fxx (x). a

Eliminating the last integral with the aid of the original equation, we arrive at an equation of the form 2.3.18: (Aλ + Bµ + Cβ)y(x) x    + A(λ2 – β 2 ) sinh[λ(x – t)] + B(µ2 – β 2 ) sinh[µ(x – t)] y(t) dt = fxx (x) – β 2 f (x). a

51.

In the special case Aλ + Bµ + Cβ = 0, this is an equation of the form 1.3.49. x  sinh2 [λ(x – t)]y(t) dt = f (x), f (a) = fx (a) = fxx (a) = 0. a

Differentiating yields an equation of the form 1.3.40: x 1 sinh[2λ(x – t)]y(t) dt = fx (x). λ a  (x) – 2fx (x). Solution: y(x) = 12 λ–2 fxxx

32

LINEAR EQUATIONS OF THE FIRST KIND WITH VARIABLE LIMIT OF INTEGRATION



x

52.

 sinh2 (λx) – sinh2 (λt) y(t) dt = f (x),

a

  fx (x) 1 d . λ dx sinh(2λx)

Solution: y(x) =

x

53.

f (a) = fx (a) = 0.

 A sinh2 (λx) + B sinh2 (λt) y(t) dt = f (x).

a

This is a special case of equation 1.9.4 with g(x) = sinh2 (λx). For B = –A, see equation 1.3.52. Solution:

x – 2A

– 2B  d 1 A+B A+B sinh(λx) sinh(λt) ft (t) dt . y(x) = A + B dx a

x

54.

 A sinh2 (λx) + B sinh2 (µt) y(t) dt = f (x).

a

This is a special case of equation 1.9.6 with g(x) = A sinh2 (λx) and h(t) = B sinh2 (µt).

x

sinh[λ(x – t)] sinh[λ(x + t)]y(t) dt = f (x).

55. a

Using the formula sinh(α – β) sinh(α + β) = 12 [cosh(2α) – cosh(2β)],

α = λx,

β = λt,

we reduce the original equation to an equation of the form 1.3.5:

x

[cosh(2λx) – cosh(2λt)]y(t) dt = 2f (x). a

Solution: y(x) =

x

56.

  fx (x) 1 d . λ dx sinh(2λx)

 A sinh(λx) sinh(µt) + B sinh(βx) sinh(γt) y(t) dt = f (x).

a

This is a special case of equation 1.9.15 with g1 (x) = A sinh(λx), h1 (t) = sinh(µt), g2 (x) = B sinh(βx), and h2 (t) = sinh(γt).

x

57.

  f (a) = fx (a) = fxx (a) = fxxx (a) = 0.

sinh3 [λ(x – t)]y(t) dt = f (x),

a

Using the formula sinh3 β = a



x

58. a

x

1 4

1 4

sinh 3β –

3 4

sinh[3λ(x – t)] –

sinh β, we arrive at an equation of the form 1.3.49: 3 4

 sinh[λ(x – t)] y(t) dt = f (x).

 sinh3 (λx) – sinh3 (λt) y(t) dt = f (x),

f (a) = fx (a) = 0.

This is a special case of equation 1.9.2 with g(x) = sinh3 (λx).

1.3. EQUATIONS WHOSE KERNELS CONTAIN HYPERBOLIC FUNCTIONS



x

59.

33

 A sinh3 (λx) + B sinh3 (λt) y(t) dt = f (x).

a

This is a special case of equation 1.9.4 with g(x) = sinh3 (λx). Solution:

x – 3A

– 3B d 1 sinh(λx) A+B y(x) = sinh(λt) A+B ft (t) dt . A + B dx a

x

60.

 A sinh2 (λx) sinh(µt) + B sinh(βx) sinh2 (γt) y(t) dt = f (x).

a

61.

This is a special case of equation 1.9.15 with g1 (x) = A sinh2 (λx), h1 (t) = sinh(µt), g2 (x) = B sinh(βx), and h2 (t) = sinh2 (γt). x sinh4 [λ(x – t)]y(t) dt = f (x). a  (a) = 0. It is assumed that f (a) = fx (a) = · · · = fxxxx Let us transform the kernel of the integral equation using the formula

sinh4 β =

1 8

cosh 4β –

1 2

cosh 2β + 38 ,

where β = λ(x – t),

and differentiate the resulting equation with respect to x. Then we arrive at an equation of the form 1.3.49: x 1   λ 2 sinh[4λ(x – t)] – sinh[2λ(x – t)] y(t) dt = fx (x). a



x

62.

sinhn [λ(x – t)]y(t) dt = f (x),

n = 2, 3, . . .

a

It is assumed that f (a) = fx (a) = · · · = fx(n) (a) = 0. 1◦ . Let us differentiate the equation with respect to x twice and transform the kernel of the resulting integral equation using the formula cosh2 β = 1 + sinh2 β, where β = λ(x – t). Then we have x x  λ2 n2 sinhn [λ(x – t)]y(t) dt + λ2 n(n – 1) sinhn–2 [λ(x – t)]y(t) dt = fxx (x). a

a

Eliminating the first term on the left-hand side with the aid of the original equation, we obtain x

  1 fxx (x) – λ2 n2 f (x) . sinhn–2 [λ(x – t)]y(t) dt = 2 λ n(n – 1) a This equation has the same form as the original equation, but the exponent of the kernel has been reduced by two. By applying this technique sufficiently many times, we finally arrive at simple integral equations of the form 1.1.1 (for even n) or 1.3.40 (for odd n). 2◦ . Solution: y(x) =

1 n λ n!



    d d d + nλ + (n – 2)λ . . . – nλ f (x). dx dx dx

Reference: A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 436).

34

LINEAR EQUATIONS OF THE FIRST KIND WITH VARIABLE LIMIT OF INTEGRATION



x

63.

 √  sinh λ x – t y(t) dt = f (x).

a

Solution: 2 d2 y(x) = πλ dx2



 √  cos λ x – t √ f (t) dt. x–t

x

a

References: A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 437), S. G. Samko, A. A. Kilbas, and O. I. Marichev (1993).





64.

 √  sinh λ t – x y(t) dt = f (x).

x

Solution: 2 d2 y(x) = πλ dx2



 √  cos λ t – x √ f (t) dt. t–x



x

References: H. M. Srivastava and R. G. Buschman (1977), A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 439), S. G. Samko, A. A. Kilbas, and O. I. Marichev (1993).



x

65.



sinh x – sinh t y(t) dt = f (x).

a

Solution: y(x) =

x

66. a



y(t) dt sinh x – sinh t

= f (x).

Solution:



x

67.

 1 d 2 x cosh t f (t) dt 2 √ . cosh x π cosh x dx sinh x – sinh t a

1 d y(x) = π dx

a

(sinh x – sinh t)λ y(t) dt = f (x),

x

cosh t f (t) dt √ . sinh x – sinh t

0 < λ < 1.

a

Solution:  y(x) = k cosh x

x

68.

d 2 1 cosh x dx



x a

cosh t f (t) dt , (sinh x – sinh t)λ

k=

sin(πλ) . πλ

(sinhµ x – sinhµ t)y(t) dt = f (x).

a

69.

This is a special case of equation 1.9.2 with g(x) = sinhµ x.  1 d  fx (x) Solution: y(x) = . µ dx cosh x sinhµ–1 x x

 A sinhµ (λx) + B sinhµ (λt) y(t) dt = f (x). a

This is a special case of equation 1.9.4 with g(x) = sinhµ (λx). Solution with B ≠ –A:

x – Aµ

– Bµ d 1 sinh(λx) A+B y(x) = sinh(λt) A+B ft (t) dt . A + B dx a

1.3. EQUATIONS WHOSE KERNELS CONTAIN HYPERBOLIC FUNCTIONS



x

70. a

y(t) dt = f (x), (sinh x – sinh t)λ

Solution: y(x) =

0 < λ < 1.

sin(πλ) d π dx

x

a

x

cosh t f (t) dt . (sinh x – sinh t)1–λ

 f (a) = fx (a) = fxx (a) = 0.

(x – t) sinh[λ(x – t)]y(t) dt = f (x),

71.

35

a

Double differentiation yields



x 2

cosh[λ(x – t)]y(t) dt + λ

2λ a

x

 (x – t) sinh[λ(x – t)]y(t) dt = fxx (x).

a

Eliminating the second term on the left-hand side with the aid of the original equation, we arrive at an equation of the form 1.3.1:

x

cosh[λ(x – t)]y(t) dt = a

Solution:

 1  fxx (x) – λ2 f (x) . 2λ

1  f (x) – λfx (x) + 12 λ3 y(x) = 2λ xxx



x

f (t) dt. a

Reference: A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 436).



x

72. a

sinh[λ(x – t)] y(t) dt = f (x), √ x–t

Solution: y(x) =

2 πλ



x

a

f (a) = fx (a) = 0. sinh[λ(x – t)]  √ [ftt (t) – λ2 f (t)] dt. x–t

Reference: A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 436).



x

73.

sinh[λ(x – t)] (x – t)3/2

a

Solution:

y(t) dt = f (x),

2 y(x) = πλ

a

x

f (a) = fx (a) = 0.

sinh[λ(x – t)]   f  (t)  √ dt. ftt (t) – λ2 f (t) + x–t x–t

Reference: A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 437).



x

74. a

√  sinh λ (x – t)(x – t + γ) y(t) dt = f (x). √ x–t+γ

  It is assumed that f (a) = fx (a) = fxx (a) = fxxx (a) = 0. Solution:

√  t 2  d2 2 x cosh λ (x – t)(x – t – γ) √ y(x) = sinh[λ(t – s)] + λ2 f (s) ds dt. 2 2 πλ a ds x–t a

References: A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 438), S. G. Samko, A. A. Kilbas, and O. I. Marichev (1993).

36

LINEAR EQUATIONS OF THE FIRST KIND WITH VARIABLE LIMIT OF INTEGRATION



x

75.

Axβ + B sinhγ (λt) + C]y(t) dt = f (x).

a

This is a special case of equation 1.9.6 with g(x) = Axβ and h(t) = B sinhγ (λt) + C.

x

76.

A sinhγ (λx) + Btβ + C]y(t) dt = f (x).

a

This is a special case of equation 1.9.6 with g(x) = A sinhγ (λx) and h(t) = Btβ + C.

x

77.

  Axλ sinhµ t + Btβ sinhγ x y(t) dt = f (x).

a

This is a special case of equation 1.9.15 with g1 (x) = Axλ , h1 (t) = sinhµ t, g2 (x) = B sinhγ x, and h2 (t) = tβ . 1.3-3. Kernels Containing Hyperbolic Tangent.

x

78.

 tanh(λx) – tanh(λt) y(t) dt = f (x).

a

79.

This is a special case of equation 1.9.2 with g(x) = tanh(λx).  1 cosh2 (λx)fx (x) x . Solution: y(x) = λ x

 A tanh(λx) + B tanh(λt) y(t) dt = f (x). a

This is a special case of equation 1.9.4 For B = –A, see equation 1.3.78. with g(x) =Atanh(λx).

x –

– B  d 1 tanh(λx) A+B tanh(λt) A+B ft (t) dt . Solution: y(x) = A + B dx a

x

80.

 A tanh(λx) + B tanh(µt) + C y(t) dt = f (x).

a

This is a special case of equation 1.9.6 with g(x) = A tanh(λx) and h(t) = B tanh(µt) + C.

x

81.

 tanh2 (λx) – tanh2 (λt) y(t) dt = f (x).

a

This is a special case of equation 1.9.2 withg(x) = tanh2 (λx).  3 d cosh (λx)fx (x) . Solution: y(x) = dx 2λ sinh(λx)

x

82.

 A tanh2 (λx) + B tanh2 (λt) y(t) dt = f (x).

a 2 This is a special case of equation 1.9.4 For B = –A, see equation 1.3.81. with g(x) =2Atanh

(λx). x – – 2B  d 1 A+B A+B tanh(λx) tanh(λt) ft (t) dt . Solution: y(x) = A + B dx a



x

83.

 A tanh2 (λx) + B tanh2 (µt) + C y(t) dt = f (x).

a

This is a special case of equation 1.9.6 with g(x) = A tanh2 (λx) and h(t) = B tanh2 (µt) + C.

1.3. EQUATIONS WHOSE KERNELS CONTAIN HYPERBOLIC FUNCTIONS



x

84.

tanh(λx) – tanh(λt)

n

y(t) dt = f (x),

37

n = 1, 2, . . .

a

The right-hand side of the equation is assumed to satisfy the conditions f (a) = fx (a) = · · · = fx(n) (a) = 0. n+1  d 1 2 cosh (λx) f (x). Solution: y(x) = dx λn n! cosh2 (λx)

x

85.



tanh x – tanh t y(t) dt = f (x).

a

Solution: y(x) =

x

86. a



 d 2 2 2 cosh x dx π cosh2 x

y(t) dt tanh x – tanh t

y(x) = x

87.

x

f (t) dt √ . cosh t tanh x – tanh t 2

a

= f (x).

Solution:





1 d π dx

a

(tanh x – tanh t)λ y(t) dt = f (x),

x

cosh2 t

f (t) dt √ . tanh x – tanh t

0 < λ < 1.

a

Solution: d 2 sin(πλ)  2 cosh x y(x) = dx πλ cosh2 x

x

88.

a

x

f (t) dt . cosh t (tanh x – tanh t)λ 2

(tanhµ x – tanhµ t)y(t) dt = f (x).

a

This is a special case of equation 1.9.2 with g(x) = tanhµ x.   1 d coshµ+1 xfx (x) Solution: y(x) = . µ dx sinhµ–1 x

x

89.

  A tanhµ x + B tanhµ t y(t) dt = f (x).

a

This is a special case of equation 1.9.4 with g(x) = tanhµ x. For B = –A, see equation 1.3.88. Solution:

x – Aµ

– Bµ  d 1 A+B A+B tanh(λx) tanh(λt) ft (t) dt . y(x) = A + B dx a

x

90. a

y(t) dt = f (x), [tanh(λx) – tanh(λt)]µ

0 < µ < 1.

This is a special case of equation 1.9.44 with g(x) = tanh(λx) and h(x) ≡ 1. Solution: x f (t) dt λ sin(πµ) d y(x) = . 2 π dx a cosh (λt)[tanh(λx) – tanh(λt)]1–µ

38

LINEAR EQUATIONS OF THE FIRST KIND WITH VARIABLE LIMIT OF INTEGRATION



x

91.

Axβ + B tanhγ (λt) + C]y(t) dt = f (x).

a

This is a special case of equation 1.9.6 with g(x) = Axβ and h(t) = B tanhγ (λt) + C.

x

92.

A tanhγ (λx) + Btβ + C]y(t) dt = f (x).

a

This is a special case of equation 1.9.6 with g(x) = A tanhγ (λx) and h(t) = Btβ + C.

x

93.

  Axλ tanhµ t + Btβ tanhγ x y(t) dt = f (x).

a

This is a special case of equation 1.9.15 with g1 (x) = Axλ , h1 (t) = tanhµ t, g2 (x) = B tanhγ x, and h2 (t) = tβ . 1.3-4. Kernels Containing Hyperbolic Cotangent.

x

94.

 coth(λx) – coth(λt) y(t) dt = f (x).

a

95.

This is a special case of equation 1.9.2 with g(x) = coth(λx).  1 d sinh2 (λx)fx (x) . Solution: y(x) = – λ dx x

 A coth(λx) + B coth(λt) y(t) dt = f (x). a

This is a special case of equation 1.9.4 For B = –A, see equation 1.3.94. with g(x)A= coth(λx).

x 

 B  d 1 tanh(λx) A+B tanh(λt) A+B ft (t) dt . Solution: y(x) = A + B dx a

x

96.

 A coth(λx) + B coth(µt) + C y(t) dt = f (x).

a

This is a special case of equation 1.9.6 with g(x) = A coth(λx) and h(t) = B coth(µt) + C.

x

97.

 coth2 (λx) – coth2 (λt) y(t) dt = f (x).

a 2 This is a special case of equation 1.9.2 with g(x)   = coth (λx). 3  d sinh (λx)fx (x) . Solution: y(x) = – dx 2λ cosh(λx)



x

98.

 A coth2 (λx) + B coth2 (λt) y(t) dt = f (x).

a 2 This is a special case of equation 1.9.4 = coth = –A, see equation 1.3.97. with g(x)2A

x(λx). For B 2B 

 1 d  A+B A+B tanh(λx) tanh(λt) ft (t) dt . Solution: y(x) = A + B dx a



x

99.

 A coth2 (λx) + B coth2 (µt) + C y(t) dt = f (x).

a

This is a special case of equation 1.9.6 with g(x) = A coth2 (λx) and h(t) = B coth2 (µt) + C.

1.3. EQUATIONS WHOSE KERNELS CONTAIN HYPERBOLIC FUNCTIONS



x

100.

coth(λx) – coth(λt)

n

y(t) dt = f (x),

39

n = 1, 2, . . .

a

The right-hand side of the equation is assumed to satisfy the conditions f (a) = fx (a) = · · · = fx(n) (a) = 0. n+1  d (–1)n 2 sinh (λx) f (x). Solution: y(x) = n dx λ n! sinh2 (λx)

x

101.

(cothµ x – cothµ t)y(t) dt = f (x).

a µ This is a special case of equation with g(x)  1.9.2  = coth x. µ+1  1 d sinh xfx (x) Solution: y(x) = – . µ dx coshµ–1 x x   A cothµ x + B cothµ t y(t) dt = f (x). 102. a

This is a special case of equation 1.9.4 with g(x) = cothµ x. For B = –A, see equation 1.3.101. Solution:

x Aµ Bµ  1 d A+B A+B y(x) = tanh x tanh t ft (t) dt . A + B dx a

x

103.

Axβ + B cothγ (λt) + C]y(t) dt = f (x).

a

This is a special case of equation 1.9.6 with g(x) = Axβ and h(t) = B cothγ (λt) + C.

x

104.

A cothγ (λx) + Btβ + C]y(t) dt = f (x).

a

This is a special case of equation 1.9.6 with g(x) = A cothγ (λx) and h(t) = Btβ + C.

x

105.

  Axλ cothµ t + Btβ cothγ x y(t) dt = f (x).

a

This is a special case of equation 1.9.15 with g1 (x) = Axλ , h1 (t) = cothµ t, g2 (x) = B cothγ x, and h2 (t) = tβ . 1.3-5. Kernels Containing Combinations of Hyperbolic Functions.

x

106.



 cosh[λ(x – t)] + A sinh[µ(x – t)] y(t) dt = f (x).

a

Let us differentiate the equation with respect to x and then eliminate the integral with the hyperbolic cosine. As a result, we arrive at an equation of the form 2.3.16: x 2 y(x) + (λ – A µ) sinh[µ(x – t)]y(t) dt = fx (x) – Aµf (x). a



x

107.

 A cosh(λx) + B sinh(µt) + C y(t) dt = f (x).

a

This is a special case of equation 1.9.6 with g(x) = A cosh(λx) and h(t) = B sinh(µt) + C.

40

LINEAR EQUATIONS OF THE FIRST KIND WITH VARIABLE LIMIT OF INTEGRATION



x

108.

 A cosh2 (λx) + B sinh2 (µt) + C y(t) dt = f (x).

a

This is a special case of equation 1.9.6 with g(x) = A cosh2 (λx) and h(t) = B sinh2 (µt) + C.

x

sinh[λ(x – t)] cosh[λ(x + t)]y(t) dt = f (x).

109. a

Using the formula sinh(α – β) cosh(α + β) =

1 2

 sinh(2α) – sinh(2β) ,

α = λx,

β = λt,

we reduce the original equation to an equation of the form 1.3.45:

x

 sinh(2λx) – sinh(2λt) y(t) dt = 2f (x).

a

Solution: y(x) =

  fx (x) 1 d . λ dx cosh(2λx)

x

cosh[λ(x – t)] sinh[λ(x + t)]y(t) dt = f (x).

110. a

Using the formula cosh(α – β) sinh(α + β) =

1 2

 sinh(2α) + sinh(2β) ,

α = λx,

β = λt,

we reduce the original equation to an equation of the form 1.3.46 with A = B = 1:

x

 sinh(2λx) + sinh(2λt) y(t) dt = 2f (x).

a



x

111.

 A cosh(λx) sinh(µt) + B cosh(βx) sinh(γt) y(t) dt = f (x).

a

This is a special case of equation 1.9.15 with g1 (x) = A cosh(λx), h1 (t) = sinh(µt), g2 (x) = B cosh(βx), and h2 (t) = sinh(γt).

x

112.

 sinh(λx) cosh(µt) + sinh(βx) cosh(γt) y(t) dt = f (x).

a

This is a special case of equation 1.9.15 with g1 (x) = sinh(λx), h1 (t) = cosh(µt), g2 (x) = sinh(βx), and h2 (t) = cosh(γt).

x

113.

 cosh(λx) cosh(µt) + sinh(βx) sinh(γt) y(t) dt = f (x).

a

This is a special case of equation 1.9.15 with g1 (x) = cosh(λx), h1 (t) = cosh(µt), g2 (x) = sinh(βx), and h2 (t) = sinh(γt).

x

114.

 A coshβ (λx) + B sinhγ (µt) y(t) dt = f (x).

a

This is a special case of equation 1.9.6 with g(x) = A coshβ (λx) and h(t) = B sinhγ (µt).

41

1.3. EQUATIONS WHOSE KERNELS CONTAIN HYPERBOLIC FUNCTIONS



x

115.

 A sinhβ (λx) + B coshγ (µt) y(t) dt = f (x).

a

This is a special case of equation 1.9.6 with g(x) = A sinhβ (λx) and h(t) = B coshγ (µt).

x

116.

  Axλ coshµ t + Btβ sinhγ x y(t) dt = f (x).

a

This is a special case of equation 1.9.15 with g1 (x) = Axλ , h1 (t) = coshµ t, g2 (x) = B sinhγ x, and h2 (t) = tβ .

x

117.



 (x – t) sinh[λ(x – t)] – λ(x – t)2 cosh[λ(x – t)] y(t) dt = f (x).

a



Solution:

x

g(t) dt,

y(x) = a

where

 g(t) =



x



118.

sinh[λ(x – t)] x–t

a

π 1 2λ 64λ5

1 y(x) = 2λ4

x

d2 – λ2 dt2

6

t

5

(t – τ ) 2 I 5 [λ(t – τ )] f (τ ) dτ . 2

a

– λ cosh[λ(x – t)] y(t) dt = f (x).

Solution:

119.





3

d2 – λ2 dx2

x

sinh[λ(x – t)]f (t) dt. a

√  √   √  sinh λ x – t – λ x – t cosh λ x – t y(t) dt = f (x),

a

Solution: 4 d3 y(x) = – 3 3 πλ dx

x

120.

a

x

f (a) = fx (a) = 0.

 √  cos λ x – t √ f (t) dt. x–t

  Axλ sinhµ t + Btβ coshγ x y(t) dt = f (x).

a

This is a special case of equation 1.9.15 with g1 (x) = Axλ , h1 (t) = sinhµ t, g2 (x) = B coshγ x, and h2 (t) = tβ .

x

121.

 A tanh(λx) + B coth(µt) + C y(t) dt = f (x).

a

This is a special case of equation 1.9.6 with g(x) = A tanh(λx) and h(t) = B coth(µt) + C.

x

122.

 A tanh2 (λx) + B coth2 (µt) y(t) dt = f (x).

a

This is a special case of equation 1.9.6 with g(x) = A tanh2 (λx) and h(t) = B coth2 (µt).

x

123.

 tanh(λx) coth(µt) + tanh(βx) coth(γt) y(t) dt = f (x).

a

This is a special case of equation 1.9.15 with g1 (x) = tanh(λx), h1 (t) = coth(µt), g2 (x) = tanh(βx), and h2 (t) = coth(γt).

42

LINEAR EQUATIONS OF THE FIRST KIND WITH VARIABLE LIMIT OF INTEGRATION



x

124.

 coth(λx) tanh(µt) + coth(βx) tanh(γt) y(t) dt = f (x).

a

This is a special case of equation 1.9.15 with g1 (x) = coth(λx), h1 (t) = tanh(µt), g2 (x) = coth(βx), and h2 (t) = tanh(γt). x

 125. tanh(λx) tanh(µt) + coth(βx) coth(γt) y(t) dt = f (x). a

This is a special case of equation 1.9.15 with g1 (x) = tanh(λx), h1 (t) = tanh(µt), g2 (x) = coth(βx), and h2 (t) = coth(γt). x

 126. A tanhβ (λx) + B cothγ (µt) y(t) dt = f (x). a

This is a special case of equation 1.9.6 with g(x) = A tanhβ (λx) and h(t) = B cothγ (µt). x

 127. A cothβ (λx) + B tanhγ (µt) y(t) dt = f (x). a

This is a special case of equation 1.9.6 with g(x) = A cothβ (λx) and h(t) = B tanhγ (µt). x   128. Axλ tanhµ t + Btβ cothγ x y(t) dt = f (x). a

This is a special case of equation 1.9.15 with g1 (x) = Axλ , h1 (t) = tanhµ t, g2 (x) = B cothγ x, and h2 (t) = tβ . x   129. Axλ cothµ t + Btβ tanhγ x y(t) dt = f (x). a

This is a special case of equation 1.9.15 with g1 (x) = Axλ , h1 (t) = cothµ t, g2 (x) = B tanhγ x, and h2 (t) = tβ .

1.4. Equations Whose Kernels Contain Logarithmic Functions 1.4-1. Kernels Containing Logarithmic Functions.

x

(ln x – ln t)y(t) dt = f (x).

1. a

2.

This is a special case of equation 1.9.2 with g(x) = ln x.  Solution: y(x) = xfxx (x) + fx (x). x ln(x – t)y(t) dt = f (x). 0

Solution:



x

y(x) = – 0



where C = lim 1 + k→∞

the gamma function.

ftt (t) dt





0

1 1 + ··· + 2 k+1

∞ z –Cz (x – t)z e–Cz x e dz – fx (0) dz, Γ(z + 1) Γ(z + 1) 0  – ln k = 0.5772 . . . is the Euler constant and Γ(z) is

References: M. L. Krasnov, A. I. Kisilev, and G. I. Makarenko (1971), A. G. Butkovskii (1979).

1.4. EQUATIONS WHOSE KERNELS CONTAIN LOGARITHMIC FUNCTIONS



43

x

[ln(x – t) + A]y(t) dt = f (x).

3. a

Solution: d y(x) = – dx



x

νA (x – t)f (t) dt, a

d νA (x) = dx





0

xz e(A–C)z dz, Γ(z + 1)

where C = 0.5772 . . . is the Euler constant and Γ(z) is the gamma function. For a = 0, the solution can be written in the form x ∞ ∞ z (A–C)z (x – t)z e(A–C)z x e y(x) = – dz – fx (0) dz. ftt (t) dt Γ(z + 1) Γ(z + 1) 0 0 0 Reference: S. G. Samko, A. A. Kilbas, and O. I. Marichev (1993).



x

(A ln x + B ln t)y(t) dt = f (x).

4. a

This is a special case of equation 1.9.4 with g(x) = ln x. For B = –A, see equation 1.4.1. Solution:

x – A – B  sign(ln x) d ln t A+B ft (t) dt . y(x) = ln x A+B A + B dx a

x

(A ln x + B ln t + C)y(t) dt = f (x).

5. a

6.

This is a special case of equation 1.9.5 with g(x) = x. x

2  ln (λx) – ln2 (λt) y(t) dt = f (x), f (a) = fx (a) = 0. a

  d xfx (x) . Solution: y(x) = dx 2 ln(λx)

x

7.

 A ln2 (λx) + B ln2 (λt) y(t) dt = f (x).

a

This is a special case of equation 1.9.4 with g(x) = ln2 (λx). For B = –A, see equation 1.4.6. Solution:

x – 2A 2B d 1 ln(λt) – A+B ft (t) dt . ln(λx) A+B y(x) = A + B dx a

x

8.

 A ln2 (λx) + B ln2 (µt) + C y(t) dt = f (x).

a

9.

This is a special case of equation 1.9.6 with g(x) = A ln2 (λx) and h(t) = B ln2 (µt) + C. x

n ln(x/t) y(t) dt = f (x), n = 1, 2, . . . a

The right-hand side of the equation is assumed to satisfy the conditions f (a) = fx (a) = · · · = fx(n) (a) = 0.  n+1 d 1 x f (x). Solution: y(x) = n! x dx

44

LINEAR EQUATIONS OF THE FIRST KIND WITH VARIABLE LIMIT OF INTEGRATION



x

10.

 2 n ln x – ln2 t y(t) dt = f (x),

n = 1, 2, . . .

a

The right-hand side of the equation is assumed to satisfy the conditions f (a) = fx (a) = · · · = fx(n) (a) = 0.  n+1 x d ln x f (x). Solution: y(x) = n 2 n! x ln x dx



x

11.

ln

x+b t+b

a

 y(t) dt = f (x).

This is a special case of equation 1.9.2 with g(x) = ln(x + b).  (x) + fx (x). Solution: y(x) = (x + b)fxx

x

12.

 ln(x/t) y(t) dt = f (x).

a

Solution: y(x) =

x

13. a

 2 x f (t) dt d 2  x . πx dx a t ln(x/t)

y(t) dt  = f (x). ln(x/t)

Solution: y(x) =

x

14.

1 d π dx



x a

f (t) dt  . t ln(x/t)

 lnµ (λx) – lnµ (λt) y(t) dt = f (x).

a

This is a special case of equation 1.9.2 with g(x) = lnµ (λx).  1 d x ln1–µ (λx)fx (x) . Solution: y(x) = µ dx

x

15.

 A lnβ (λx) + B lnγ (µt) + C y(t) dt = f (x).

a

This is a special case of equation 1.9.6 with g(x) = A lnβ (λx) and h(t) = B lnγ (µt) + C.

x

16.

[ln(x/t)]λ y(t) dt = f (x),

0 < λ < 1.

a

Solution:



x

17. a

y(t) dt [ln(x/t)]λ

 2 x f (t) dt d k x , y(x) = λ x dx t[ln(x/t)] a = f (x),

k=

sin(πλ) . πλ

0 < λ < 1.

This is a special case of equation 1.9.44 with g(x) = ln x and h(x) ≡ 1. Solution: x sin(πλ) d f (t) dt y(x) = . π dx a t[ln(x/t)]1–λ

45

1.4. EQUATIONS WHOSE KERNELS CONTAIN LOGARITHMIC FUNCTIONS

√ x+ x–t ln √ √ y(t) dt = f (x). x– x–t 0 Solution: √ t d 1 d x √ f (t) dt. y(x) = π dx 0 x – t dt

18.



x

Reference: A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 451).

√ √ t+ t–x ln √ √ y(t) dt = f (x). t– t–x x Solution: t d 1 1 d ∞ √ f (t) dt. y(x) = √ π x dx x t – x dt

19.



Reference: A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 452).

1.4-2. Kernels Containing Power-Law and Logarithmic Functions.

x

20.

 (x – t) ln(x – t) + A y(t) dt = f (x).

a

Solution: y(x) = –

d2 dx2



x

νA (x – t)f (t) dt,

νA (x) =

a

d dx



∞ 0

xz e(A–C)z dz, Γ(z + 1)

where C = 0.5772 . . . is the Euler constant and Γ(z) is the gamma function. Reference: S. G. Samko, A. A. Kilbas, and O. I. Marichev (1993).



21.

x

ln(x – t) + A y(t) dt = f (x), 0 < λ < 1. (x – t)λ a Solution: x x F (t) dt sin(πλ) d , F (x) = νh (x – t)f (t) dt, y(x) = – π dx a (x – t)1–λ a ∞ z hz x e d dz, h = A + ψ(1 – λ), νh (x) = dx 0 Γ(z + 1)

 where Γ(z) is the gamma function and ψ(z) = Γ(z) z is the logarithmic derivative of the gamma function. Reference: S. G. Samko, A. A. Kilbas, and O. I. Marichev (1993).

22.

x

(x – t)α–1

Γ(α) Solution:

[ln(x – t) + A]y(t) dt = f (x),

α > 0.

a

1 y(x) = – Γ([α] – α + 1)



d dx

[α]+1

a

x

F (t) dt , (x – t)α–[α]

F (x) =



x

νh (x – t)f (t) dt, a

xz ehz dz, h = A + ψ(α), Γ(z + 1) 0

 where Γ(z) is the gamma function and ψ(z) = Γ(z) z is the logarithmic derivative of the gamma function. νh (x) =

d dx

Reference: S. G. Samko, A. A. Kilbas, and O. I. Marichev (1993, p. 483).

46

LINEAR EQUATIONS OF THE FIRST KIND WITH VARIABLE LIMIT OF INTEGRATION



x

23.

 β λ t ln x – xβ lnλ t)y(t) dt = f (x).

a

This is a special case of equation 1.9.11 with g(x) = lnλ x and h(t) = tβ .

x

24.

 β λ At ln x + Bxµ lnγ t)y(t) dt = f (x).

a

This is a special case of equation 1.9.15 with g1 (x) = A lnλ x, h1 (t) = tβ , g2 (x) = Bxµ , and h2 (t) = lnγ t.



x

ln

25. a

xµ + b ctλ + s

 y(t) dt = f (x).

This is a special case of equation 1.9.6 with g(x) = ln(xµ + b) and h(t) = – ln(ctλ + s).

1.5. Equations Whose Kernels Contain Trigonometric Functions 1.5-1. Kernels Containing Cosine.

x

cos[λ(x – t)]y(t) dt = f (x).

1. a

Solution: y(x) = fx (x) + λ2



x

f (x) dx. a

References: H. M. Srivastava and R. G. Buschman (1977), A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 442).



x

2.



 cos[λ(x – t)] – 1 y(t) dt = f (x),

 f (a) = fx (a) = fxx (a) = 0.

a

Solution: y(x) = –

x

3.



1  f (x) – fx (x). λ2 xxx

 cos[λ(x – t)] + b y(t) dt = f (x).

a

For b = 0, see equation 1.5.1. For b = –1, see equation 1.5.2. For λ = 0, see equation 1.1.1. Differentiating the equation with respect to x, we arrive at an equation of the form 2.5.16: y(x) –



λ b+1

x

sin[λ(x – t)]y(t) dt = a

fx (x) . b+1

1◦ . Solution with b(b + 1) > 0: λ2 f  (x) + y(x) = x b+1 k(b + 1)2





x

sin[k(x –

t)]ft (t) dt,

where k = λ

a

b . b+1

2◦ . Solution with b(b + 1) < 0: λ2 f  (x) + y(x) = x b+1 k(b + 1)2





x

sinh[k(x – a

t)]ft (t) dt,

where k = λ

–b . b+1

47

1.5. EQUATIONS WHOSE KERNELS CONTAIN TRIGONOMETRIC FUNCTIONS



x

cos(λx + βt)y(t) dt = f (x).

4. a

Differentiating the equation with respect to x twice yields x cos[(λ + β)x]y(x) – λ sin(λx + βt)y(t) dt = fx (x), (1) a x    cos[(λ + β)x]y(x) x – λ sin[(λ + β)x]y(x) – λ2 cos(λx + βt)y(t) dt = fxx (x). (2) a

Eliminating the integral term from (2) with the aid of the original equation, we arrive at the first-order linear ordinary differential equation  (x) + λ2 f (x), wx – λ tan[(λ + β)x]w = fxx

w = cos[(λ + β)x]y(x).

(3)

Setting x = a in (1) yields the initial condition w(a) = fx (a). On solving equation (3) under this condition, after some transformations we obtain the solution of the original integral equation in the form y(x) =



x

5.

1 λ sin[(λ + β)x] fx (x) + f (x) cos[(λ + β)x] cos2 [(λ + β)x] x λβ – f (t) cosk–2 [(λ + β)t] dt, cosk+1 [(λ + β)x] a

k=

λ . λ+β

 cos(λx) – cos(λt) y(t) dt = f (x).

a

6.

This is a special case of equation  1.9.2 with g(x) = cos(λx). fx (x) 1 d . Solution: y(x) = – λ dx sin(λx) x

 A cos(λx) + B cos(λt) y(t) dt = f (x). a

7.

This is a special case of equation 1.9.4 with g(x) = cos(λx). For B = –A, see equation 1.5.5. Solution with B ≠ –A:

x – A – B  sign cos(λx) d A+B A+B y(x) = cos(λx) cos(λt) ft (t) dt . A+B dx a x

 A cos(λx) + B cos(µt) + C y(t) dt = f (x).

8.

This is a special case of equation 1.9.6 with g(x) = A cos(λx) and h(t) = B cos(µt) + C. x   A1 cos[λ1 (x – t)] + A2 cos[λ2 (x – t)] y(t) dt = f (x).

a

a

The equation is equivalent to the equation x   B1 sin[λ1 (x – t)] + B2 sin[λ2 (x – t)] y(t) dt = F (x), a x A1 A2 B1 = , B2 = , F (x) = f (t) dt, λ1 λ2 a which has the form 1.5.41. (Differentiation of this equation yields the original integral equation.)

48

LINEAR EQUATIONS OF THE FIRST KIND WITH VARIABLE LIMIT OF INTEGRATION



x

9.

cos2 [λ(x – t)]y(t) dt = f (x).

a

Differentiating yields an equation of the form 2.5.16: x y(x) – λ sin[2λ(x – t)]y(t) dt = fx (x). a

Solution: y(x) = fx (x) +

x

10.

Solution: y(x) = –

x



x

√ where k = λ 2.

sin[k(x – t)]ft (t) dt,

a

 cos2 (λx) – cos2 (λt) y(t) dt = f (x),

a

11.

2λ2 k

f (a) = fx (a) = 0.

   1 d fx (x) . λ dx sin(2λx)

 A cos2 (λx) + B cos2 (λt) y(t) dt = f (x).

a

12.

This is a special case of equation 1.9.4 with g(x) = cos2 (λx). For B = –A, see equation 1.5.10. Solution:

x – 2A

– 2B  1 d A+B A+B y(x) = cos(λx) cos(λt) ft (t) dt . A + B dx a x

 A cos2 (λx) + B cos2 (µt) + C y(t) dt = f (x). a

13.

This is a special case of equation 1.9.6 with g(x) = A cos2 (λx) and h(t) = B cos2 (µt) + C. x cos[λ(x – t)] cos[λ(x + t)]y(t) dt = f (x). a

Using the trigonometric formula cos(α – β) cos(α + β) =

1 2

 cos(2α) + cos(2β) ,

α = λx,

β = λt,

we reduce the original equation to an equation of the form 1.5.6 with A = B = 1: x

 cos(2λx) + cos(2λt) y(t) dt = 2f (x). a

Solution with cos(2λx) > 0: y(x) =

  x f  (t) dt 1 d √ √t . dx cos(2λx) a cos(2λt)

x

cos[λ(x – t)] cos[µ(x – t)]y(t) dt = f (x),

14. a

Solution: 1

y(x) =  λ2 – µ2



d2 + (λ + µ)2 dx2



d2 + (λ – µ)2 dx2

f (a) = fx (a) = 0. 

x t

sin a

  λ2 + µ2 (t – s) f (s) ds dt.

a

Reference: A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 444).

1.5. EQUATIONS WHOSE KERNELS CONTAIN TRIGONOMETRIC FUNCTIONS



x

15.

49

 A cos(λx) cos(µt) + B cos(βx) cos(γt) y(t) dt = f (x).

a

This is a special case of equation 1.9.15 with g1 (x) = A cos(λx), h1 (t) = cos(µt), g2 (x) = B cos(βx), and h2 (t) = cos(γt).

x

16.

cos3 [λ(x – t)]y(t) dt = f (x).

a

Using the formula cos3 β =

x

a



x

17.

1 4

1 4

cos 3β +

3 4

cos β, we arrive at an equation of the form 1.5.8:

cos[3λ(x – t)] +

3 4

 cos[λ(x – t)] y(t) dt = f (x).

 cos3 (λx) – cos3 (λt) y(t) dt = f (x),

a

f (a) = fx (a) = 0.

  fx (x) 1 d . Solution: y(x) = – 3λ dx sin(λx) cos2 (λx)

x

18.

 A cos3 (λx) + B cos3 (λt) y(t) dt = f (x).

a

This is a special case of equation 1.9.4 with g(x) = cos3 (λx). For B = –A, see equation 1.5.17. Solution:

x – 3A

– 3B  d 1 A+B A+B cos(λx) cos(λt) ft (t) dt . y(x) = A + B dx a

x

19.

 cos2 (λx) cos(µt) + cos(βx) cos2 (γt) y(t) dt = f (x).

a

This is a special case of equation 1.9.15 with g1(x) = cos2 (λx), h1 (t) = cos(µt), g2 (x) = cos(βx), and h2 (t) = cos2 (γt).

x

20.

cos4 [λ(x – t)]y(t) dt = f (x).

a

Let us transform the kernel of the integral equation using the trigonometric formula cos4 β = 1 1 3 8 cos 4β + 2 cos 2β + 8 , where β = λ(x – t), and differentiate the resulting equation with respect to x. Then we arrive at an equation of the form 2.5.18:

x

y(x) – λ a



x

21.

cos(λx) – cos(λt)

n

1 2

 sin[4λ(x – t)] + sin[2λ(x – t)] y(t) dt = fx (x).

y(t) dt = f (x),

n = 1, 2, . . .

a

The right-hand side of the equation is assumed to satisfy the conditions f (a) = fx (a) = · · · = fx(n) (a) = 0. n+1  (–1)n d 1 f (x). Solution: y(x) = n sin(λx) λ n! sin(λx) dx

50

LINEAR EQUATIONS OF THE FIRST KIND WITH VARIABLE LIMIT OF INTEGRATION



x

22.



cos t – cos x y(t) dt = f (x).

a

23.

24.

This is a special case of equation 1.9.40 with g(x) = 1 – cos x. Solution:  1 d 2 x sin t f (t) dt 2 √ y(x) = sin x . π sin x dx cos t – cos x a x y(t) dt = f (x). √ cos t – cos x a Solution: x sin t f (t) dt 1 d √ . y(x) = π dx a cos t – cos x x (cos t – cos x)λ y(t) dt = f (x), 0 < λ < 1. a

Solution:



x

25.

 1 d 2 x sin t f (t) dt y(x) = k sin x , λ sin x dx a (cos t – cos x)

k=

sin(πλ) . πλ

(cosµ x – cosµ t)y(t) dt = f (x).

a

26.

µ This is a special case of equation with g(x)  1.9.2  = cos x.  fx (x) 1 d . Solution: y(x) = – µ dx sin x cosµ–1 x x   A cosµ x + B cosµ t y(t) dt = f (x). a

27.

28.

This is a special case of equation 1.9.4 with g(x) = cosµ x. For B = –A, see equation 1.5.25. Solution:

x – Aµ – Bµ  1 d A+B A+B y(x) = cos x cos t ft (t) dt . A + B dx a x y(t) dt = f (x), 0 < λ < 1. λ a (cos t – cos x) Solution: x sin t f (t) dt sin(πλ) d y(x) = . π dx a (cos t – cos x)1–λ x (x – t) cos[λ(x – t)]y(t) dt = f (x), f (a) = fx (a) = 0. a

Differentiating the equation twice yields x x 2  sin[λ(x – t)]y(t) dt – λ (x – t) cos[λ(x – t)]y(t) dt = fxx (x). y(x) – 2λ a

a

Eliminating the third term on the left-hand side with the aid of the original equation, we arrive at an equation of the form 2.5.16: x  y(x) – 2λ sin[λ(x – t)]y(t) dt = fxx (x) + λ2 f (x). a

1.5. EQUATIONS WHOSE KERNELS CONTAIN TRIGONOMETRIC FUNCTIONS



29.

51

x

cos[λ(x – t)] √ y(t) dt = f (x), f (a) = fx (a) = 0. x–t a Solution: x sin[λ(x – t)]  2 √ [ftt (t) + λ2 f (t)] dt. y(x) = πλ a x–t Reference: A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 445).



x

30.



 √  x – t cos λ x – t y(t) dt = f (x).

a

Solution: y(x) =

1 π

a

x

 √  cosh λ x – t  √ ft (t) dt. x–t

References: A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, pp. 445–446), S. G. Samko, A. A. Kilbas, and O. I. Marichev (1993).

 √  cos λ x – t y(t) dt = f (x). √ x–t a Solution:  √  x cosh λ x – t 1 d √ y(x) = f (t) dt. π dx a x–t

31.

x

References: A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 446), S. G. Samko, A. A. Kilbas, and O. I. Marichev (1993).

 √  cos λ t – x y(t) dt = f (x). √ t–x x Solution:  √  ∞ cosh λ t – x 1 d √ y(x) = – f (t) dt. π dx x t–x

32.



References: H. M. Srivastava and R. G. Buschman (1977), A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 448), S. G. Samko, A. A. Kilbas, and O. I. Marichev (1993).

  √ cos λ x2 – t2 y(t) dt = f (x). √ x2 – t2 0 Solution:  √  x cosh λ x2 – t2 2 d √ t f (t) dt. y(x) = π dx 0 x2 – t2   √ ∞ cos λ t2 – x2 y(t) dt = f (x). √ t2 – x2 x Solution:  √  ∞ cosh λ t2 – x2 2 d √ t f (t) dt. y(x) = – π dx x t2 – x2   √ x cos λ xt – t2 y(t) dt = f (x). √ x–t 0 Solution:  √  x cosh λ x2 – xt 1 √ [f (t)/2 + tft (t)] dt. y(x) = πx 0 x–t

33.

34.

35.

x

Reference: A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 446).

52

LINEAR EQUATIONS OF THE FIRST KIND WITH VARIABLE LIMIT OF INTEGRATION



x

36. 0

 √  cos λ x2 – xt y(t) dt = f (x). √ x–t

Solution:

  √   x cosh λ xt – t2 x d √ √ f (t) dt . x y(x) = π dx x–t 0 √

Reference: A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 446).



x

37. a

√  cos λ (x – t)(x – t + γ) y(t) dt = f (x). √ x–t

  (a) = fxxx (a) = 0. It is assumed that f (a) = fx (a) = fxx Solution:

√  t 2  2 x sin λ (x – t)(x – t – γ) 2 d 2 √ y(x) = sin[λ(t – s)] +λ f (s) ds dt. πλ2 a x–t–γ ds 2 a

Reference: A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 447).



x

38.

Axβ + B cosγ (λt) + C]y(t) dt = f (x).

a

This is a special case of equation 1.9.6 with g(x) = Axβ and h(t) = B cosγ (λt) + C.

x

39.

A cosγ (λx) + Btβ + C]y(t) dt = f (x).

a

This is a special case of equation 1.9.6 with g(x) = A cosγ (λx) and h(t) = Btβ + C.

x

40.

  Axλ cosµ t + Btβ cosγ x y(t) dt = f (x).

a

This is a special case of equation 1.9.15 with g1 (x) = Axλ , h1 (t) = cosµ t, g2 (x) = B cosγ x, and h2 (t) = tβ . 1.5-2. Kernels Containing Sine.

x

f (a) = fx (a) = 0.

sin[λ(x – t)]y(t) dt = f (x),

41. a

Solution: y(x) =

1  f (x) + λf (x). λ xx

References: H. M. Srivastava and R. G. Buschman (1977), A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 442).



x

42.



 sin[λ(x – t)] + b y(t) dt = f (x).

a

For b = 0, see equation 1.5.41. Assume that b ≠ 0. Differentiating the equation with respect to x yields an equation of the form 2.5.3: y(x) +

λ b



x

cos[λ(x – t)]y(t) dt = a

1  f (x). b x

53

1.5. EQUATIONS WHOSE KERNELS CONTAIN TRIGONOMETRIC FUNCTIONS



x

sin(λx + βt)y(t) dt = f (x).

43. a

For β = –λ, see equation 1.5.41. Assume that β ≠ –λ. Differentiating the equation with respect to x twice yields

x

cos(λx + βt)y(t) dt = fx (x), (1) x    sin[(λ + β)x]y(x) x + λ cos[(λ + β)x]y(x) – λ2 sin(λx + βt)y(t) dt = fxx (x). (2) sin[(λ + β)x]y(x) + λ

a

a

Eliminating the integral term from (2) with the aid of the original equation, we arrive at the first-order linear ordinary differential equation  wx + λ cot[(λ + β)x]w = fxx (x) + λ2 f (x),

w = sin[(λ + β)x]y(x).

(3)

Setting x = a in (1) yields the initial condition w(a) = fx (a). On solving equation (3) under this condition, after some transformation we obtain the solution of the original integral equation in the form y(x) =



x

44.

1 λ cos[(λ + β)x] f  (x) – f (x) sin[(λ + β)x] x sin2 [(λ + β)x] x λβ – f (t) sink–2 [(λ + β)t] dt, sink+1 [(λ + β)x] a

k=

λ . λ+β

 sin(λx) – sin(λt) y(t) dt = f (x).

a

This is a special case of equation   1.9.2with g(x) = sin(λx). 1 d fx (x) . Solution: y(x) = λ dx cos(λx)

x

45.

 A sin(λx) + B sin(λt) y(t) dt = f (x).

a

This is a special case of equation 1.9.4 with g(x) = sin(λx). For B = –A, see equation 1.5.44. Solution with B ≠ –A:

x – A B sign sin(λx) d sin(λt) – A+B ft (t) dt . sin(λx) A+B y(x) = A+B dx a

x

46.

 A sin(λx) + B sin(µt) + C y(t) dt = f (x).

a

This is a special case of equation 1.9.6 with g(x) = A sin(λx) and h(t) = B sin(µt) + C.

x

47.



 µ sin[λ(x – t)] – λ sin[µ(x – t)] y(t) dt = f (x).

a   It is assumed that f (a) = fx (a) = fxx (a) = fxxx (a) = 0. Solution:  + λ2 µ2 f f  + (λ2 + µ2 )fxx , y(x) = xxxx 3 3 λµ – λ µ

f = f (x).

54

LINEAR EQUATIONS OF THE FIRST KIND WITH VARIABLE LIMIT OF INTEGRATION



x

48. a



 A1 sin[λ1 (x – t)] + A2 sin[λ2 (x – t)] y(t) dt = f (x),

f (a) = fx (a) = 0.

This equation can be solved in the same manner as equation 1.3.49, i.e., by reducing it to a second-order linear ordinary differential equation with constant coefficients. Let A1 λ2 + A2 λ1 ∆ = –λ1 λ2 . A1 λ1 + A2 λ2 1◦ . Solution for ∆ > 0:  (A1 λ1 + A2 λ2 )y(x) = fxx (x) + Bf (x) + C

k=

√ ∆,



B = ∆ + λ21 + λ22 ,

 1 C = √ ∆2 + (λ21 + λ22 )∆ + λ21 λ22 . ∆

 (A1 λ1 + A2 λ2 )y(x) = fxx (x) + Bf (x) + C



sinh[k(x – t)]f (t) dt, a

2◦ . Solution for ∆ < 0:

k=

x



x

sin[k(x – t)]f (t) dt, a

–∆,

B = ∆ + λ21 + λ22 ,

 1 2 C= √ ∆ + (λ21 + λ22 )∆ + λ21 λ22 . –∆

3◦ . Solution for ∆ = 0:  (A1 λ1 + A2 λ2 )y(x) = fxx (x) + (λ21 + λ22 )f (x) + λ21 λ22



x

(x – t)f (t) dt. a

4◦ . Solution for ∆ = ∞: y(x) = –

  + (λ21 + λ22 )fxx + λ21 λ22 f fxxxx , 3 3 A1 λ1 + A2 λ2

f = f (x).

In the last case, the relation A1 λ1 + A2 λ2 = 0 holds and the right-hand side of the integral   equation is assumed to satisfy the conditions f (a) = fx (a) = fxx (a) = fxxx (a) = 0.

49.

Remark. The solution can be obtained from the solution of equation 1.3.49 in which the change of variables λk → iλk , Ak → –iAk , i2 = –1 (k = 1, 2), should be made. x   A sin[λ(x – t)] + B sin[µ(x – t)] + C sin[β(x – t)] y(t) dt = f (x). a

It is assumed that f (a) = fx (a) = 0. Differentiating the integral equation twice yields x  2  (Aλ + Bµ + Cβ)y(x) – Aλ sin[λ(x – t)] + Bµ2 sin[µ(x – t)] y(t) dt a x 2  – Cβ sin[β(x – t)]y(t) dt = fxx (x). a

Eliminating the last integral with the aid of the original equation, we arrive at an equation of the form 2.5.18: x  (Aλ + Bµ + Cβ)y(x) + A(β 2 – λ2 ) sin[λ(x – t)] a   + B(β 2 – µ2 ) sin[µ(x – t)] y(t) dt = fxx (x) + β 2 f (x). In the special case Aλ + Bµ + Cβ = 0, this is an equation of the form 1.5.41.

55

1.5. EQUATIONS WHOSE KERNELS CONTAIN TRIGONOMETRIC FUNCTIONS



x

50.

 f (a) = fx (a) = fxx (a) = 0.

sin2 [λ(x – t)]y(t) dt = f (x),

a

Differentiation yields an equation of the form 1.5.41:

x

sin[2λ(x – t)]y(t) dt = a

1  f (x). λ x

 (x) + 2fx (x). Solution: y(x) = 12 λ–2 fxxx



x

51.

 sin2 (λx) – sin2 (λt) y(t) dt = f (x),

f (a) = fx (a) = 0.

a

   fx (x) 1 d . Solution: y(x) = λ dx sin(2λx)

x

52.

 A sin2 (λx) + B sin2 (λt) y(t) dt = f (x).

a

This is a special case of equation 1.9.4 with g(x) = sin2 (λx). For B = –A, see equation 1.5.51. Solution:

x – 2A – 2B  d 1 A+B A+B sin(λx) sin(λt) ft (t) dt . y(x) = A + B dx a

x

53.

 A sin2 (λx) + B sin2 (µt) + C y(t) dt = f (x).

a

This is a special case of equation 1.9.6 with g(x) = A sin2 (λx) and h(t) = B sin2 (µt) + C.

x

sin[λ(x – t)] sin[λ(x + t)]y(t) dt = f (x),

54. a

f (a) = fx (a) = 0.

Using the trigonometric formula sin(α – β) sin(α + β) =

1 2

 cos(2β) – cos(2α) ,

α = λx,

β = λt,

we reduce the original equation to an equation of the form 1.5.5:

x

 cos(2λx) – cos(2λt) y(t) dt = –2f (x).

a

   fx (x) 1 d . Solution: y(x) = λ dx sin(2λx)

x

sin[λ(x – t)] sin[µ(x – t)]y(t) dt = f (x),

55. a

Solution:

 y(x) =

d2 + (λ + µ)2 dx2



 f (a) = fx (a) = fxx (a) = 0.

 x d2 1 2 + (λ – µ) f (t) dt. dx2 2λµ a

Reference A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 443).

56

LINEAR EQUATIONS OF THE FIRST KIND WITH VARIABLE LIMIT OF INTEGRATION



x

56.

 sin(λx) sin(µt) + sin(βx) sin(γt) y(t) dt = f (x).

a

This is a special case of equation 1.9.15 with g1 (x) = sin(λx), h1 (t) = sin(µt), g2 (x) = sin(βx), and h2 (t) = sin(γt).

x

57.

sin3 [λ(x – t)]y(t) dt = f (x).

a   (a) = fxxx (a) = 0. It is assumed that f (a) = fx (a) = fxx 1 3 3 Using the formula sin β = – 4 sin 3β + 4 sin β, we arrive at an equation of the form 1.5.48:



x

a



x

58.

– 14 sin[3λ(x – t)] +

 sin3 (λx) – sin3 (λt) y(t) dt = f (x),

a

3 4

 sin[λ(x – t)] y(t) dt = f (x). f (a) = fx (a) = 0.

This is a special case of equation 1.9.2 with g(x) = sin3 (λx).

x

59.

 A sin3 (λx) + B sin3 (λt) y(t) dt = f (x).

a

This is a special case of equation 1.9.4 with g(x) = sin3 (λx). For B = –A, see equation 1.5.58. Solution:

x – 3A 3B sign sin(λx) d sin(λt) – A+B f  (t) dt . y(x) = sin(λx) A+B t A+B dx a

x

60.

 sin2 (λx) sin(µt) + sin(βx) sin2 (γt) y(t) dt = f (x).

a

This is a special case of equation 1.9.15 with g1 (x) = sin2 (λx), h1 (t) = sin(µt), g2 (x) = sin(βx), and h2 (t) = sin2 (γt).

x

61.

sin4 [λ(x – t)]y(t) dt = f (x).

a  (a) = 0. It is assumed that f (a) = fx (a) = · · · = fxxxx Let us transform the kernel of the integral equation using the trigonometric formula sin4 β = 18 cos 4β – 12 cos 2β + 38 , where β = λ(x – t), and differentiate the resulting equation with respect to x. Then we obtain an equation of the form 1.5.48:

λ a



x

62.

x

 – 12 sin[4λ(x – t)] + sin[2λ(x – t)] y(t) dt = fx (x).

sinn [λ(x – t)]y(t) dt = f (x),

n = 2, 3, . . .

a

It is assumed that f (a) = fx (a) = · · · = fx(n) (a) = 0. 1◦ . Let us differentiate the equation with respect to x twice and transform the kernel of the resulting integral equation using the formula cos2 β = 1 – sin2 β, where β = λ(x – t). We have



x

sinn [λ(x – t)]y(t) dt + λ2 n(n – 1)

–λ2 n2 a

a

x

 sinn–2 [λ(x – t)]y(t) dt = fxx (x).

1.5. EQUATIONS WHOSE KERNELS CONTAIN TRIGONOMETRIC FUNCTIONS

57

Eliminating the first term on the left-hand side with the aid of the original equation, we obtain

x

sinn–2 [λ(x – t)]y(t) dt = a

1 λ2 n(n

– 1)

  fxx (x) + λ2 n2 f (x) .

This equation has the same form as the original equation, but the degree characterizing the kernel has been reduced by two. By applying this technique sufficiently many times, we finally arrive at simple integral equations of the form 1.1.1 (for even n) or 1.5.41 (for odd n). 2◦ . Solution: 1 y(x) = n λ n!



d dx

1–α  β  k=1

 d2 2 f (x), + (2k + α)λ dx2

where α = n – 2[n/2], β = [(n + 1)/2], [A] denotes the integer part of number A. Reference: A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 443).



x

 f (a) = fx (a) = fxx (a) = 0.

(x – t) sin[λ(x – t)]y(t) dt = f (x),

63. a

Solution: 1 y(x) = 2λ



d2 + λ2 dx2

2

x

f (t) dt. a

References: A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 444), S. G. Samko, A. A. Kilbas, and O. I. Marichev (1993).



x

64.

 √  sin λ x – t y(t) dt = f (x).

a

Solution: 2 d2 y(x) = πλ dx2



x a

 √  cosh λ x – t √ f (t) dt. x–t

See also Example 2 in Section 10.4. References: A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 445), S. G. Samko, A. A. Kilbas, and O. I. Marichev (1993).





65.

√ sin(λ t – x)y(t) dt = f (x).

x

Solution: y(x) =

2 d2 πλ dx2



∞ x

√ cos(λ t – x ) √ f (t) dt. t–x

Reference: A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 447).



x

66. a

sin[λ(x – t)] y(t) dt = f (x), √ x–t

Solution:

2 y(x) = πλ

a

f (a) = fx (a) = 0. x

cos[λ(x – t)]  √ [ftt (t) + λ2 f (t)] dt. x–t

References: A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 445), S. G. Samko, A. A. Kilbas, and O. I. Marichev (1993).

58

LINEAR EQUATIONS OF THE FIRST KIND WITH VARIABLE LIMIT OF INTEGRATION



67.

x

sin[λ(x – t)] y(t) dt = f (x), f (a) = fx (a) = 0. (x – t)3/2 a Solution:   x sin[λ(x – t)]  2 f  (t) 2 √ dt. y(x) = f (t) + λ f (t) + tt πλ2 a x–t x–t References: A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 445), S. G. Samko, A. A. Kilbas, and O. I. Marichev (1993).



x

68. a

√  sin λ (x – t)(x – t + γ) y(t) dt = f (x). √ x–t+γ

  It is assumed that f (a) = fx (a) = fxx (a) = fxxx (a) = 0. Solution:

√  t 2  2 x cos λ (x – t)(x – t – γ) 2 d 2 √ y(x) = sin[λ(t – s)] + λ f (s) ds dt. πλ2 a ds 2 x–t a

Reference: A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 447).



x

69.



sin x – sin t y(t) dt = f (x).

a

Solution: y(x) = 70.

x



y(t) dt

sin x – sin t Solution: a



x

71.

 2 x cos t f (t) dt 1 d 2 √ cos x . π cos x dx sin x – sin t a

= f (x). 1 d y(x) = π dx

(sin x – sin t)λ y(t) dt = f (x),

a

x

cos t f (t) dt √ . sin x – sin t

0 < λ < 1.

a

Solution:



x

72.

 1 d 2 x cos t f (t) dt y(x) = k cos x , λ cos x dx a (sin x – sin t)

k=

sin(πλ) . πλ

(sinµ x – sinµ t)y(t) dt = f (x).

a

This is a special case of equation 1.9.2 with g(x) = sinµ x.  fx (x) 1 d . Solution: y(x) = µ dx cos x sinµ–1 x

x

73.



 A| sin(λx)|µ + B| sin(λt)|µ y(t) dt = f (x).

a

This is a special case of equation 1.9.4 with g(x) = | sin(λx)|µ . Solution:

x – Aµ Bµ d 1 sin(λt) – A+B ft (t) dt . sin(λx) A+B y(x) = A + B dx a

1.5. EQUATIONS WHOSE KERNELS CONTAIN TRIGONOMETRIC FUNCTIONS



x

74. a

y(t) dt = f (x), [sin(λx) – sin(λt)]µ

59

0 < µ < 1.

This is a special case of equation 1.9.44 with g(x) = sin(λx) and h(x) ≡ 1. Solution: x λ sin(πµ) d cos(λt)f (t) dt y(x) = . π dx a [sin(λx) – sin(λt)]1–µ

x

 f (a) = fx (a) = fxx (a) = 0.

(x – t) sin[λ(x – t)]y(t) dt = f (x),

75. a

Double differentiation yields



x

cos[λ(x – t)]y(t) dt – λ2

2λ a

x

 (x – t) sin[λ(x – t)]y(t) dt = fxx (x).

a

Eliminating the second integral on the left-hand side of this equation with the aid of the original equation, we arrive at an equation of the form 1.5.1:

x

cos[λ(x – t)]y(t) dt = a

 1  f (x) + λ2 f (x) . 2λ xx

Solution: y(x) =

x

76. a

1  1 fxxx (x) + λfx (x) + λ3 2λ 2

| sin(λ(x – t))|y(t) dt = f (x),

Solution: y(x) =

1 λ



x



x

f (t) dt. a

 f (a) = fx (a) = fxx (a) = 0.

   (–1)[λ(x–t)/π] fttt (t) + λ2 ft (t) dt,

a

where [A] denotes the integer part of number A. References: H. M. Srivastava and R. G. Buschman (1977), A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 443).



x

77.

Axβ + B sinγ (λt) + C]y(t) dt = f (x).

a

This is a special case of equation 1.9.6 with g(x) = Axβ and h(t) = B sinγ (λt) + C.

x

78.

A sinγ (λx) + Btβ + C]y(t) dt = f (x).

a

This is a special case of equation 1.9.6 with g(x) = A sinγ (λx) and h(t) = Btβ + C.

x

79.

  Axλ sinµ t + Btβ sinγ x y(t) dt = f (x).

a

This is a special case of equation 1.9.15 with g1 (x) = Axλ , h1 (t) = sinµ t, g2 (x) = B sinγ x, and h2 (t) = tβ .

60

LINEAR EQUATIONS OF THE FIRST KIND WITH VARIABLE LIMIT OF INTEGRATION

1.5-3. Kernels Containing Tangent.

x

80.

 tan(λx) – tan(λt) y(t) dt = f (x).

a

81.

This is a special case of equation 1.9.2 with g(x) = tan(λx).  1 d 2 cos (λx)fx (x) . Solution: y(x) = λ dx x

 A tan(λx) + B tan(λt) y(t) dt = f (x). a

This is a special case of equation 1.9.4 with g(x) = tan(λx). For B = –A, see equation 1.5.80.

x – A

– B  d 1 tan(λx) A+B tan(λt) A+B ft (t) dt . Solution: y(x) = A + B dx a

x

82.

 A tan(λx) + B tan(µt) + C y(t) dt = f (x).

a

This is a special case of equation 1.9.6 with g(x) = A tan(λx) and h(t) = B tan(µt) + C.

x

83.

 tan2 (λx) – tan2 (λt) y(t) dt = f (x).

a 2 This is a special case of equation 1.9.2 with   g(x) = tan (λx). 3  d cos (λx)fx (x) . Solution: y(x) = dx 2λ sin(λx)



x

84.

 A tan2 (λx) + B tan2 (λt) y(t) dt = f (x).

a 2 This is a special case of equation 1.9.4 = tan equation 1.5.83. with g(x) x(λx). For B2B= –A, see

2A – – d 1 tan(λt) A+B ft (t) dt . tan(λx) A+B Solution: y(x) = A + B dx a



x

85.

 A tan2 (λx) + B tan2 (µt) + C y(t) dt = f (x).

a

This is a special case of equation 1.9.6 with g(x) = A tan2 (λx) and h(t) = B tan2 (µt) + C.

x

86.

tan(λx) – tan(λt)

n

y(t) dt = f (x),

n = 1, 2, . . .

a

The right-hand side of the equation is assumed to satisfy the conditions f (a) = fx (a) = · · · = fx(n) (a) = 0.  n+1 d 1 2 cos (λx) f (x). Solution: y(x) = n λ n! cos2 (λx) dx

x

87.



tan x – tan t y(t) dt = f (x).

a

Solution: y(x) =

 d 2 2 2 cos x π cos2 x dx

a

x

cos2

f (t) dt √ . t tan x – tan t

1.5. EQUATIONS WHOSE KERNELS CONTAIN TRIGONOMETRIC FUNCTIONS



x

88. a



y(t) dt tan x – tan t

= f (x).

Solution: 1 d y(x) = π dx

x

89.

61

(tan x – tan t)λ y(t) dt = f (x),



x

a

cos2

f (t) dt √ . t tan x – tan t

0 < λ < 1.

a

Solution: y(x) =

x

90.

sin(πλ)  2 d 2 cos x πλ cos2 x dx



x a

f (t) dt . cos2 t(tan x – tan t)λ

(tanµ x – tanµ t)y(t) dt = f (x).

a

This is a special case of equation 1.9.2 with  g(x) = tanµ x.  µ+1  1 d cos xfx (x) . Solution: y(x) = µ dx sinµ–1 x

x

91.

  A tanµ x + B tanµ t y(t) dt = f (x).

a

This is a special case of equation 1.9.4 with g(x) = tanµ x. For B = –A, see equation 1.5.90. Solution: y(x) =

x

92. a



x – Aµ

– Bµ d 1 tan(λx) A+B tan(λt) A+B ft (t) dt . A + B dx a

y(t) dt [tan(λx) – tan(λt)]µ

= f (x),

0 < µ < 1.

This is a special case of equation 1.9.44 with g(x) = tan(λx) and h(x) ≡ 1. Solution: x f (t) dt λ sin(πµ) d . y(x) = π dx a cos2 (λt)[tan(λx) – tan(λt)]1–µ

x

93.

Axβ + B tanγ (λt) + C]y(t) dt = f (x).

a

This is a special case of equation 1.9.6 with g(x) = Axβ and h(t) = B tanγ (λt) + C.

x

94.

A tanγ (λx) + Btβ + C]y(t) dt = f (x).

a

This is a special case of equation 1.9.6 with g(x) = A tanγ (λx) and h(t) = Btβ + C.

x

95.

  Axλ tanµ t + Btβ tanγ x y(t) dt = f (x).

a

This is a special case of equation 1.9.15 with g1 (x) = Axλ , h1 (t) = tanµ t, g2 (x) = B tanγ x, and h2 (t) = tβ .

62

LINEAR EQUATIONS OF THE FIRST KIND WITH VARIABLE LIMIT OF INTEGRATION

1.5-4. Kernels Containing Cotangent.

x

96.

 cot(λx) – cot(λt) y(t) dt = f (x).

a

97.

This is a special case of equation 1.9.2 with g(x) = cot(λx).  1 d 2 sin (λx)fx (x) . Solution: y(x) = – λ dx x

 A cot(λx) + B cot(λt) y(t) dt = f (x). a

This is a special case of equation 1.9.4 with g(x) = cot(λx). For B = –A, see equation 1.5.96. x  A  B d 1 tan(λx) A+B tan(λt) A+B ft (t) dt . Solution: y(x) = A + B dx a

x

98.

 A cot(λx) + B cot(µt) + C y(t) dt = f (x).

a

This is a special case of equation 1.9.6 with g(x) = A cot(λx) and h(t) = B cot(µt) + C.

x

99.

 cot2 (λx) – cot2 (λt) y(t) dt = f (x).

a 2 This is a special case of equation  3 1.9.2withg(x) = cot (λx). d sin (λx)fx (x) . Solution: y(x) = – dx 2λ cos(λx)



x

100.

 A cot2 (λx) + B cot2 (λt) y(t) dt = f (x).

a

This is a special case of equation 1.9.4 = cot2 (λx). For B = –A, see with g(x)

equation 1.5.99. x 2A 2B  1 d tan(λt) A+B ft (t) dt . tan(λx) A+B Solution: y(x) = A + B dx a

x

101.

 A cot2 (λx) + B cot2 (µt) + C y(t) dt = f (x).

a

This is a special case of equation 1.9.6 with g(x) = A cot2 (λx) and h(t) = B cot2 (µt) + C.

x

102.

cot(λx) – cot(λt)

n

y(t) dt = f (x),

n = 1, 2, . . .

a

The right-hand side of the equation is assumed to satisfy the conditions f (a) = fx (a) = · · · = fx(n) (a) = 0.  n+1 d (–1)n 2 sin (λx) f (x). Solution: y(x) = n λ n! sin2 (λx) dx

x

103.

(cotµ x – cotµ t)y(t) dt = f (x).

a µ This is a special case of equation  1.9.2 with g(x)  = cot x. 1 d sinµ+1 xfx (x) . Solution: y(x) = – µ dx cosµ–1 x

1.5. EQUATIONS WHOSE KERNELS CONTAIN TRIGONOMETRIC FUNCTIONS



x

104.

63

  A cotµ x + B cotµ t y(t) dt = f (x).

a

This is a special case of equation 1.9.4 with g(x) = cotµ x. For B = –A, see equation 1.5.103. Solution:

x Aµ Bµ  d 1 A+B A+B tan x tan t ft (t) dt . y(x) = A + B dx a

x

105.

Axβ + B cotγ (λt) + C]y(t) dt = f (x).

a

This is a special case of equation 1.9.6 with g(x) = Axβ and h(t) = B cotγ (λt) + C.

x

106.

A cotγ (λx) + Btβ + C]y(t) dt = f (x).

a

This is a special case of equation 1.9.6 with g(x) = A cotγ (λx) and h(t) = Btβ + C.

x

107.

  Axλ cotµ t + Btβ cotγ x y(t) dt = f (x).

a

This is a special case of equation 1.9.15 with g1 (x) = Axλ , h1 (t) = cotµ t, g2 (x) = B cotγ x, and h2 (t) = tβ .

1.5-5. Kernels Containing Combinations of Trigonometric Functions.

x

108.



 cos[λ(x – t)] + A sin[µ(x – t)] y(t) dt = f (x).

a

Differentiating the equation with respect to x followed by eliminating the integral with the cosine yields an equation of the form 2.3.16:

x

y(x) – (λ + A2 µ)

sin[µ(x – t)] y(t) dt = fx (x) – Aµf (x).

a



x

109.

 A cos(λx) + B sin(µt) + C y(t) dt = f (x).

a

This is a special case of equation 1.9.6 with g(x) = A cos(λx) and h(t) = B sin(µt) + C.

x

110.

 A sin(λx) + B cos(µt) + C y(t) dt = f (x).

a

This is a special case of equation 1.9.6 with g(x) = A sin(λx) and h(t) = B cos(µt) + C.

x

111.

 A cos2 (λx) + B sin2 (µt) y(t) dt = f (x).

a

This is a special case of equation 1.9.6 with g(x) = A cos2 (λx) and h(t) = B sin2 (µt).

64

LINEAR EQUATIONS OF THE FIRST KIND WITH VARIABLE LIMIT OF INTEGRATION



x

sin[λ(x – t)] cos[λ(x + t)]y(t) dt = f (x),

112. a

f (a) = fx (a) = 0.

Using the trigonometric formula sin(α – β) cos(α + β) =

1 2

 sin(2α) – sin(2β) ,

α = λx,

β = λt,

we reduce the original equation to an equation of the form 1.5.44:

x

 sin(2λx) – sin(2λt) y(t) dt = 2f (x).

a

   fx (x) 1 d . Solution: y(x) = λ dx cos(2λx)

x

cos[λ(x – t)] sin[λ(x + t)]y(t) dt = f (x).

113. a

Using the trigonometric formula cos(α – β) sin(α + β) =

1 2

 sin(2α) + sin(2β) ,

α = λx,

β = λt,

we reduce the original equation to an equation of the form 1.5.45 with A = B = 1:

x

 sin(2λx) + sin(2λt) y(t) dt = 2f (x).

a

Solution with sin(2λx) > 0:   x d ft (t) dt 1 √ √ y(x) = . dx sin(2λx) a sin(2λt)

x

sin[λ(x – t)] cos[µ(x – t)]y(t) dt = f (x),

114. a

 f (a) = fx (a) = fxx (a) = 0.

Solution with µ < λ: y(x) =

 2  2  x

  d d 1 2 2  + (λ + µ) + (λ – µ) sin λ2 – µ2 (x – t) f (t) dt. 2 2 dx λ λ2 – µ2 dx a

Solution with µ > λ: y(x) =

1

 λ λ2 – µ2



d2 + (λ + µ)2 dx2



d2 + (λ – µ)2 dx2



x

  sinh µ2 – λ2 (x – t) f (t) dt.

a

Reference: A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 444).



x

115.

 A cos(λx) sin(µt) + B cos(βx) sin(γt) y(t) dt = f (x).

a

This is a special case of equation 1.9.15 with g1 (x) = A cos(λx), h1 (t) = sin(µt), g2 (x) = B cos(βx), and h2 (t) = sin(γt).

1.5. EQUATIONS WHOSE KERNELS CONTAIN TRIGONOMETRIC FUNCTIONS



x

116.

65

 A sin(λx) cos(µt) + B sin(βx) cos(γt) y(t) dt = f (x).

a

This is a special case of equation 1.9.15 with g1 (x) = A sin(λx), h1 (t) = cos(µt), g2 (x) = B sin(βx), and h2 (t) = cos(γt). x

 A cos(λx) cos(µt) + B sin(βx) sin(γt) y(t) dt = f (x). 117. a

This is a special case of equation 1.9.15 with g1 (x) = A cos(λx), h1 (t) = cos(µt), g2 (x) = B sin(βx), and h2 (t) = sin(γt). x

 118. A cosβ (λx) + B sinγ (µt) y(t) dt = f (x). a

This is a special case of equation 1.9.6 with g(x) = A cosβ (λx) and h(t) = B sinγ (µt). x

 119. A sinβ (λx) + B cosγ (µt) y(t) dt = f (x). a

This is a special case of equation 1.9.6 with g(x) = A sinβ (λx) and h(t) = B cosγ (µt). x   120. Axλ cosµ t + Btβ sinγ x y(t) dt = f (x). a

This is a special case of equation 1.9.15 with g1 (x) = Axλ , h1 (t) = cosµ t, g2 (x) = B sinγ x, and h2 (t) = tβ . x   121. Axλ sinµ t + Btβ cosγ x y(t) dt = f (x). a

This is a special case of equation 1.9.15 with g1 (x) = Axλ , h1 (t) = sinµ t, g2 (x) = B cosγ x, and h2 (t) = tβ . x   122. (x – t) sin[λ(x – t)] – λ(x – t)2 cos[λ(x – t)] y(t) dt = f (x). a



Solution:

x

y(x) =

g(t) dt, a

where



6 t d2 2 + λ (t – τ )5/2 J5/2 [λ(t – τ )] f (τ ) dτ . dt2 a

x sin[λ(x – t)] 123. – λ cos[λ(x – t)] y(t) dt = f (x). x–t a g(t) =

π 1 2λ 64λ5



Solution: y(x) =

x

124.

1 2λ4



d2 + λ2 dx2

3

x

sin[λ(x – t)]f (t) dt. a

√  √   √  sin λ x – t – λ x – t cos λ x – t y(t) dt = f (x),

a

Solution: 4 d3 y(x) = πλ3 dx3

a

x

 √  cosh λ x – t √ f (t) dt. x–t

f (a) = fx (a) = 0.

66

LINEAR EQUATIONS OF THE FIRST KIND WITH VARIABLE LIMIT OF INTEGRATION



x

125.

 A tan(λx) + B cot(µt) + C y(t) dt = f (x).

a

This is a special case of equation 1.9.6 with g(x) = A tan(λx) and h(t) = B cot(µt) + C. x

 A tan2 (λx) + B cot2 (µt) y(t) dt = f (x). 126. a

This is a special case of equation 1.9.6 with g(x) = A tan2 (λx) and h(t) = B cot2 (µt). x

 tan(λx) cot(µt) + tan(βx) cot(γt) y(t) dt = f (x). 127. a

This is a special case of equation 1.9.15 with g1 (x) = tan(λx), h1 (t) = cot(µt), g2 (x) = tan(βx), and h2 (t) = cot(γt). x

 128. cot(λx) tan(µt) + cot(βx) tan(γt) y(t) dt = f (x). a

This is a special case of equation 1.9.15 with g1 (x) = cot(λx), h1 (t) = tan(µt), g2 (x) = cot(βx), and h2 (t) = tan(γt). x

 129. tan(λx) tan(µt) + cot(βx) cot(γt) y(t) dt = f (x). a

This is a special case of equation 1.9.15 with g1 (x) = tan(λx), h1 (t) = tan(µt), g2 (x) = cot(βx), and h2 (t) = cot(γt). x

 130. A tanβ (λx) + B cotγ (µt) y(t) dt = f (x). a

This is a special case of equation 1.9.6 with g(x) = A tanβ (λx) and h(t) = B cotγ (µt). x

 A cotβ (λx) + B tanγ (µt) y(t) dt = f (x). 131. a

This is a special case of equation 1.9.6 with g(x) = A cotβ (λx) and h(t) = B tanγ (µt). x   Axλ tanµ t + Btβ cotγ x y(t) dt = f (x). 132. a

This is a special case of equation 1.9.15 with g1 (x) = Axλ , h1 (t) = tanµ t, g2 (x) = B cotγ x, and h2 (t) = tβ . x   133. Axλ cotµ t + Btβ tanγ x y(t) dt = f (x). a

This is a special case of equation 1.9.15 with g1 (x) = Axλ , h1 (t) = cotµ t, g2 (x) = B tanγ x, and h2 (t) = tβ .

1.6. Equations Whose Kernels Contain Inverse Trigonometric Functions 1.6-1. Kernels Containing Arccosine. x

 arccos(λx) – arccos(λt) y(t) dt = f (x). 1. a

This is a special case of equation 1.9.2 with g(x) = arccos(λx).  1 d √ Solution: y(x) = – 1 – λ2 x2 fx (x) . λ dx

1.6. EQUATIONS WHOSE KERNELS CONTAIN INVERSE TRIGONOMETRIC FUNCTIONS



x

2.

67

 A arccos(λx) + B arccos(λt) y(t) dt = f (x).

a

This is a special case of equation 1.9.4 with g(x) = arccos(λx). For B = –A, see equation 1.6.1. Solution:

x – A

– B  d 1 A+B A+B arccos(λx) arccos(λt) y(x) = ft (t) dt . A + B dx a

x

3.

 A arccos(λx) + B arccos(µt) + C y(t) dt = f (x).

a

This is a special case of equation 1.9.6 with g(x) = A arccos(λx) and h(t) = B arccos(µt) + C.

x

4.

arccos(λx) – arccos(λt)

n

y(t) dt = f (x),

n = 1, 2, . . .

a

The right-hand side of the equation is assumed to satisfy the conditions f (a) = fx (a) = · · · = fx(n) (a) = 0. Solution: n+1 √ (–1)n d 2 2 √ y(x) = 1–λ x f (x). dx λn n! 1 – λ2 x2

x

5.

 arccos(λt) – arccos(λx) y(t) dt = f (x).

a

6.

This is a special case of equation 1.9.40 with g(x) = 1 – arccos(λx). Solution:  2 x ϕ(t)f (t) dt 1 d 1 2 √ , ϕ(x) = √ . y(x) = ϕ(x) π ϕ(x) dx arccos(λt) – arccos(λx) 1 – λ2 x2 a x y(t) dt = f (x). √ arccos(λt) – arccos(λx) a Solution: y(x) =

x

7.

λ d π dx

a

x

ϕ(t)f (t) dt √ , arccos(λt) – arccos(λx)

arccos(λt) – arccos(λx)



y(t) dt = f (x),

1 ϕ(x) = √ . 1 – λ2 x2

0 < µ < 1.

a

Solution: 

2

x

ϕ(t)f (t) dt , µ a [arccos(λt) – arccos(λx)] 1 sin(πµ) . ϕ(x) = √ , k= 2 2 πµ 1–λ x

1 d y(x) = kϕ(x) ϕ(x) dx



x

8.

 arccosµ (λx) – arccosµ (λt) y(t) dt = f (x).

a

This is a special case of equation 1.9.2√with g(x) = arccosµ (λx).   1 d fx (x) 1 – λ2 x2 . Solution: y(x) = – λµ dx arccosµ–1 (λx)

68

LINEAR EQUATIONS OF THE FIRST KIND WITH VARIABLE LIMIT OF INTEGRATION



x

9. a

y(t) dt

µ = f (x), arccos(λt) – arccos(λx)

0 < µ < 1.

Solution: y(x) =

x

10.

λ sin(πµ) d π dx



x a

ϕ(t)f (t) dt , [arccos(λt) – arccos(λx)]1–µ

1 ϕ(x) = √ . 1 – λ2 x2

 A arccosβ (λx) + B arccosγ (µt) + C y(t) dt = f (x).

a

This is a special case of equation 1.9.6 with g(x) = A arccosβ (λx) and h(t) = B arccosγ (µt)+C.

1.6-2. Kernels Containing Arcsine.

x

11.

 arcsin(λx) – arcsin(λt) y(t) dt = f (x).

a

This is a special case of equation 1.9.2 with g(x) = arcsin(λx).  1 d √ Solution: y(x) = 1 – λ2 x2 fx (x) . λ dx

x

12.

 A arcsin(λx) + B arcsin(λt) y(t) dt = f (x).

a

This is a special case of equation 1.9.4 with g(x) = arcsin(λx). For B = –A, see equation 1.6.11. Solution: y(x) =

x

13.



x – A B sign x d arcsin(λt) – A+B ft (t) dt . arcsin(λx) A+B A + B dx a

 A arcsin(λx) + B arcsin(µt) + C y(t) dt = f (x).

a

This is a special case of equation 1.9.6 with g(x) = A arcsin(λx) and h(t) = B arcsin(µt) + C.

x

14.

arcsin(λx) – arcsin(λt)

n

y(t) dt = f (x),

n = 1, 2, . . .

a

The right-hand side of the equation is assumed to satisfy the conditions f (a) = fx (a) = · · · = fx(n) (a) = 0. Solution: n+1 √ d 1 2 2 √ 1–λ x f (x). y(x) = dx λn n! 1 – λ2 x2

x

15.

 arcsin(λx) – arcsin(λt) y(t) dt = f (x).

a

Solution: y(x) =

 2 x ϕ(t)f (t) dt 1 d 2 √ ϕ(x) , π ϕ(x) dx arcsin(λx) – arcsin(λt) a

1 ϕ(x) = √ . 1 – λ2 x2

1.6. EQUATIONS WHOSE KERNELS CONTAIN INVERSE TRIGONOMETRIC FUNCTIONS



x



16. a

y(t) dt

69

= f (x).

arcsin(λx) – arcsin(λt)

Solution: y(x) =

x

17.

λ d π dx



x

ϕ(t)f (t) dt √ , arcsin(λx) – arcsin(λt)

a

arcsin(λx) – arcsin(λt)



y(t) dt = f (x),

1 ϕ(x) = √ . 1 – λ2 x2

0 < µ < 1.

a

Solution: 

2

x

ϕ(t)f (t) dt , µ a [arcsin(λx) – arcsin(λt)] 1 sin(πµ) . ϕ(x) = √ , k= 2 2 πµ 1–λ x

1 d y(x) = kϕ(x) ϕ(x) dx



x

18.

 arcsinµ (λx) – arcsinµ (λt) y(t) dt = f (x).

a

This is a special case of equation 1.9.2 with g(x) = arcsinµ (λx).   √  1 d fx (x) 1 – λ2 x2 . Solution: y(x) = λµ dx arcsinµ–1 (λx)

x

y(t) dt

µ = f (x), arcsin(λx) – arcsin(λt)

19. a

0 < µ < 1.

Solution: y(x) =

x

20.

λ sin(πµ) d π dx

a

x

ϕ(t)f (t) dt , [arcsin(λx) – arcsin(λt)]1–µ

1 ϕ(x) = √ . 1 – λ2 x2

 A arcsinβ (λx) + B arcsinγ (µt) + C y(t) dt = f (x).

a

This is a special case of equation 1.9.6 with g(x) = A arcsinβ (λx) and h(t) = B arcsinγ (µt)+C.



x

21.

arcsin

1–

0

t y(t) dt = f (x). x

Solution:



2 1 d y(x) = √ π x dx

0

x

t d √ f (t) dt. dt x–t

Reference: A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 452).







22.

arcsin x

1–

x t

y(t) dt = f (x).

Solution: 2 d y(x) = π dx





x

√ t d √ f (t) dt. t – x dt

Reference: A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 453).

70

LINEAR EQUATIONS OF THE FIRST KIND WITH VARIABLE LIMIT OF INTEGRATION

1.6-3. Kernels Containing Arctangent.

x

23.

 arctan(λx) – arctan(λt) y(t) dt = f (x).

a

24.

This is a special case of equation 1.9.2 with g(x) = arctan(λx).  1 d Solution: y(x) = (1 + λ2 x2 ) fx (x) . λ dx x

 A arctan(λx) + B arctan(λt) y(t) dt = f (x). a

This is a special case of equation 1.9.4 with g(x) = arctan(λx). For B = –A, see equation 1.6.21. Solution:

x – A B sign x d arctan(λt) – A+B ft (t) dt . arctan(λx) A+B y(x) = A + B dx a

x

25.

 A arctan(λx) + B arctan(µt) + C y(t) dt = f (x).

a

26.

This is a special case of equation 1.9.6 with g(x) = A arctan(λx) and h(t) = B arctan(µt) + C. x

n arctan(λx) – arctan(λt) y(t) dt = f (x), n = 1, 2, . . . a

The right-hand side of the equation is assumed to satisfy the conditions f (a) = fx (a) = · · · = fx(n) (a) = 0. Solution:  n+1 1 2 2 d (1 + λ y(x) = n x ) f (x). λ n! (1 + λ2 x2 ) dx

x

27.

 arctan(λx) – arctan(λt) y(t) dt = f (x).

a

Solution:  2 x 2 ϕ(t)f (t) dt 1 d √ y(x) = ϕ(x) , π ϕ(x) dx arctan(λx) – arctan(λt) a

x

28. a



y(t) dt arctan(λx) – arctan(λt)

Solution: λ d y(x) = π dx

x

29. a



 t arctan

x–t t

a

x

ϕ(x) =

1 . 1 + λ2 x2

= f (x).

ϕ(t)f (t) dt √ , arctan(λx) – arctan(λt)

ϕ(x) =

1 . 1 + λ2 x2

 y(t) dt = f (x).

The equation can be rewritten in terms of the Gaussian hypergeometric function in the form x  x y(t) dt = f (x), where α = 12 , β = 1, γ = 32 . (x – t)γ–1 F α, β, γ; 1 – t a See 1.8.135 for the solution of this equation.

1.6. EQUATIONS WHOSE KERNELS CONTAIN INVERSE TRIGONOMETRIC FUNCTIONS



x

30.

arctan(λx) – arctan(λt)



y(t) dt = f (x),

71

0 < µ < 1.

a

Solution: 

2

x

ϕ(t)f (t) dt , [arctan(λx) – arctan(λt)]µ a 1 sin(πµ) . ϕ(x) = , k= 2 2 1+λ x πµ

y(x) = kϕ(x)



x

31.

1 d ϕ(x) dx

 arctanµ (λx) – arctanµ (λt) y(t) dt = f (x).

a

This is a special case of equation with g(x) = arctanµ (λx).  1.9.2 2 2  1 d (1 + λ x )fx (x) . Solution: y(x) = λµ dx arctanµ–1 (λx)

x

32. a

y(t) dt

µ = f (x), arctan(λx) – arctan(λt)

0 < µ < 1.

Solution: λ sin(πµ) d y(x) = π dx

x

33.

a

x

ϕ(t)f (t) dt , [arctan(λx) – arctan(λt)]1–µ

ϕ(x) =

1 . 1 + λ2 x2

 A arctanβ (λx) + B arctanγ (µt) + C y(t) dt = f (x).

a

This is a special case of equation 1.9.6 with g(x) = A arctanβ (λx) and h(t) = B arctanγ (µt)+C.

1.6-4. Kernels Containing Arccotangent.

x

34.

 arccot(λx) – arccot(λt) y(t) dt = f (x).

a

This is a special case of equation 1.9.2 with g(x) = arccot(λx).  1 d Solution: y(x) = – (1 + λ2 x2 ) fx (x) . λ dx

x

35.

 A arccot(λx) + B arccot(λt) y(t) dt = f (x).

a

This is a special case of equation 1.9.4 with g(x) = arccot(λx). For B = –A, see equation 1.6.34. Solution:

x – A

– B  1 d A+B A+B arccot(λx) arccot(λt) ft (t) dt . y(x) = A + B dx a

x

36.

 A arccot(λx) + B arccot(µt) + C y(t) dt = f (x).

a

This is a special case of equation 1.9.6 with g(x) = A arccot(λx) and h(t) = B arccot(µt) + C.

72

LINEAR EQUATIONS OF THE FIRST KIND WITH VARIABLE LIMIT OF INTEGRATION



x

37.

arccot(λx) – arccot(λt)

n

y(t) dt = f (x),

n = 1, 2, . . .

a

The right-hand side of the equation is assumed to satisfy the conditions f (a) = fx (a) = · · · = fx(n) (a) = 0. Solution:  n+1 (–1)n 2 2 d y(x) = n (1 + λ x ) f (x). λ n! (1 + λ2 x2 ) dx

x

38.

 arccot(λt) – arccot(λx) y(t) dt = f (x).

a

Solution:  2 x ϕ(t)f (t) dt 1 d 2 √ , y(x) = ϕ(x) π ϕ(x) dx arccot(λt) – arccot(λx) a

x

39. a



y(t) dt arccot(λt) – arccot(λx)

ϕ(x) =

1 . 1 + λ2 x2

= f (x).

Solution: y(x) =

x

40.

λ d π dx



x

ϕ(t)f (t) dt √ , arccot(λt) – arccot(λx)

a

arccot(λt) – arccot(λx)



y(t) dt = f (x),

ϕ(x) =

1 . 1 + λ2 x2

0 < µ < 1.

a

Solution: 

2

x

ϕ(t)f (t) dt , µ a [arccot(λt) – arccot(λx)] 1 sin(πµ) . ϕ(x) = , k= 1 + λ2 x2 πµ

1 d y(x) = kϕ(x) ϕ(x) dx



x

41.

 arccotµ (λx) – arccotµ (λt) y(t) dt = f (x).

a µ This is a special case of equation  1.9.2 with g(x) = arccot (λx). 1 d (1 + λ2 x2 )fx (x) . Solution: y(x) = – λµ dx arccotµ–1 (λx)



x

42. a

y(t) dt

µ = f (x), arccot(λt) – arccot(λx)

0 < µ < 1.

Solution: y(x) =

x

43.

λ sin(πµ) d π dx



x a

ϕ(t)f (t) dt , [arccot(λt) – arccot(λx)]1–µ

ϕ(x) =

1 . 1 + λ2 x2

 A arccotβ (λx) + B arccotγ (µt) + C y(t) dt = f (x).

a

This is a special case of equation 1.9.6 with g(x) = A arccotβ (λx) and h(t) = B arccotγ (µt)+C.

1.7. EQUATIONS WHOSE KERNELS CONTAIN COMBINATIONS OF ELEMENTARY FUNCTIONS

73

1.7. Equations Whose Kernels Contain Combinations of Elementary Functions 1.7-1. Kernels Containing Exponential and Hyperbolic Functions.

x

1. a

  eµ(x–t) A1 cosh[λ1 (x – t)] + A2 cosh[λ2 (x – t)] y(t) dt = f (x).

The substitution w(x) = e–µx y(x) leads to an equation of the form 1.3.8: x   A1 cosh[λ1 (x – t)] + A2 cosh[λ2 (x – t)] w(t) dt = e–µx f (x). a



x

2.

eµ(x–t) cosh2 [λ(x – t)]y(t) dt = f (x).

a

Solution: 2λ2 k

y(x) = ϕ(x) –

x

3.



x

eµ(x–t) sinh[k(x – t)]ϕ(x) dt,

√ k = λ 2,

ϕ(x) = fx (x) – µf (x).

a

eµ(x–t) cosh3 [λ(x – t)]y(t) dt = f (x).

a

The substitution w(x) = e–µx y(x) leads to an equation of the form 1.3.15: x cosh3 [λ(x – t)]w(t) dt = e–µx f (x). a



x

4.

eµ(x–t) cosh4 [λ(x – t)]y(t) dt = f (x).

a

The substitution w(x) = e–µx y(x) leads to an equation of the form 1.3.19: x cosh4 [λ(x – t)]w(t) dt = e–µx f (x). a



x

5.

n eµ(x–t) cosh(λx) – cosh(λt) y(t) dt = f (x),

n = 1, 2, . . .

a

Solution: y(x) =

x

6.

eµ(x–t)



n+1  d 1 1 µx e sinh(λx) Fµ (x), λn n! sinh(λx) dx

cosh x – cosh t y(t) dt = f (x),

Fµ (x) = e–µx f (x).

f (a) = 0.

a

Solution: y(x) =

7.

 1 d 2 x e–µt sinh t f (t) dt 2 µx √ e sinh x . π sinh x dx cosh x – cosh t a

x

eµ(x–t) y(t) dt = f (x). √ cosh x – cosh t a Solution: y(x) =

1 µx d e π dx

a

x

e–µt sinh t f (t) dt √ . cosh x – cosh t

74

LINEAR EQUATIONS OF THE FIRST KIND WITH VARIABLE LIMIT OF INTEGRATION



x

8.

eµ(x–t) (cosh x – cosh t)λ y(t) dt = f (x),

0 < λ < 1.

a

The substitution w(x) = e–µx y(x) leads to an equation of the form 1.3.23: x (cosh x – cosh t)λ w(t) dt = e–µx f (x). a



x

9.

 Aeµ(x–t) + B coshλ x y(t) dt = f (x).

a

10.

This is a special case of equation 1.9.15 with g1 (x) = Aeµx , h1 (t) = e–µt , g2 (x) = B coshλ x, and h2 (t) = 1. x

µ(x–t)  Ae + B coshλ t y(t) dt = f (x).

11.

This is a special case of equation 1.9.15 with g1 (x) = Aeµx , h1 (t) = e–µt , g2 (x) = B, and h2 (t) = coshλ t. x eµ(x–t) (coshλ x – coshλ t)y(t) dt = f (x).

a

a

The substitution w(x) = e–µx y(x) leads to an equation of the form 1.3.24: x (coshλ x – coshλ t)w(t) dt = e–µx f (x). a



x

12.

  eµ(x–t) A coshλ x + B coshλ t y(t) dt = f (x).

a

The substitution w(x) = e–µx y(x) leads to an equation of the form 1.3.25: x   A coshλ x + B coshλ t w(t) dt = e–µx f (x). a

13.

x

eµ(x–t) y(t) dt

(cosh x – cosh t)λ Solution:

= f (x),

0 < λ < 1.

a

y(x) =

x

14. a

sin(πλ) µx d e π dx

a

x

e–µt sinh t f (t) dt . (cosh x – cosh t)1–λ

  eµ(x–t) A1 sinh[λ1 (x – t)] + A2 sinh[λ2 (x – t)] y(t) dt = f (x).

The substitution w(x) = e–µx y(x) leads to an equation of the form 1.3.49: x   A1 sinh[λ1 (x – t)] + A2 sinh[λ2 (x – t)] w(t) dt = e–µx f (x). a



x

15.

eµ(x–t) sinh2 [λ(x – t)]y(t) dt = f (x).

a

The substitution w(x) = e–µx y(x) leads to an equation of the form 1.3.51: x sinh2 [λ(x – t)]w(t) dt = e–µx f (x). a

1.7. EQUATIONS WHOSE KERNELS CONTAIN COMBINATIONS OF ELEMENTARY FUNCTIONS



x

16.

eµ(x–t) sinh3 [λ(x – t)]y(t) dt = f (x).

a

The substitution w(x) = e–µx y(x) leads to an equation of the form 1.3.57: x sinh3 [λ(x – t)]w(t) dt = e–µx f (x). a



x

17.

eµ(x–t) sinhn [λ(x – t)]y(t) dt = f (x),

n = 2, 3, . . .

a

The substitution w(x) = e–µx y(x) leads to an equation of the form 1.3.62: x sinhn [λ(x – t)]w(t) dt = e–µx f (x). a



x

18.

 √  eµ(x–t) sinh k x – t y(t) dt = f (x).

a

Solution: 2 µx d2 y(x) = e πk dx2

x

19.

eµ(x–t)





x

a

 √  e–µt cos k x – t √ f (t) dt. x–t

sinh x – sinh t y(t) dt = f (x).

a

Solution: y(x) =

20.

 1 d 2 x e–µt cosh t f (t) dt 2 µx √ e cosh x . π cosh x dx sinh x – sinh t a

x

eµ(x–t) y(t) dt = f (x). √ sinh x – sinh t a Solution: y(x) =

x

21.

1 µx d e π dx



x a

eµ(x–t) (sinh x – sinh t)λ y(t) dt = f (x),

e–µt cosh t f (t) dt √ . sinh x – sinh t 0 < λ < 1.

a

The substitution w(x) = e–µx y(x) leads to an equation of the form 1.3.67: x (sinh x – sinh t)λ w(t) dt = e–µx f (x). a



x

22.

eµ(x–t) (sinhλ x – sinhλ t)y(t) dt = f (x).

a

The substitution w(x) = e–µx y(x) leads to an equation of the form 1.3.68: x (sinhλ x – sinhλ t)w(t) dt = e–µx f (x). a



x

23.

  eµ(x–t) A sinhλ x + B sinhλ t y(t) dt = f (x).

a

The substitution w(x) = e–µx y(x) leads to an equation of the form 1.3.69: x   A sinhλ x + B sinhλ t w(t) dt = e–µx f (x). a

75

76

LINEAR EQUATIONS OF THE FIRST KIND WITH VARIABLE LIMIT OF INTEGRATION



x

24.

 Aeµ(x–t) + B sinhλ x y(t) dt = f (x).

a

This is a special case of equation 1.9.15 with g1 (x) = Aeµx , h1 (t) = e–µt , g2 (x) = B sinhλ x, and h2 (t) = 1.

x

25.

 Aeµ(x–t) + B sinhλ t y(t) dt = f (x).

a

This is a special case of equation 1.9.15 with g1 (x) = Aeµx , h1 (t) = e–µt , g2 (x) = B, and h2 (t) = sinhλ t.

x

26. a

eµ(x–t) y(t) dt

= f (x),

(sinh x – sinh t)λ

0 < λ < 1.

Solution: sin(πλ) µx d e y(x) = π dx

x

27.



x a

e–µt cosh t f (t) dt . (sinh x – sinh t)1–λ

  eµ(x–t) A tanhλ x + B tanhλ t y(t) dt = f (x).

a

The substitution w(x) = e–µx y(x) leads to an equation of the form 1.3.89:

x

 A tanhλ x + B tanhλ t w(t) dt = e–µx f (x).

a



x

28.

  eµ(x–t) A tanhλ x + B tanhβ t + C y(t) dt = f (x).

a

The substitution w(x) = e–µx y(x) leads to an equation of the form 1.9.6 with g(x) = A tanhλ x, h(t) = B tanhβ t + C:

x

 A tanhλ x + B tanhβ t + C w(t) dt = e–µx f (x).

a



x

29.

 Aeµ(x–t) + B tanhλ x y(t) dt = f (x).

a

This is a special case of equation 1.9.15 with g1 (x) = Aeµx , h1 (t) = e–µt , g2 (x) = B tanhλ x, and h2 (t) = 1.

x

30.

 Aeµ(x–t) + B tanhλ t y(t) dt = f (x).

a

This is a special case of equation 1.9.15 with g1 (x) = Aeµx , h1 (t) = e–µt , g2 (x) = B, and h2 (t) = tanhλ t.

x

31.

  eµ(x–t) A cothλ x + B cothλ t y(t) dt = f (x).

a

The substitution w(x) = e–µx y(x) leads to an equation of the form 1.3.102: a

x

 A cothλ x + B cothλ t w(t) dt = e–µx f (x).

1.7. EQUATIONS WHOSE KERNELS CONTAIN COMBINATIONS OF ELEMENTARY FUNCTIONS



x

32.

77

  eµ(x–t) A cothλ x + B cothβ t + C y(t) dt = f (x).

a

The substitution w(x) = e–µx y(x) leads to an equation of the form 1.9.6 with g(x) = A cothλ x, h(t) = B cothβ t + C:

x

 A cothλ x + B cothβ t + C w(t) dt = e–µx f (x).

a



x

33.

 Aeµ(x–t) + B cothλ x y(t) dt = f (x).

a

This is a special case of equation 1.9.15 with g1 (x) = Aeµx , h1 (t) = e–µt , g2 (x) = B cothλ x, and h2 (t) = 1.

x

34.

 Aeµ(x–t) + B cothλ t y(t) dt = f (x).

a

This is a special case of equation 1.9.15 with g1 (x) = Aeµx , h1 (t) = e–µt , g2 (x) = B, and h2 (t) = cothλ t. 1.7-2. Kernels Containing Exponential and Logarithmic Functions.

x

35.

eλ(x–t) (ln x – ln t)y(t) dt = f (x).

a

Solution:

x

36.

 y(x) = eλx xϕxx (x) + ϕx (x) ,

ϕ(x) = e–λx f (x).

eλ(x–t) ln(x – t)y(t) dt = f (x).

0

The substitution w(x) = e–λx y(x) leads to an equation of the form 1.4.2:

x

ln(x – t)w(t) dt = e–λx f (x). 0



x

37.

eλ(x–t) (A ln x + B ln t)y(t) dt = f (x).

a

The substitution w(x) = e–λx y(x) leads to an equation of the form 1.4.4:

x

(A ln x + B ln t)w(t) dt = e–λx f (x). a



x

38.

 eµ(x–t) A ln2 (λx) + B ln2 (λt) y(t) dt = f (x).

a

The substitution w(x) = e–λx y(x) leads to an equation of the form 1.4.7: a

x

 A ln2 (λx) + B ln2 (λt) w(t) dt = e–λx f (x).

78

LINEAR EQUATIONS OF THE FIRST KIND WITH VARIABLE LIMIT OF INTEGRATION



x

39.

n eλ(x–t) ln(x/t) y(t) dt = f (x),

n = 1, 2, . . .

a

Solution:



x

40.

eλ(x–t)

n+1  d 1 λx e x y(x) = Fλ (x), n! x dx

Fλ (x) = e–λx f (x).

 ln(x/t) y(t) dt = f (x).

a

Solution:



x

41. a

 2 x –λt d 2eλx e f (t) dt  x y(x) = . πx dx a t ln(x/t)

eλ(x–t)  y(t) dt = f (x). ln(x/t)

Solution: d 1 y(x) = eλx π dx

x

42.



x

a

e–λt f (t) dt  . t ln(x/t)

 Aeµ(x–t) + B lnν (λx) y(t) dt = f (x).

a

43.

This is a special case of equation 1.9.15 with g1 (x) = Aeµx , h1 (t) = e–µt , g2 (x) = B lnν (λx), and h2 (t) = 1. x

µ(x–t)  Ae + B lnν (λt) y(t) dt = f (x). a

44.

This is a special case of equation 1.9.15 with g1 (x) = Aeµx , h1 (t) = e–µt , g2 (x) = B, and h2 (t) = lnν (λt). x eµ(x–t) [ln(x/t)]λ y(t) dt = f (x), 0 < λ < 1. a

The substitution w(x) = e–µx y(x) leads to an equation of the form 1.4.16: x [ln(x/t)]λ w(t) dt = e–µx f (x). a



x

45. a

eµ(x–t) [ln(x/t)]λ

Solution:

y(t) dt = f (x),

0 < λ < 1.

sin(πλ) µx d e y(x) = π dx

a

x

f (t) dt . teµt [ln(x/t)]1–λ

1.7-3. Kernels Containing Exponential and Trigonometric Functions.

x

46.

eµ(x–t) cos[λ(x – t)]y(t) dt = f (x).

a

Solution: y(x) = fx (x) – µf (x) + λ2



x

eµ(x–t) f (t) dt. a

1.7. EQUATIONS WHOSE KERNELS CONTAIN COMBINATIONS OF ELEMENTARY FUNCTIONS



x

47. a

79

  eµ(x–t) A1 cos[λ1 (x – t)] + A2 cos[λ2 (x – t)] y(t) dt = f (x).

The substitution w(x) = e–µx y(x) leads to an equation of the form 1.5.8: x   A1 cos[λ1 (x – t)] + A2 cos[λ2 (x – t)] w(t) dt = e–µx f (x). a



x

48.

eµ(x–t) cos2 [λ(x – t)]y(t) dt = f (x).

a

The substitution w(x) = e–µx y(x) leads to an equation of the form 1.5.9. Solution: √ 2λ2 x µ(x–t) y(x) = ϕ(x) + e sin[k(x – t)]ϕ(t) dt, k = λ 2, ϕ(x) = fx (x) – µf (x). k a

x

49.

eµ(x–t) cos3 [λ(x – t)]y(t) dt = f (x).

a

The substitution w(x) = e–µx y(x) leads to an equation of the form 1.5.16: x cos3 [λ(x – t)]w(t) dt = e–µx f (x). a



x

50.

eµ(x–t) cos4 [λ(x – t)]y(t) dt = f (x).

a

The substitution w(x) = e–µx y(x) leads to an equation of the form 1.5.20: x cos4 [λ(x – t)]w(t) dt = e–µx f (x). a



x

51.

n eµ(x–t) cos(λx) – cos(λt) y(t) dt = f (x),

n = 1, 2, . . .

a

The right-hand side of the equation is assumed to satisfy the conditions f (a) = fx (a) = · · · = fx(n) (a) = 0. Solution: n+1  d (–1)n 1 y(x) = n eµx sin(λx) Fµ (x), Fµ (x) = e–µx f (x). λ n! sin(λx) dx

x

52.

eµ(x–t)



cos t – cos x y(t) dt = f (x).

a

Solution: y(x) =

53.

 1 d 2 x e–µt sin t f (t) dt 2 µx √ e sin x . π sin x dx cos t – cos x a

x

eµ(x–t) y(t) dt = f (x). √ cos t – cos x a Solution: y(x) =

1 µx d e π dx

a

x

e–µt sin t f (t) dt √ . cos t – cos x

80

LINEAR EQUATIONS OF THE FIRST KIND WITH VARIABLE LIMIT OF INTEGRATION



x

54.

eµ(x–t) (cos t – cos x)λ y(t) dt = f (x),

0 < λ < 1.

a

Solution:  1 d 2 x e–µt sin t f (t) dt , y(x) = keµx sin x λ sin x dx a (cos t – cos x)

x

55.

k=

sin(πλ) . πλ

eµ(x–t) (cosλ x – cosλ t)y(t) dt = f (x).

a

The substitution w(x) = e–µx y(x) leads to an equation of the form 1.5.25:

x

(cosλ x – cosλ t)w(t) dt = e–µx f (x). a



x

56.

  eµ(x–t) A cosλ x + B cosλ t y(t) dt = f (x).

a

The substitution w(x) = e–µx y(x) leads to an equation of the form 1.5.26:

x

 A cosλ x + B cosλ t w(t) dt = e–µx f (x).

a



x

57. a

eµ(x–t) y(t) dt (cos t – cos x)λ

= f (x),

0 < λ < 1.

The substitution w(x) = e–µx y(x) leads to an equation of the form 1.5.27: a



x

58.

x

w(t) dt = e–µx f (x). (cos t – cos x)λ

 Aeµ(x–t) + B cosν (λx) y(t) dt = f (x).

a

This is a special case of equation 1.9.15 with g1 (x) = Aeµx , h1 (t) = e–µt , g2 (x) = B cosν (λx), and h2 (t) = 1.

x

59.

 Aeµ(x–t) + B cosν (λt) y(t) dt = f (x).

a

This is a special case of equation 1.9.15 with g1 (x) = Aeµx , h1 (t) = e–µt , g2 (x) = B, and h2 (t) = cosν (λt).

x

60.

eµ(x–t) sin[λ(x – t)]y(t) dt = f (x),

a

Solution: y(x) =

x

61. a

1 λ

f (a) = fx (a) = 0.

  fxx (x) – 2µfx (x) + (λ2 + µ2 )f (x) .

  eµ(x–t) A1 sin[λ1 (x – t)] + A2 sin[λ2 (x – t)] y(t) dt = f (x).

The substitution w(x) = e–µx y(x) leads to an equation of the form 1.5.48: a

x

 A1 sin[λ1 (x – t)] + A2 sin[λ2 (x – t)] w(t) dt = e–µx f (x).

1.7. EQUATIONS WHOSE KERNELS CONTAIN COMBINATIONS OF ELEMENTARY FUNCTIONS



x

62.

eµ(x–t) sin2 [λ(x – t)]y(t) dt = f (x).

a

The substitution w(x) = e–µx y(x) leads to an equation of the form 1.5.50: x sin2 [λ(x – t)]w(t) dt = e–µx f (x). a



x

63.

eµ(x–t) sin3 [λ(x – t)]y(t) dt = f (x).

a

The substitution w(x) = e–µx y(x) leads to an equation of the form 1.5.57: x sin3 [λ(x – t)]w(t) dt = e–µx f (x). a



x

64.

eµ(x–t) sinn [λ(x – t)]y(t) dt = f (x),

n = 2, 3, . . .

a

The substitution w(x) = e–µx y(x) leads to an equation of the form 1.5.62: x sinn [λ(x – t)]w(t) dt = e–µx f (x). a



x

65.

 √  eµ(x–t) sin k x – t y(t) dt = f (x).

a

Solution: 2 µx d2 y(x) = e πk dx2

x

66.

eµ(x–t)





 √  e–µt cosh k x – t √ f (t) dt. x–t

x

a

sin x – sin t y(t) dt = f (x).

a

Solution:



67.

 1 d 2 x e–µt cos t f (t) dt 2 µx √ . y(x) = e cos x π cos x dx sin x – sin t a

x

eµ(x–t) y(t) dt = f (x). √ sin x – sin t a Solution: y(x) =

x

68.

1 µx d e π dx



eµ(x–t) (sin x – sin t)λ y(t) dt = f (x),

a

x

e–µt cos t f (t) dt √ . sin x – sin t 0 < λ < 1.

a

Solution:



x

69.

 1 d 2 x e–µt cos t f (t) dt , y(x) = keµx cos x cos x dx (sin x – sin t)λ a

k=

eµ(x–t) (sinλ x – sinλ t)y(t) dt = f (x).

a

The substitution w(x) = e–µx y(x) leads to an equation of the form 1.5.72: x (sinλ x – sinλ t)w(t) dt = e–µx f (x). a

sin(πλ) . πλ

81

82

LINEAR EQUATIONS OF THE FIRST KIND WITH VARIABLE LIMIT OF INTEGRATION



x

70.

  eµ(x–t) A sinλ x + B sinλ t y(t) dt = f (x).

a

The substitution w(x) = e–µx y(x) leads to an equation of the form 1.9.4 with g(x) = sinλ x:

x

 A sinλ x + B sinλ t w(t) dt = e–µx f (x).

a



x

71. a

eµ(x–t) y(t) dt (sin x – sin t)λ

= f (x),

0 < λ < 1.

The substitution w(x) = e–µx y(x) leads to an equation of the form 1.5.74: a



x

72.

x

w(t) dt = e–µx f (x). (sin x – sin t)λ

 Aeµ(x–t) + B sinν (λx) y(t) dt = f (x).

a

This is a special case of equation 1.9.15 with g1 (x) = Aeµx , h1 (t) = e–µt , g2 (x) = B sinν (λx), and h2 (t) = 1.

x

73.

 Aeµ(x–t) + B sinν (λt) y(t) dt = f (x).

a

This is a special case of equation 1.9.15 with g1 (x) = Aeµx , h1 (t) = e–µt , g2 (x) = B, and h2 (t) = sinν (λt).

x

74.

  eµ(x–t) A tanλ x + B tanλ t y(t) dt = f (x).

a

The substitution w(x) = e–µx y(x) leads to an equation of the form 1.5.91:

x

 A tanλ x + B tanλ t w(t) dt = e–µx f (x).

a



x

75.

  eµ(x–t) A tanλ x + B tanβ t + C y(t) dt = f (x).

a

The substitution w(x) = e–µx y(x) leads to an equation of the form 1.9.6:

x

 A tanλ x + B tanβ t + C w(t) dt = e–µx f (x).

a



x

76.

 Aeµ(x–t) + B tanν (λx) y(t) dt = f (x).

a

This is a special case of equation 1.9.15 with g1 (x) = Aeµx , h1 (t) = e–µt , g2 (x) = B tanν (λx), and h2 (t) = 1.

x

77.

 Aeµ(x–t) + B tanν (λt) y(t) dt = f (x).

a

This is a special case of equation 1.9.15 with g1 (x) = Aeµx , h1 (t) = e–µt , g2 (x) = B, and h2 (t) = tanν (λt).

1.7. EQUATIONS WHOSE KERNELS CONTAIN COMBINATIONS OF ELEMENTARY FUNCTIONS



x

78.

83

  eµ(x–t) A cotλ x + B cotλ t y(t) dt = f (x).

a

The substitution w(x) = e–µx y(x) leads to an equation of the form 1.5.104:

x

 A cotλ x + B cotλ t w(t) dt = e–µx f (x).

a



x

79.

  eµ(x–t) A cotλ x + B cotβ t + C y(t) dt = f (x).

a

The substitution w(x) = e–µx y(x) leads to an equation of the form 1.9.6:

x

 A cotλ x + B cotβ t + C w(t) dt = e–µx f (x).

a



x

80.

 Aeµ(x–t) + B cotν (λx) y(t) dt = f (x).

a

This is a special case of equation 1.9.15 with g1 (x) = Aeµx , h1 (t) = e–µt , g2 (x) = B cotν (λx), and h2 (t) = 1.

x

81.

 Aeµ(x–t) + B cotν (λt) y(t) dt = f (x).

a

This is a special case of equation 1.9.15 with g1 (x) = Aeµx , h1 (t) = e–µt , g2 (x) = B, and h2 (t) = cotν (λt). 1.7-4. Kernels Containing Hyperbolic and Logarithmic Functions.

x

82.

 A coshβ (λx) + B lnγ (µt) + C y(t) dt = f (x).

a

This is a special case of equation 1.9.6 with g(x) = A coshβ (λx) and h(t) = B lnγ (µt) + C.

x

83.

 A coshβ (λt) + B lnγ (µx) + C y(t) dt = f (x).

a

This is a special case of equation 1.9.6 with g(x) = B lnγ (µx) + C and h(t) = A coshβ (λt).

x

84.

 A sinhβ (λx) + B lnγ (µt) + C y(t) dt = f (x).

a

This is a special case of equation 1.9.6 with g(x) = A sinhβ (λx) and h(t) = B lnγ (µt) + C.

x

85.

 A sinhβ (λt) + B lnγ (µx) + C y(t) dt = f (x).

a

This is a special case of equation 1.9.6 with g(x) = B lnγ (µx) and h(t) = A sinhβ (λt) + C.

x

86.

 A tanhβ (λx) + B lnγ (µt) + C y(t) dt = f (x).

a

This is a special case of equation 1.9.6 with g(x) = A tanhβ (λx) and h(t) = B lnγ (µt) + C.

84

LINEAR EQUATIONS OF THE FIRST KIND WITH VARIABLE LIMIT OF INTEGRATION



x

87.

 A tanhβ (λt) + B lnγ (µx) + C y(t) dt = f (x).

a

This is a special case of equation 1.9.6 with g(x) = B lnγ (µx) and h(t) = A tanhβ (λt) + C.

x

88.

 A cothβ (λx) + B lnγ (µt) + C y(t) dt = f (x).

a

This is a special case of equation 1.9.6 with g(x) = A cothβ (λx) and h(t) = B lnγ (µt) + C.

x

89.

 A cothβ (λt) + B lnγ (µx) + C y(t) dt = f (x).

a

This is a special case of equation 1.9.6 with g(x) = B lnγ (µx) and h(t) = A cothβ (λt) + C. 1.7-5. Kernels Containing Hyperbolic and Trigonometric Functions.

x

90.

 A coshβ (λx) + B cosγ (µt) + C y(t) dt = f (x).

a

This is a special case of equation 1.9.6 with g(x) = A coshβ (λx) and h(t) = B cosγ (µt) + C.

x

91.

 A coshβ (λt) + B sinγ (µx) + C y(t) dt = f (x).

a

This is a special case of equation 1.9.6 with g(x) = B sinγ (µx) + C and h(t) = A coshβ (λt).

x

92.

 A coshβ (λx) + B tanγ (µt) + C y(t) dt = f (x).

a

This is a special case of equation 1.9.6 with g(x) = A coshβ (λx) and h(t) = B tanγ (µt) + C.

x

93.

 A sinhβ (λx) + B cosγ (µt) + C y(t) dt = f (x).

a

This is a special case of equation 1.9.6 with g(x) = A sinhβ (λx) and h(t) = B cosγ (µt) + C.

x

94.

 A sinhβ (λt) + B sinγ (µx) + C y(t) dt = f (x).

a

This is a special case of equation 1.9.6 with g(x) = B sinγ (µx) and h(t) = A sinhβ (λt) + C.

x

95.

 A sinhβ (λx) + B tanγ (µt) + C y(t) dt = f (x).

a

This is a special case of equation 1.9.6 with g(x) = A sinhβ (λx) and h(t) = B tanγ (µt) + C.

x

96.

 A tanhβ (λx) + B cosγ (µt) + C y(t) dt = f (x).

a

This is a special case of equation 1.9.6 with g(x) = A tanhβ (λx) and h(t) = B cosγ (µt) + C.

x

97.

 A tanhβ (λx) + B sinγ (µt) + C y(t) dt = f (x).

a

This is a special case of equation 1.9.6 with g(x) = A tanhβ (λx) and h(t) = B sinγ (µt) + C.

1.7. EQUATIONS WHOSE KERNELS CONTAIN COMBINATIONS OF ELEMENTARY FUNCTIONS



x

sinh[λ(x – t)] – sin[λ(x – t)]y(t) dt = f (x).

98. a

  It is assumed that f (a) = fx (a) = fxx (a) = fxxx (a) = 0. Solution:  4  1 d 4 y(x) = – λ f (x). 2λ3 dx4

Reference: A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 449).



x

 f (a) = fx (a) = fxx (a) = 0.

sinh[λ(x – t)] sin[λ(x – t)]y(t) dt = f (x),

99. a

Solution: 1 y(x) = 2λ2



d4 + 4λ4 dx4



x

f (t) dt. a

Reference: A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 449).



x

sinh[λ(x – t)] cos[λ(x – t)]y(t) dt = f (x).

100. a

  It is assumed that f (a) = fx (a) = fxx (a) = fxxx (a) = 0. Solution:  4  x √ 1 d 4 y(x) = √ + 4λ sinh[ 2λ(x – t)]f (t) dt. 4 2 2λ dx a

Reference: A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 449).



x

cosh[λ(x – t)] sin[λ(x – t)]y(t) dt = f (x).

101. a

  It is assumed that f (a) = fx (a) = fxx (a) = fxxx (a) = 0. Solution:  4  x √ d 1 4 + 4λ sin[ 2λ(x – t)]f (t) dt. y(x) = √ 4 2λ2 dx a

Reference: A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 450).



x

cosh[λ(x – t)] cos[λ(x – t)]y(t) dt = f (x),

102.

f (a) = 0.

a

Solution: y(x) =

1 2



d4 + 4λ4 dx4



x

(x – t)2 f (t) dt. a

Reference: A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 450).

1.7-6. Kernels Containing Logarithmic and Trigonometric Functions.

x

103.

 A cosβ (λx) + B lnγ (µt) + C y(t) dt = f (x).

a

This is a special case of equation 1.9.6 with g(x) = A cosβ (λx) and h(t) = B lnγ (µt) + C.

85

86

LINEAR EQUATIONS OF THE FIRST KIND WITH VARIABLE LIMIT OF INTEGRATION



x

104.

 A cosβ (λt) + B lnγ (µx) + C y(t) dt = f (x).

a

This is a special case of equation 1.9.6 with g(x) = B lnγ (µx) + C and h(t) = A cosβ (λt).

x

105.

 A sinβ (λx) + B lnγ (µt) + C y(t) dt = f (x).

a

This is a special case of equation 1.9.6 with g(x) = A sinβ (λx) and h(t) = B lnγ (µt) + C.

x

106.

 A sinβ (λt) + B lnγ (µx) + C y(t) dt = f (x).

a

This is a special case of equation 1.9.6 with g(x) = B lnγ (µx) and h(t) = A sinβ (λt) + C.

1.8. Equations Whose Kernels Contain Special Functions∗ 1.8-1. Kernels Containing Error Function or Exponential Integral.

x

1.

√ erf(λ x – t)y(t) dt = f (x),

0

f (0) = fx (0) = 0.

Here erf z is the error function (see Supplement 11.2-1). Solution: x λ2 t 2 d e 1 √ ft (t) dt. y(x) = √ e–λ x πλ dx 0 x–t References: H. M. Srivastava and R. G. Buschman (1977), A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 458).





2.

√ erf(λ t – x)y(t) dt = f (x).

x

Solution:

2 d 1 y(x) = √ eλ x πλ dx





x

eλ2 t

1 √ ft (t) dt. t–x

References: H. M. Srivastava and R. G. Buschman (1977), A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 459).



x

Ei(λ(t – x))y(t) dt = f (x),

3. 0

f (0) = fx (0) = 0.

Here Ei(z) is the exponential integral (see Supplement 11.2-2). Solution:   2 1 x λ(t–x) d d f (t) dt, y(x) = – e ν(λ(x – t)) + λ λ 0 dt2 dt ∞ ξ z dξ where ν(z) = . Γ(ξ + 1) 0 References: H. M. Srivastava and R. G. Buschman (1977), A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 455). * For notation and properties of special functions, see Supplement 11.

87

1.8. EQUATIONS WHOSE KERNELS CONTAIN SPECIAL FUNCTIONS

1.8-2. Kernels Containing Sine and Cosine Integrals.

x

f (0) = fx (0) = 0.

[sin(x – t) Si(x – t) – cos(x – t) ci(x – t)]y(t) dt = f (x),

4. 0

Here Si(z) is the sine integral and ci(z) is the cosine integral (see Supplements 11.3-1 and 11.3-2). Solution:  2  x d ν(x – t) + 1 f (t) dt, y(x) = dt2 0 ∞ ξ z dξ . where ν(z) = Γ(ξ + 1) 0 References: H. M. Srivastava and R. G. Buschman (1977), A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 458).



x

[cos(x – t) Si(x – t) – sin(x – t) ci(x – t)]y(t) dt = f (x),

5. 0

Solution:



x

y(x) = 0

where ν(z) = 0



 f (0) = fx (0) = fxx (0) = 0.

 d3 d f (t) dt, ν(x – t) + dt3 dt 

z ξ dξ . Γ(ξ + 1)

Reference: A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 458).

1.8-3. Kernels Containing Fresnel Integrals.

x

S(x – t)y(t) dt = f (x),

6. 0

  f (0) = fx (0) = fxx (0) = fxxx (0) = 0.

Here S(z) is the Fresnel sine integral (see Supplement 11.3-3). Solution:   4 x d d2 y(t) dt. C(x – t) + y(x) = 4 dt4 dt2 0 References: H. M. Srivastava and R. G. Buschman (1977), A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 460).



x

C(x – t)y(t) dt = f (x),

7. 0

  f (0) = fx (0) = fxx (0) = fxxx (0) = 0.

Here C(z) is the Fresnel cosine integral (see Supplement 11.3-3). Solution:   4 x d2 d y(t) dt. y(x) = 4 S(x – t) + dt4 dt2 0 References: H. M. Srivastava and R. G. Buschman (1977), A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 460).

88

LINEAR EQUATIONS OF THE FIRST KIND WITH VARIABLE LIMIT OF INTEGRATION

1.8-4. Kernels Containing Incomplete Gamma Functions. x 8. γ(ν, λ(x – t))y(t) dt = f (x). 0

Here γ(ν, z) is the incomplete gamma function (see Supplement 11.5-1). 1◦ . Let Re ν > 0, m = [Re ν] + 1, where [Re ν] denotes the integer part of the number Re ν, and f (0) = fx (0) = · · · = fx(m) (0) = 0. Then the solution is  m x eλt λ–ν d –λx e y(x) = f  (t) dt. ν–m+1 t Γ(ν)Γ(m – ν) dx (x – t) 0 2◦ . Let ν = n/2, where n is a positive integer, and f (0) = f  (0) = · · · = f (n+1) (0) = 0. Then the solution is n x   d2  d λ–n n y(x) = 2 , λ(x – t) + λ γ f (t) dt. Γ (n/2) 0 2 dt2 dt References: H. M. Srivastava and R. G. Buschman (1977), A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 461).





γ(ν, λ(t – x))y(t) dt = f (x).

9. x

m ∞  d λ–ν e–λt eλx – f  (t) dt, ν–m+1 t Γ(ν)Γ(m – ν) dx (t – x) x where Re ν > 0, m = [Re ν] + 1, and [Re ν] denotes the integer part of the number Re ν.

Solution:

y(x) = –

References: H. M. Srivastava and R. G. Buschman (1977), A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 462).



x

10.

Γ(ν, λ(x – t))y(t) dt = f (x).

0

   d2  e–λx x  d  λt ν e f (t) dt, y(x) = Eν [λ(x – t)] –λ 2 Γ(ν) 0 dt dt where Re ν > 0 and Eν (z) are the Weber function, 1 π sin(νt – z sin t) dt. Eν (z) = π 0

Solution:

References: H. M. Srivastava and R. G. Buschman (1977), A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 462).

1.8-5. Kernels Containing Bessel Functions. x 11. J0 (λ(x – t))y(t) dt = f (x). a

This is a special case of equation 1.8.17 with n = 0 and J0 (z) is the Bessel function (see Supplement 11.6-1). If f (a) = fx (a) = 0 then the solution is   2 x d 2 y(x) = f (t) dt. J0 (λ(x – t)) + λ dt2 a Example. In the special case λ = 1 and f (x) = A sin x, the solution has the form y(x) = AJ0 (x). References: H. M. Srivastava and R. G. Buschman (1977), A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 470). * For notation and properties of special functions, see Supplement 11.

1.8. EQUATIONS WHOSE KERNELS CONTAIN SPECIAL FUNCTIONS



x

[J0 (λx) – J0 (λt)]y(t) dt = f (x).

12. a

13.

89

   fx (x) d . Solution: y(x) = – dx λJ1 (λx) x [AJ0 (λx) + BJ0 (λt)]y(t) dt = f (x). a

For B = –A, see equation 1.8.12. We consider the interval [a, x] in which J0 (λx) does not change its sign. Solution with B ≠ –A:

x – A – B  1 d A+B A+B y(x) = ± J0 (λx) J0 (λt) ft (t) dt . A + B dx a

14.

Here the sign of J0 (λx) should be taken. x (x – t) J0 (λ(x – t))y(t) dt = f (x). a

 (a) = 0 then the This is a special case of equation 1.8.18 with n = 0. If f (a) = fx (a) = fxx solution is 2  2 x t d 2 y(x) = J0 (λ(x – t)) +λ F (t) dt, F (t) = f (s) ds. dt2 a a

Reference: A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 472).



x

(x – t)J1 (λ(x – t))y(t) dt = f (x).

15. a

This is a special case of equation 1.8.17 with n = 1. If f (a) = fx (a) = 0 then the solution is   2 fx (x) 1 x d 2 + J0 (λ(x – t)) + λ f (t) dt. y(x) = λ λ a dt2 Reference: A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 471).



x

16. a

(x – t)2 J1 (λ(x – t))y(t) dt = f (x).

 (a) = 0 then This is a special case of equation 1.8.18 with n = 1. If f (a) = fx (a) = · · · = fxxxx the solution is 3  2 x t 1 d 2 y(x) = J0 (λ(x – t)) +λ F (t) dt, F (t) = f (s) ds. 3λ a dt2 a

Reference: A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 472).



x

17. a

(x – t)n Jn (λ(x – t))y(t) dt = f (x),

n = 0, 1, 2, . . .

If f (a) = fx (a) = · · · = fx(2n+1) (a) = 0 then the solution is n+1  2 x 2n n! d 2 y(x) = J (λ(x – t)) + λ f (t) dt. 0 (2n)!λn a dt2 References: H. M. Srivastava and R. G. Buschman (1977), A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, pp. 471–472).

90

LINEAR EQUATIONS OF THE FIRST KIND WITH VARIABLE LIMIT OF INTEGRATION



x

18. a

(x – t)n+1 Jn (λ(x – t))y(t) dt = f (x),

n = 0, 1, 2, . . .

If f (a) = fx (a) = · · · = fx(2n+2) (a) = 0 then the solution is n+2  2 d 2n+1 (n + 1)! x 2 J0 (λ(x – t)) +λ F (t) dt, y(x) = (2n + 2)!λn a dt2

F (t) =

t

f (s) ds. a

References: H. M. Srivastava and R. G. Buschman (1977), A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 472).



x

19. a

(x – t)1/2 J1/2 (λ(x – t))y(t) dt = f (x).

This is a special case of equation 1.8.23 with n = 1. If f (a) = fx (a) = 0 then the solution is   π  f (x) + λ2 f (x) . y(x) = 2λ xx Reference: A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 471).



x

20. a

(x – t)3/2 J1/2 (λ(x – t))y(t) dt = f (x).

 This is a special case of equation 1.8.24 with n = 1. Let f (a) = fx (a) = fxx (a) = 0. Then the solution is 2 x √  2 π d 2 y(x) = √ + λ f (t) dt. 2 2λ dx2 a

Reference: A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 472).



x

21. a

(x – t)3/2 J3/2 (λ(x – t))y(t) dt = f (x).

  This is a special case of equation 1.8.23 with n = 2. If f (a) = fx (a) = fxx (a) = fxxx (a) = 0 then the solution is  2 2 √ π d 2 y(x) = + λ f (x). (2λ)3/2 dx2

Reference: A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 471).



x

22. a

(x – t)5/2 J3/2 (λ(x – t))y(t) dt = f (x).

 This is a special case of equation 1.8.24 with n = 2. Let f (a) = fx (a) = · · · = fxxxx (a) = 0. Then the solution is  2 3 x √ π d 2 y(x) = + λ f (t) dt. 4(2λ)3/2 dx2 a

Reference: A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 472).



x

(x – t)

23. a

2n–1 2

J 2n–1 (λ(x – t))y(t) dt = f (x),

n = 1, 2, 3, . . .

2

Let f (a) = fx (a) = · · · = f (2n–1) (a) = 0. Then the solution is  2 n √ π d 2 y(x) = +λ f (x). 2n–1 dx2 (2λ) 2 (n – 1)! References: H. M. Srivastava and R. G. Buschman (1977), A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 471).

1.8. EQUATIONS WHOSE KERNELS CONTAIN SPECIAL FUNCTIONS



x

(x – t)

24.

2n+1 2

a

J 2n–1 (λ(x – t))y(t) dt = f (x),

91

n = 1, 2, 3, . . .

2

Let f (a) = fx (a) = · · · = f (2n) (a) = 0. Then the solution is  2 n+1 x √ π d 2 + λ f (t) dt. y(x) = 2(2λ)n–1/2n! dx2 a References: H. M. Srivastava and R. G. Buschman (1977), A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 472).



x

25. a

[Jν (λx) – Jν (λt)]y(t) dt = f (x).

26.

This is a special case of equation 1.9.2 with g(x) = Jν (λx), where Jν (z) is the Bessel function (see Supplement 11.6-1).   xfx (x) d . Solution: y(x) = dx νJν (λx) – λxJν+1 (λx) x [AJν (λx) + BJν (λt)]y(t) dt = f (x).

27.

For B = –A, see equation 1.8.25. We consider the interval [a, x] in which Jν (λx) does not change its sign. Solution with B ≠ –A:

x – A – B  1 d A+B A+B y(x) = ± Jν (λx) Jν (λt) ft (t) dt . A + B dx a Here the sign of Jν (λx) should be taken. x [AJν (λx) + BJµ (βt)]y(t) dt = f (x).

28.

This is a special case of equation 1.9.6 with g(x) = AJν (λx) and h(t) = BJµ (βt). x (x – t)ν Jν (λ(x – t))y(t) dt = f (x).

a

a

a ◦

1 . Let Re ν > –1/2 and f (a) = fx (a) = . . . = fx(2n–1) (a) = 0, where n = [Re ν + 1/2] + 1 and [A] stands for the integer part of the number A. Then the solution is n  2 x π(2λ)1–n d n–ν–1 2 y(x) = (x – t) Jn–ν–1 (λ(x – t)) +λ f (t) dt. Γ(ν + 1/2)Γ(n – ν – 1/2) a dt2 2◦ . For ν = n and ν = n – 1/2 (n is an integer) see equations 1.8.17 and 1.8.23. References: H. M. Srivastava and R. G. Buschman (1977), A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 471), S. G. Samko, A. A. Kilbas, and O. I. Marichev (1993).



x

29. a

(x – t)ν+1 Jν (λ(x – t))y(t) dt = f (x).

1◦ . Let Re ν > –1 and f (a) = fx (a) = · · · = f (2n–2) (a) = 0, where n = [Re ν + 3/2] + 1 and [A] stands for the integer part of the number A. Then the solution is n  2 x 21–n λ2–n π d 2 y(x) = (x – t)n–ν–2 Jn–ν–2 (λ(x – t)) + λ F (t) dt, Γ(ν + 3/2)Γ(n – ν – 3/2) a dt2 t

where F (t) =

a

f (s) ds.

2◦ . For ν = n and ν = n – 1/2 (n is an integer) see equations 1.8.18 and 1.8.24. References: H. M. Srivastava and R. G. Buschman (1977), A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 472), S. G. Samko, A. A. Kilbas, and O. I. Marichev (1993).

92

LINEAR EQUATIONS OF THE FIRST KIND WITH VARIABLE LIMIT OF INTEGRATION



x

30. 0

Jν (λ(x – t)) y(t) dt = f (x), x–t

Re ν > 0.

1◦ . If ν = n is a positive integer number and f (0) = fx (0) = · · · = fx(n) (0) = 0 then

y(x) =

 n–2k  2 k [n/2] d n  2k d 2 C + λ f (x) n λn dx dx2 k=0

+

n λn







[(n–1)/2]

x

J0 (λ(x – t)) 0

Cn2k+1

k=0

d dt

n–2k–1 

where [A] stands for the integer part of the number A and Cnk =

d2 + λ2 dt2

k+1 f (t) dt,

n! are binomial k! (n – k)!

coefficients (0! = 1). 2◦ . If ν is not an integer, [Re ν] + 1 = m > 1, and f (0) = fx (0) = · · · = fx(m) (0) = 0 then ν y(x) = m λ







[(m–1)/2]

x

Jm–ν (λ(x – t)) 0

2k+1 Cm

k=0

ν(m – ν) + λm

0

x

d dt

m–2k–1 

d2 + λ2 dt2

k+1 f (t) dt

 m–2k  2 k [m/2] Jm–ν (λ(x – t))  2k d d 2 Cm +λ f (t) dt. x–t dt dt2 k=0

Reference: A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, pp. 470–471).



x

31. a

 √  J0 λ x – t y(t) dt = f (x).

This is a special case of equation 1.8.38 with n = 0. If f (a) = fx (a) = 0 then the solution is d2 y(x) = dx2



x

 √  I0 λ x – t f (t) dt.

a

Reference: A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 472).



x

32.

a

 √   √  AJν λ x + BJν λ t y(t) dt = f (x).

 √  We consider the interval [a, x] in which Jν λ x does not change its sign. Solution with B ≠ –A:

x A  √  – B  1 d  √  – A+B A+B Jν λ x Jν λ t ft (t) dt . y(x) = ± A + B dx a  √  Here the sign Jν λ x should be taken.

x

33. a

 √   √  AJν λ x + BJµ β t y(t) dt = f (x).

 √   √  This is a special case of equation 1.9.6 with g(x) = AJν λ x and h(t) = BJµ β t .

1.8. EQUATIONS WHOSE KERNELS CONTAIN SPECIAL FUNCTIONS



x

34. a



93

 √  x – t J1 λ x – t y(t) dt = f (x).

 (a) = 0 This is a special case of equation 1.8.38 with n = 1. If the conditions f (a) = fx (a) = fxx are satisfied, then the solution is x  √  2 d3 y(x) = I0 λ x – t f (t) dt. 3 λ dx a

References: H. M. Srivastava and R. G. Buschman (1977), A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 472).



x

35. a

 √  (x – t)1/4 J1/2 λ x – t y(t) dt = f (x).

36.

This is a special case of equation 1.8.39 with n = 1. If the conditions f (a) = fx (a) = 0 are satisfied, then the solution is   √  x cosh λ x – t 2 d2 √ y(x) = f (t) dt. πλ dx2 a x–t x  √  (x – t)3/4 J3/2 λ x – t y(t) dt = f (x).

37.

 (a) = 0 This is a special case of equation 1.8.39 with n = 2. If the conditions f (a) = fx (a) = fxx are satisfied, then the solution is  √  x cosh λ x – t 23/2 d3 √ y(x) = √ 3/2 3 f (t) dt. dx a πλ x–t x  √  (x – t)–1/4 J–1/2 λ x – t y(t) dt = f (x).

38.

This is a special case of equation 1.8.39 with n = 0. If the condition f (a) = 0 is satisfied, then the solution is   √  x cosh λ x – t λ d √ y(x) = f (t) dt. 2π dx a x–t x  √  (x – t)n/2 Jn λ x – t y(t) dt = f (x), n = 0, 1, 2, . . .

a

a

a

This is a special case of equation 1.8.40 with ν = n and m = n + 2. If the conditions f (a) = fx (a) = · · · = fx(n+1) (a) = 0 are satisfied, then the solution is  2 n dn+2 x  √  I0 λ x – t f (t) dt. y(x) = n+2 λ dx a References: H. M. Srivastava and R. G. Buschman (1977), A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 472).



x

(x – t)

39. a

2n–1 4

 √  J 2n–1 λ x – t y(t) dt = f (x),

n = 0, 1, 2, . . .

2

This is a special case of equation 1.8.40 with ν = n – 1/2 and m = n + 1. If the conditions f (a) = fx (a) = · · · = fx(n) (a) = 0 are satisfied, then the solution is 1 y(x) = √ π

 √    2n–1 n+1 x 2 cosh λ x – t d 2 √ f (t) dt. λ dxn+1 a x–t

94

LINEAR EQUATIONS OF THE FIRST KIND WITH VARIABLE LIMIT OF INTEGRATION



x

40. a

 √  (x – t)ν/2 Jν λ x – t y(t) dt = f (x),

Re ν > –1.

1◦ . Let f (a) = fx (a) = · · · = fx(m–1) (a) = 0, where m = [Re ν + 1] + 1 and [A] stands for the integer part of the number A. Then the solution is  2 m–2 dm x   m–ν–2  √  2 y(x) = x – t λ x – t f (t) dt. I m–ν–2 λ dxm a 2◦ . For ν = n and ν = n – 1/2 (n is an integer) see equations 1.8.38 and 1.8.39. References: H. M. Srivastava and R. G. Buschman (1977), A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 472).





41. x

 √  (t – x)ν/2 Jν λ t – x y(t) dt = f (x),

Re ν > –1.

Solution: m ∞  2–m   √  d λ – y(x) = (t – x)(m–ν)/2–1 Im–ν–2 λ t – x f (t) dt, 2 dx x where m = [Re ν + 1] + 1 and [A] stands for the integer part of the number A. References: H. M. Srivastava and R. G. Buschman (1977), A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 474), S. G. Samko, A. A. Kilbas, and O. I. Marichev (1993).



x

42.

   (x – t)ν/2 Jν λ t(x – t) y(t) dt = f (x).

0

Solution: y(x) =

λ –1/2 x 2



x

   (x – t)–(ν+1)/2 J–ν–1 λ x(x – t) tν+1 d(t–ν f (t)),

0

where –1 < Re ν < 0. References: A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 473), S. G. Samko, A. A. Kilbas, and O. I. Marichev (1993).



x

43.

   (x – t)ν/2 Jν λ x(x – t) y(t) dt = f (x).

0

Solution:   x    λ –ν d ν+1 ν/2 –(ν+1)/2 x t (x – t) I–ν–1 λ t(x – t) f (t) dt , y(x) = x 2 dx 0 where –1 < Re ν < 0. References: A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 473), S. G. Samko, A. A. Kilbas, and O. I. Marichev (1993).

44.

x

 √  J0 λ x2 – t2 y(t) dt = f (x).

0

Solution: y(x) = fx (x) + λ

d dx

0

x

 √  t √ I1 λ x2 – t2 f (t) dt. 2 2 x –t

Reference: S. Feny¨o and H. W. Stolle (1984, p. 328).

1.8. EQUATIONS WHOSE KERNELS CONTAIN SPECIAL FUNCTIONS



x

45.

95

 2 2  –1/4  √  x –t J–1/2 λ x2 – t2 y(t) dt = f (x).

0

Solution:



2λ d π dx

y(x) =



x

0

 √  cosh λ x2 – t2 √ t f (t) dt. x2 – t2

Reference: S. G. Samko, A. A. Kilbas, and O. I. Marichev (1993).





46. x

 2  –1/4  √  t – x2 J–1/2 λ t2 – x2 y(t) dt = f (x).

Solution:



2λ d π dx

y(x) = –

x

47.





x

 √  cosh λ t2 – x2 √ t f (t) dt. t2 – x2

 2 2  ν/2  √  x –t Jν λ x2 – t2 y(t) dt = f (x),

–1 < ν < 0.

0

Solution: d y(x) = λ dx



x

–(ν+1)/2  √   t x2 – t2 I–ν–1 λ x2 – t2 f (t) dt.

0

Reference: S. G. Samko, A. A. Kilbas, and O. I. Marichev (1993).





48. x

 2  ν/2  √  t – x2 Jν λ t2 – x2 y(t) dt = f (x),

Solution: y(x) = –λ

d dx





–1 < ν < 0.

–(ν+1)/2  √   t t2 – x2 I–ν–1 λ t2 – x2 f (t) dt.

x

Reference: S. G. Samko, A. A. Kilbas, and O. I. Marichev (1993).



x

49. a

[Atk Jν (λx) + Bxm Jµ (λt)]y(t) dt = f (x).

This is a special case of equation 1.9.15 with g1 (x) = AJν (λx), h1 (t) = tk , g2 (x) = Bxm , and h2 (t) = Jµ (λt).

x

50. a

[AJν2 (λx) + BJν2 (λt)]y(t) dt = f (x).

Solution with B ≠ –A:

x – 2A – 2B  d 1 A+B A+B Jν (λx) Jν (λt) y(x) = ft (t) dt . A + B dx a

x

51. a

 AJνk (λx) + BJµm (βt) y(t) dt = f (x).

This is a special case of equation 1.9.6 with g(x) = AJνk (λx) and h(t) = BJµm (βt).

96

LINEAR EQUATIONS OF THE FIRST KIND WITH VARIABLE LIMIT OF INTEGRATION



x



52. 0

x–t x–t+γ

 ν/2 Jν (λ



(x – t)(x – t + γ))y(t) dt = f (x).

Let –1 < Re ν < m + 1 < 2n + 1 (n and m are the minimal integer numbers), and f (0) = fx (0) = · · · = fx(2n+m+1) (0) = 0. Then (m–ν)/2    x–t Jm–ν λ (x – t)(x – t – γ) x–t–γ 0  n–(j+1)/2 n+1 m–j  2 t m j Cm t–s d d 2 Jn–(j+1)/2 (λ(t–s)) +λ f (s) ds dt, × 2 Γ(n – j/2) 2λ ds ds 0

y(x) =

√ –m πλ

x

j=0

where Cnk are binomial coefficients. Reference: A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 473), S. G. Samko, A. A. Kilbas, and O. I. Marichev (1993).



x

53. a

[Y0 (λx) – Y0 (λt)]y(t) dt = f (x).

Solution: y(x) = –

x

54. a

[Yν (λx) – Yν (λt)]y(t) dt = f (x).

Solution: y(x) =

   d fx (x) . dx λY1 (λx)

  xfx (x) d . dx νYν (λx) – λxYν+1 (λx)

x

55. a

[AYν (λx) + BYν (λt)]y(t) dt = f (x).

For B = –A, see equation 1.8.54. We consider the interval [a, x] in which Yν (λx) does not change its sign. Solution with B ≠ –A:

x – A – B  d 1 A+B A+B Yν (λx) y(x) = ± Yν (λt) ft (t) dt . A + B dx a Here the sign of Yν (λx) should be taken.

x

56. a

[Atk Yν (λx) + Bxm Yµ (λt)]y(t) dt = f (x).

This is a special case of equation 1.9.15 with g1 (x) = AYν (λx), h1 (t) = tk , g2 (x) = Bxm , and h2 (t) = Yµ (λt).

x

57. a

[AJν (λx)Yµ (βt) + BJν (λt)Yµ (βx)]y(t) dt = f (x).

This is a special case of equation 1.9.15 with g1 (x) = AJν (λx), h1 (t) = Yµ (βt), g2 (x) = BYµ (βx), and h2 (t) = Jν (λt).

1.8. EQUATIONS WHOSE KERNELS CONTAIN SPECIAL FUNCTIONS

97

1.8-6. Kernels Containing Modified Bessel Functions.

x

I0 (λ(x – t))y(t) dt = f (x).

58. a

This is a special case of equation 1.8.64 with n = 0 and I0 (z) is the modified Bessel function (see Supplement 11.7-1). If f (a) = fx (a) = 0 then the solution is   2 x d 2 I0 (λ(x – t)) – λ f (t) dt. y(x) = dt2 a Reference: A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 481).



x

[I0 (λx) – I0 (λt)]y(t) dt = f (x),

59. a

Solution: y(x) =

f (a) = fx (a) = 0.

   fx (x) d . dx λI1 (λx)

x

[AI0 (λx) + BI0 (λt)]y(t) dt = f (x).

60. a

For B = –A, see equation 1.8.59. Solution with B ≠ –A:

x – A – B  d 1 A+B A+B I0 (λx) I0 (λt) ft (t) dt . y(x) = ± A + B dx a

61.

Here the sign of Iν (λx) should be taken. x (x – t)I0 (λ(x – t))y(t) dt = f (x). a

 (a) = 0 then the This is a special case of equation 1.8.65 with n = 0. If f (a) = fx (a) = fxx solution is 2  2 x t d 2 y(x) = I0 (λ(x – t)) – λ F (t) dt, F (t) = f (s) ds. dt2 a a



x

(x – t)I1 (λ(x – t))y(t) dt = f (x).

62. a

This is a special case of equation 1.8.64 with n = 1. If f (a) = fx (a) = 0 then the solution is   2 x d 2 f (t) dt. I0 (λ(x – t)) – λ y(x) = λ–1 fx (x) + λ–1 dt2 a Reference: A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 481).



x

63. a

(x – t)2 I1 (λ(x – t))y(t) dt = f (x).

 (a) = 0 then This is a special case of equation 1.8.65 with n = 1. If f (a) = fx (a) = · · · = fxxxx the solution is 3  2 x t 1 d 2 y(x) = I0 (λ(x – t)) – λ F (t) dt, F (t) = f (s) ds. 3λ a dt2 a

98

LINEAR EQUATIONS OF THE FIRST KIND WITH VARIABLE LIMIT OF INTEGRATION



x

64. a

(x – t)n In (λ(x – t))y(t) dt = f (x),

n = 0, 1, 2, . . .

If f (a) = fx (a) = · · · = fx(2n+1) (a) = 0 then the solution is n+1  2 x d 2n n! 2 I0 (λ(x – t)) –λ f (t) dt. y(x) = (2n)!λn a dt2 References: H. M. Srivastava and R. G. Buschman (1977), A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 481), S. G. Samko, A. A. Kilbas, and O. I. Marichev (1993).



x

65. a

(x – t)n+1 In (λ(x – t))y(t) dt = f (x),

n = 0, 1, 2, . . .

This is a special case of equation 1.8.78 with ν = n and m = n + 2. If f (a) = fx (a) = · · · = fx(2n+2) (a) = 0 then the solution is n+2  2 t d 2n+1 (n + 1)! x 2 I (λ(x – t)) – λ F (t) dt, F (t) = f (s) ds. y(x) = 0 (2n + 2)! λn a dt2 a References: H. M. Srivastava and R. G. Buschman (1977), A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 482), S. G. Samko, A. A. Kilbas, and O. I. Marichev (1993).



x

66. a

(x – t)1/2 I1/2 (λ(x – t))y(t) dt = f (x).

This is a special case of equation 1.8.70 with n = 1. If f (a) = fx (a) = 0 then the solution is   π  fxx (x) – λ2 f (x) . y(x) = 2λ Reference: A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 481).



x

67. a

68.

(x – t)3/2 I1/2 (λ(x – t))y(t) dt = f (x).

 (a) = 0 then the This is a special case of equation 1.8.71 with n = 1. If f (a) = fx (a) = fxx solution is  2 2 x √ π d 2 y(x) = – λ f (t) dt. 2(2λ)1/2 dx2 a x (x – t)3/2 I3/2 (λ(x – t))y(t) dt = f (x). a

  (a) = fxxx (a) = 0 This is a special case of equation 1.8.70 with n = 2. If f (a) = fx (a) = fxx then the solution is  2 2 √ π d 2 y(x) = – λ f (x). (2λ)3/2 dx2

Reference: A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 481).



x

69. a

(x – t)5/2 I3/2 (λ(x – t))y(t) dt = f (x).

 (a) = 0 then This is a special case of equation 1.8.71 with n = 2. If f (a) = fx (a) = · · · = fxxxx the solution is  2 3 x √ π d 2 y(x) = – λ f (t) dt. 4(2λ)3/2 dx2 a

99

1.8. EQUATIONS WHOSE KERNELS CONTAIN SPECIAL FUNCTIONS



x

(x – t)

70.

2n–1 2

a

I 2n–1 (λ(x – t))y(t) dt = f (x),

n = 1, 2, 3, . . .

2

If f (a) = fx (a) = · · · = fx(2n–1) (a) = 0 then the solution is  2 n √ π d 2 –λ f (x). y(x) = (2λ)n–1/2 (n – 1)! dx2 References: H. M. Srivastava and R. G. Buschman (1977), A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 481), S. G. Samko, A. A. Kilbas, and O. I. Marichev (1993).



x

(x – t)

71. a

2n+1 2

I 2n–1 (λ(x – t))y(t) dt = f (x),

n = 0, 1, 2, . . .

2

If f (a) = fx (a) = · · · = f (2n) (a) = 0 then the solution is  2 n+1 x √ π d 2 y(x) = –λ f (t) dt. 2(2λ)n–1/2 n! dx2 a References: H. M. Srivastava and R. G. Buschman (1977), A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 482).



x

72. a

73.

[Iν (λx) – Iν (λt)]y(t) dt = f (x).

This is a special case of equation 1.9.2 with g(x) = Iν (λx), where Iν (z) is the modified Bessel function (see Supplement 11.7-1). x [AIν (λx) + BIν (λt)]y(t) dt = f (x). a

Solution with B ≠ –A:



x

– A – B  d 1 A+B A+B Iν (λx) Iν (λt) ft (t) dt . y(x) = A + B dx a



x

74. a

75.

[AIν (λx) + BIµ (βt)]y(t) dt = f (x).

This is a special case of equation 1.9.6 with g(x) = AIν (λx) and h(t) = BIµ (βt). x Iν (λ(x – t))y(t) dt = f (x). 0 ◦

1 . Let –1 < Re ν < 1 and f (0) = fx (0) = 0. Then the solution is   2 x d 2 f (t) dt. I–ν (λ(x – t)) – λ y(x) = dt2 0 2◦ . Let ν = n ≥ 0 (n is an integer number) and f (0) = fx (0) = · · · = fx(n+1) (0) = 0. Then the solution is  n–2k–1  2 k+1 [(n–1)/2]  d d –n 2k+1 2 y(x) = λ Cn –λ f (x) dx dx2 k=0

–n

I0 (λ(x – t))



0





[n/2]

x

k=0

Cn2k

d dt

n–2k 

d2 – λ2 dt2

k+1 f (t) dt,

where [A] stands for the integer part of the number A and Cnk are binomial coefficients.

100

LINEAR EQUATIONS OF THE FIRST KIND WITH VARIABLE LIMIT OF INTEGRATION

3◦ . Let Re ν > –1 and f (0) = fx (0) = · · · = fx(m+1) (0) = 0, where m = [Re ν + 1]. Then the solution is  m–2k–1  2 k+1 [(m–1)/2] m – ν x Im–ν (λ(x – t))  d d 2k+1 2 y(x) = Cm –λ f (t) dt λm 0 x–t dt dt2 k=0

–m

Im–ν (λ(x – t))







[m/2]

x

0

2k Cm

k=0

d dt

m–2k 

d2 – λ2 dt2

k+1 f (t) dt.

References: H. M. Srivastava and R. G. Buschman (1977), A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, pp. 479–480), S. G. Samko, A. A. Kilbas, and O. I. Marichev (1993).



x

76.

(x – t)–1 Iν (λ(x – t))y(t) dt = f (x).

0 ◦

1 . Let Re ν > 0 and f (0) = fx (0) = · · · = fx(m) (0) = 0, where m = [Re ν] + 1 and [A] stands for the integer part of the number A. Then the solution is  m–2k–1  2 k+1 x [(m–1)/2]  d d 2k+1 2 y(x) = νλ–m Im–ν (λ(x – t)) Cm – λ f (t) dt dt dt2 0 k=0



–1

(x – t) Im–ν (λ(x – t))

+ ν(m – ν)λ





[m/2]

x

–m 0

2k Cm

k=0

d dt

m–2k 

d2 – λ2 dt2

k f (t) dt,

where Cnk are binomial coefficients. 2◦ . If ν = n > 0 (n is an integer number) and f (0) = fx (0) = · · · = fx(n) (0) = 0 then  n–2k  2 k [n/2]  d d 2 y(x) = nλ–n Cn2k – λ f (x) dx dx2 k=0

+ nλ–n





[(n–1)/2]

x

I0 (λ(x – t)) 0

k=0

Cn2k+1

d dt

n–2k–1 

d2 – λ2 dt2

k+1 f (t) dt.

References: H. M. Srivastava and R. G. Buschman (1977), A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, pp. 480–481).



x

77. a ◦

(x – t)ν Iν (λ(x – t))y(t) dt = f (x),

Re ν > –1/2.

1 . Let f (a) = fx (a) = · · · = fx(2m–1) (a) = 0, where m = [Re ν + 1/2] + 1 and [A] stands for the integer part of the number A. Then the solution is m  2 x d (2λ)1–m π m–ν–1 2 (x – t) Im–ν–1 (λ(x – t)) –λ f (t) dt. y(x) = Γ(ν + 1/2)Γ(m – ν – 1/2) a dt2 2◦ . Let f (a) = fx (a) = · · · = fx(m–1) (a) = 0, where m = [2 Re ν + 1] + 1. Then the solution is  √ x π λΓ(–ν – 1) –λx dm λx e e (x – t)m–ν–1 y(x) = 2ν+1 2 Γ(ν + 1/2) dxm a  m  (–m)k (–2ν – 2)k (k – ν – 1)Ik–ν–1 (λ(x – t))f (t) dt , × Γ(m + k – 2ν – 1)k! k=0

where (a)k = a(a + 1) . . . (a + k – 1) is the Pochhammer symbol. 3◦ . For ν = n and ν = n – 1/2 (n is an integer) see equations 1.8.64 and 1.8.70. References: H. M. Srivastava and R. G. Buschman (1977), A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 481), S. G. Samko, A. A. Kilbas, and O. I. Marichev (1993).

101

1.8. EQUATIONS WHOSE KERNELS CONTAIN SPECIAL FUNCTIONS



x

78. a

(x – t)ν+1 Iν (λ(x – t))y(t) dt = f (x).

1◦ . Let Re ν > –1 and f (a) = fx (a) = · · · = f (2m–2) (a) = 0, where m = [Re ν + 3/2] + 1 and [A] stands for the integer part of the number A. Then the solution is y(x) =

21–m λ2–m π Γ(ν + 3/2)Γ(m – ν – 3/2)





x

(x – t)m–ν–2 Im–ν–2 (λ(x – t)) a

d2 – λ2 dt2

m F (t) dt,

t

where F (t) =

a

f (s) ds.

2◦ . For ν = n and ν = n – 1/2 (n is an integer) see equations 1.8.65 and 1.8.71. References: H. M. Srivastava and R. G. Buschman (1977), A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 482), S. G. Samko, A. A. Kilbas, and O. I. Marichev (1993).



x

79. a

 √  I0 λ x – t y(t) dt = f (x).

This is a special case of equation 1.8.86 with n = 0. If the conditions f (a) = fx (a) = 0 are satisfied, then the solution is y(x) =

x

80.

a

d2 dx2



 √  J0 λ x – t f (t) dt.

x

a

 √   √  AIν λ x + BIν λ t y(t) dt = f (x).

Solution with B ≠ –A:

x A

 √ – B  d  √ – A+B 1 A+B Iν λ x Iν λ t ft (t) dt . y(x) = A + B dx a

x

81. a

 √   √  AIν λ x + BIµ β t y(t) dt = f (x).

 √   √  This is a special case of equation 1.9.6 with g(x) = AIν λ x and h(t) = BIµ β t .

x

82. a



 √  x – t I1 λ x – t y(t) dt = f (x).

 (a) = 0 This is a special case of equation 1.8.86 with n = 1. If the conditions f (a) = fx (a) = fxx are satisfied, then the solution is

2 d3 y(x) = λ dx3

x

83. a



x

 √  J0 λ x – t f (t) dt.

a

 √  (x – t)1/4 I1/2 λ x – t y(t) dt = f (x).

This is a special case of equation 1.8.87 with n = 1. If the conditions f (a) = fx (a) = 0 are satisfied, then the solution is   √  x cos λ x – t 2 d2 √ f (t) dt. y(x) = πλ dx2 a x–t

102

LINEAR EQUATIONS OF THE FIRST KIND WITH VARIABLE LIMIT OF INTEGRATION



x

84. a

 √  (x – t)3/4 I3/2 λ x – t y(t) dt = f (x).

85.

 This is a special case of equation 1.8.87 with n = 2. If the conditions f (a) = fx (a) = fxx (a) = 0 are satisfied, then the solution is  √  x cos λ x – t 23/2 d3 √ y(x) = √ 3/2 3 f (t) dt. dx a πλ x–t x  √  (x – t)–1/4 I–1/2 λ x – t y(t) dt = f (x).

86.

This is a special case of equation 1.8.87 with n = 0. If the condition f (a) = 0 is satisfied, then the solution is   √  x cos λ x – t λ d √ y(x) = f (t) dt. 2π dx a x–t x  √  (x – t)n/2 In λ x – t y(t) dt = f (x), n = 0, 1, 2, . . .

87.

This is a special case of equation 1.8.88 with ν = n and m = n + 2. If the conditions f (a) = fx (a) = · · · = fx(n+1) (a) = 0 are satisfied, then the solution is  2 n dn+2 x  √  y(x) = J0 λ x – t f (t) dt. n+2 λ dx a x   2n–1  √  x – t 4 I 2n–1 λ x – t y(t) dt = f (x), n = 0, 1, 2, . . .

a

a

a

88.

2

This is a special case of equation 1.8.88 with ν = n – 1/2 and m = n + 1. If the conditions f (a) = fx (a) = · · · = fx(n) (a) = 0 are satisfied, then the solution is  √    2n–1 n+1 x 2 cos λ x – t d 2 1 √ y(x) = √ f (t) dt. π λ dxn+1 a x–t x  √  (x – t)ν/2 Iν λ x – t y(t) dt = f (x), Re ν > –1. a

1◦ . Let f (a) = fx (a) = · · · = fx(m–1) (a) = 0, where m = [Re ν + 1] + 1 and [A] stands for the integer part of the number A. Then the solution is  2 m–2 dm x   m–ν–2  √  2 y(x) = x–t Jm–ν–2 λ x – t f (t) dt. m λ dx a 2◦ . For ν = n and ν = n – 1/2 (n is an integer) see equations 1.8.86 and 1.8.87. References: A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 482), S. G. Samko, A. A. Kilbas, and O. I. Marichev (1993).





89. x

√ (t – x)ν/2 Iν (λ t – x)y(t) dt = f (x),

Solution: y(x) =

Re ν > –1.

m ∞  2–m  √ d λ – (t – x)(m–ν)/2–1 Jm–ν–2 (λ t – x)f (t) dt, 2 dx x

where m = [Re ν + 1] + 1 and [A] stands for the integer part of the number A. References: H. M. Srivastava and R. G. Buschman (1977), A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 484), S. G. Samko, A. A. Kilbas, and O. I. Marichev (1993).

1.8. EQUATIONS WHOSE KERNELS CONTAIN SPECIAL FUNCTIONS



x

90.

103

   (x – t)ν/2 Iν λ t(x – t) y(t) dt = f (x).

0

Solution: y(x) =

λ –1/2 x 2



x

     (x – t)–(ν+1)/2 J–ν–1 λ x(x – t) tν+1 d t–ν f (t) ,

0

where –1 < Re ν < 0. References: K. Soni (1968), A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 483), S. G. Samko, A. A. Kilbas, and O. I. Marichev (1993).



x

91.

   (x – t)ν/2 Iν λ x(x – t) y(t) dt = f (x).

0

Solution:   x    λ –ν d ν+1 ν/2 –(ν+1)/2 x t (x – t) J–ν–1 λ t(x – t) f (t) dt , y(x) = x 2 dx 0 where –1 < Re ν < 0. References: K. Soni (1968), A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 483), S. G. Samko, A. A. Kilbas, and O. I. Marichev (1993).



x

92.

 2 2  –1/4  √  x –t I–1/2 λ x2 – t2 y(t) dt = f (x).

0

Solution:

 y(x) =





93. x

2λ d π dx

 y(x) = –

x

94.

x

0

 √  cos λ x2 – t2 √ t f (t) dt. x2 – t2

 2  –1/4  √  t – x2 I–1/2 λ t2 – x2 y(t) dt = f (x).

Solution:





2λ d π dx





x

 √  cos λ t2 – x2 √ t f (t) dt. t2 – x2

 2 2  ν/2  √  x –t Iν λ x2 – t2 y(t) dt = f (x),

–1 < ν < 0.

0

Solution: d y(x) = λ dx



x

–(ν+1)/2  √   t x2 – t2 J–ν–1 λ x2 – t2 f (t) dt.

0

Reference: S. G. Samko, A. A. Kilbas, and O. I. Marichev (1993).





95. x

 √  (t2 – x2 )ν/2 Iν λ t2 – x2 y(t) dt = f (x),

Solution: d y(x) = –λ dx





–1 < ν < 0.

 √  t (t2 – x2 )–(ν+1)/2 J–ν–1 λ t2 – x2 f (t) dt.

x

Reference: S. G. Samko, A. A. Kilbas, and O. I. Marichev (1993).

104

LINEAR EQUATIONS OF THE FIRST KIND WITH VARIABLE LIMIT OF INTEGRATION



x

96. 0



x–t x–t+γ

 ν/2

   Iν λ (x – t)(x – t + γ) y(t) dt = f (x).

Let –1 < Re ν < m + 1 < 2n + 1 (n and m are the minimal integer numbers), and f (0) = fx (0) = · · · = fx(2n+m+1) (0) = 0. Then (m–ν)/2    x–t Im–ν λ (x – t)(x – t + γ) x–t+γ 0  n–(j+1)/2 n+1  m–j  2 t m j Cm t–s d d 2 × In–(j+1)/2 (λ(t–s)) –λ f (s) ds dt, 2 Γ(n – j/2) 2λ ds ds 0

√ y(x) = πλ–m

x

j=0

where Cnk are binomial coefficients. References: A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, pp. 483–484), S. G. Samko, A. A. Kilbas, and O. I. Marichev (1993).



x

97. a

[Atk Iν (λx) + Bxs Iµ (λt)]y(t) dt = f (x).

This is a special case of equation 1.9.15 with g1 (x) = AIν (λx), h1 (t) = tk , g2 (x) = Bxs , and h2 (t) = Iµ (λt).

x

98. a

[AIν2 (λx) + BIν2 (λt)]y(t) dt = f (x).

Solution with B ≠ –A:

x – 2A – 2B  d 1 A+B A+B Iν (λx) Iν (λt) ft (t) dt . y(x) = A + B dx a

x

99. a

[AIνk (λx) + BIµs (βt)]y(t) dt = f (x).

This is a special case of equation 1.9.6 with g(x) = AIνk (λx) and h(t) = BIµs (βt).

x

100. a

[K0 (λx) – K0 (λt)]y(t) dt = f (x).

Solution: y(x) = –

   fx (x) d . dx λK1 (λx)

x

101. a

[Kν (λx) – Kν (λt)]y(t) dt = f (x).

This is a special case of equation 1.9.2 with g(x) = Kν (λx).

x

102. a

[AKν (λx) + BKν (λt)]y(t) dt = f (x).

Solution with B ≠ –A: y(x) =



x

– A – B d 1 Kν (λx) A+B Kν (λt) A+B ft (t) dt . A + B dx a

1.8. EQUATIONS WHOSE KERNELS CONTAIN SPECIAL FUNCTIONS



x

103. a

105

[Atk Kν (λx) + Bxs Kµ (λt)]y(t) dt = f (x).

This is a special case of equation 1.9.15 with g1 (x) = AKν (λx), h1 (t) = tk , g2 (x) = Bxs , and h2 (t) = Kµ (λt). x [AIν (λx)Kµ (βt) + BIν (λt)Kµ (βx)]y(t) dt = f (x). 104. a

This is a special case of equation 1.9.15 with g1 (x) = AIν (λx), h1 (t) = Kµ (βt), g2 (x) = BKµ (βx), and h2 (t) = Iν (λt). 1.8-7. Kernels Containing Legendre Polynomials. 105.



x

x

 y(t) dt = f (x),

f (1) = 0, x ≥ 1. t Here Pn (x) is the Legendre polynomial (see Supplement 11.11-1). Solution:  n+1 x 1 d xn+1 y(x) = (x – t)n–1 f (t) dt, (n – 1)! x dx 1 Pn

1

where n = 1, 2, 3, . . . Reference: A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, pp. 495–496).





x

Pn

106. 1

t x

 y(t) dt = f (x),

Solution:

f (1) = fx (1) = 0,



x

t2–n Pn–2

y(x) = 1

x ≥ 1.

2  

n  1 d x t f (t) dt, t t dt

where n = 2, 3, 4, . . . References: H. M. Srivastava and R. G. Buschman (1977), A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 496).





1

107. x

Pn

x t

 y(t) dt = f (x),

Solution:

f (1) = fx (1) = 0,

1

–2

y(x) = x

t

n+2

x

0 < x ≤ 1.

2    1 d 2–n t t f (t) dt, Pn–2 x t dt

where n = 2, 3, 4, . . . References: H. M. Srivastava and R. G. Buschman (1977), A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 496).





1

108. x

Pn

t x

 y(t) dt = f (x),

Solution:

f (1) = fx (1) = 0,



1

y(x) =

t x

2–n

0 < x ≤ 1.

2    1 d n x t f (t) dt, Pn–2 t t dt

where n = 2, 3, 4, . . . References: H. M. Srivastava and R. G. Buschman (1977), A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 496).

106

LINEAR EQUATIONS OF THE FIRST KIND WITH VARIABLE LIMIT OF INTEGRATION

  x x Pn 2 – 1 y(t) dt = f (x), t 0

109.

Solution:

f (0) = 0,

x > 0.

  x dn+1 xn –n n–1 x (x – t) f (t) dt , (n – 1)! dxn+1 0

y(x) = where n = 1, 2, 3, . . .

References: H. M. Srivastava and R. G. Buschman (1977), A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 497).



1

110. x

  x Pn 2 – 1 y(t) dt = f (x), t

Solution:

f (1) = 0,

x ≤ 1.

n+1    1 (t – x)n–1 d –n y(x) = x – f (t) dt , x dx (n – 1)! x n

where n = 1, 2, 3, . . . References: H. M. Srivastava and R. G. Buschman (1977), A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 497).



1

111. x

  t Pn 2 – 1 y(t) dt = f (x), x

Solution:

f (1) = 0,

x ≤ 1.

 n+1   1 d (t – x)n–1 –n–1 n+1 y(x) = – t x f (t) dt , dx (n – 1)! x

where n = 1, 2, 3, . . . References: H. M. Srivastava and R. G. Buschman (1977), A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 498).



x

Pn (cosh(x – t))y(t) dt = f (x),

112. 0

Solution:

 y(x) =

f (0) = fx (0) = 0.

d2 – (n + 1)2 dx2



x

Pn+1 (cosh(x – t))f (t) dt, 0

where n = 1, 2, 3, . . . References: H. M. Srivastava and R. G. Buschman (1977), A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 498).



x

Pn (cos(x – t))y(t) dt = f (x),

113. 0

Solution:

 y(x) =

f (0) = fx (0) = 0.

d2 + (n + 1)2 dx2



x

Pn+1 (cos(x – t))f (t) dt, 0

where n = 1, 2, 3, . . . References: H. M. Srivastava and R. G. Buschman (1977), A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 498).

1.8. EQUATIONS WHOSE KERNELS CONTAIN SPECIAL FUNCTIONS

107

1.8-8. Kernels Containing Associated Legendre Functions. 114.

x

(x2 – t2 )–µ/2 Pµ ν

x

y(t) dt = f (x), 0 ≤ a < x. t Here Pµν (x) is the modified associated Legendre function (see Supplement 11.11-3). Let 1 – n < Re µ < 1 (n = 1, 2, . . . ) and f (a) = fx (a) = · · · = fx(n–1) (a) = 0. Then the solution is     x n n+µ–2 t n+µ–1 d 1–µ 2 2 –n 2–n–µ 2 y(x) = x f (t) dt . x (x – t ) t Pν dxn x a a

References: H. M. Srivastava and R. G. Buschman (1977), A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 515), S. G. Samko, A. A. Kilbas, and O. I. Marichev (1993).



x

115. a

(x2 – t2 )–µ/2 Pµ ν

t x

y(t) dt = f (x),

0 ≤ a < x.

Let 1 – n < Re µ < 1 (n = 1, 2, . . . ) and f (a) = fx (a) = · · · = fx(n–1) (a) = 0. Then the solution is x   n+µ–2 dn 2–n–µ x 2 2 2 f (t) dt. y(x) = (x – t ) P ν dxn a t References: H. M. Srivastava and R. G. Buschman (1977), A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 515), S. G. Samko, A. A. Kilbas, and O. I. Marichev (1993).

116.



(t2 – x2 )–µ/2 Pµ ν

x

y(t) dt = f (x). t Let 1 – n < Re µ < 1 (n = 1, 2, . . . ). Then the solution is     b n n+µ–2 t n n+µ–1 d 1–µ 2 2 –n 2–n–µ 2 f (t) dt . x y(x) = (–1) x (t – x ) t Pν dxn x x x

References: A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 516), S. G. Samko, A. A. Kilbas, and O. I. Marichev (1993).

117.



(t2 – x2 )–µ/2 Pµ ν

t

y(t) dt = f (x). x Let 1 – n < Re µ < 1 (n = 1, 2, . . . ). Then the solution is b x n+µ–2 dn f (t) dt. y(x) = (–1)n n (t2 – x2 ) 2 Pν2–n–µ dx x t x

References: A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 516), S. G. Samko, A. A. Kilbas, and O. I. Marichev (1993).

1.8-9. Kernels Containing Confluent Hypergeometric Functions. 118.

x

  (x – t)b–1 Φ a, b; λ(x – t) y(t) dt = f (x).

s

Here Φ(a, b; z) is the Kummer confluent hypergeometric function (see Supplement 11.9-1). Let 0 < Re b < n (n = 1, 2, . . . ) and f (s) = fx (s) = · · · = fx(n–1) (s) = 0. Then the solution is x  dn (x – t)n–b–1  Φ –a, n – b; λ(x – t) f (t) dt. y(x) = n dx s Γ(b)Γ(n – b) References: H. M. Srivastava and R. G. Buschman (1977), A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 530), S. G. Samko, A. A. Kilbas, and O. I. Marichev (1993).

108

LINEAR EQUATIONS OF THE FIRST KIND WITH VARIABLE LIMIT OF INTEGRATION





119. x

  (t – x)b–1 Φ a, b; λ(x – t) y(t) dt = f (x).

Here Φ(a, b; z) is the Kummer confluent hypergeometric function (see Supplement 11.9-1). If 0 < Re b < n (n = 1, 2, . . . ) then the solution is ∞  (t – x)n–b–1  y(x) = Φ –a, n – b; λ(x – t) ft(n) (t) dt. Γ(b)Γ(n – b) x References: A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 530), S. G. Samko, A. A. Kilbas, and O. I. Marichev (1993).



x

120.

  (x – t)ν–1/2 Mµ,ν λ(x – t) y(t) dt = f (x).

0

Here Mµ,ν (z) is the Whittaker confluent hypergeometric function (see Supplement 11.9-3). Let –1/2 < Re ν < (n – 1)/2 and f (0) = fx (0) = · · · = fx(n–1) (0) = 0. Then solution is  x   dn (x – t)(ν–3)/2–ν λ–n/2 e–λx/2 n eλx/2 Mn/2–µ,n/2–ν–1 λ(x – t) f (t) dt. y(x) = Γ(2ν + 1) dx 0 Γ((ν – 1)/2 – ν) References: H. M. Srivastava and R. G. Buschman (1977), A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 522).





121. x

  (t – x)ν–1/2 Mµ,ν λ(t – x) y(t) dt = f (x).

Here Mµ,ν (z) is the Whittaker confluent hypergeometric function (see Supplement 11.9-3). Let –1/2 < Re ν < (n – 1)/2. Then solution is ∞   dn –λt/2  λ–n/2 (t – x)(ν–3)/2–ν λt/2 e y(x) = e Mn/2–µ,n/2–ν–1 λ(t – x) f (t) dt. n Γ(2ν + 1) x Γ((ν – 1)/2 – ν) dt Reference: A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 522).

1.8-10. Kernels Containing Hermite Polynomials.

x

122.

 √  (x – t)–1/2 H2n λ x – t y(t) dt = f (x),

f (0) = 0.

0

Here Hm (x) is the Hermite polynomial (see Supplement 11.17-3). Solution:  m x  (x – t)m–3/2  d (–1)nn! F n, m – 12 ; λ2 (x – t) f (t) dt, y(x) = √ π (2n)! dx 0 Γ(m – 1/2) where m ≥ 1 and F (a, b; x) is the Kummer confluent hypergeometric function (see Supplement 11.9-1). References: H. M. Srivastava and R. G. Buschman (1977), A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 556).

123.

x

 √  H2n+1 λ x – t y(t) dt = f (x),

0

f (0) = fx (0) = 0.

Solution:

 m x  (–1)nn! (x – t)m–5/2  d F n, m – 32 ; λ2 (x – t) f (t) dt, y(x) = √ λ π (2n + 1)! dx 0 Γ(m – 3/2) where m ≥ 2 and F (a, b; x) is the Kummer confluent hypergeometric function (see Supplement 11.9-1). References: H. M. Srivastava and R. G. Buschman (1977), A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 556).

1.8. EQUATIONS WHOSE KERNELS CONTAIN SPECIAL FUNCTIONS

109

1.8-11. Kernels Containing Chebyshev Polynomials.



x

2

2 –1/2

(x – t )

124.

x

Tn

 y(t) dt = f (x),

t

1

f (1) = 0,

x ≥ 1.

Here Tn (x) is the Chebyshev polynomials of the first kind (see Supplement 11.17-2). Solution:    2 x n 2 2 –1/2 t d 1–n t f (t) dt, y(x) = t (x – t ) Tn–1 π 1 x dt where n = 1, 2, 3, . . . References: H. M. Srivastava and R. G. Buschman (1977), A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 499).



x

125.



t

(x2 – t2 )–1/2 Tn

x

1

Solution: y(x) =

 y(t) dt = f (x),

f (1) = 0,

x ≥ 1.

    x x 2 n+1 d x x–n f (t) dt , (x2 – t2 )–1/2 Tn+1 π dx t 1

where n = 1, 2, 3, . . . References: H. M. Srivastava and R. G. Buschman (1977), A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 499).





1 2

2 –1/2

(t – x )

126. x

Solution:

Tn

x t

 y(t) dt = f (x),

f (1) = 0,

0 < x ≤ 1.

    1 t 2 –n d n+1 2 2 –1/2 x f (t) dt , y(x) = – x (t – x ) Tn+1 π dx x x

where n = 1, 2, 3, . . . References: H. M. Srivastava and R. G. Buschman (1977), A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 499).





1

127. x

(t2 – x2 )–1/2 Tn

t x

 y(t) dt = f (x),

Solution: y(x) = –

2 π



f (1) = 0,

1

t1–n (t2 – x2 )–1/2 Tn–1 x

0 < x ≤ 1.

  x d n [t f (t)] dt, t dt

where n = 1, 2, 3, . . . References: H. M. Srivastava and R. G. Buschman (1977), A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 500).

128. 0

x

  x (x – t)–1/2 Tn 2 – 1 y(t) dt = f (x), t

Solution:

f (0) = 0,

x > 0.

  x dn xn –n n–3/2 x (x – t) f (t) dt , y(x) = √ π Γ(n – 1/2) dxn 0

where n = 1, 2, 3, . . . References: H. M. Srivastava and R. G. Buschman (1977), A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 500).

110

LINEAR EQUATIONS OF THE FIRST KIND WITH VARIABLE LIMIT OF INTEGRATION



1

(t – x)

129.

–1/2

x

  x Tn 2 – 1 y(t) dt = f (x), t

f (1) = 0,

x ≤ 1.

Solution:

n    1 xn (t – x)n–3/2 d –n √ y(x) = f (t) dt , – x dx π x Γ(n – 1/2) where n = 1, 2, 3, . . . References: H. M. Srivastava and R. G. Buschman (1977), A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 501).



1

(t – x)

130. x

–1/2

  t Tn 2 – 1 y(t) dt = f (x), x

f (1) = 0,

x ≤ 1.

Solution:

 n   1 1 d (t – x)n–3/2 –n n+1/2 √ y(x) = – t f (t) dt , x πx dx x Γ(n – 1/2) where n = 1, 2, 3, . . . References: H. M. Srivastava and R. G. Buschman (1977), A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 501).

1.8-12. Kernels Containing Laguerre Polynomials.

x

f (0) = fx (0) = 0,

Ln (λ(x – t))y(t) dt = f (x),

131. 0

x > 0.

Here Ln (x) is the Laguerre polynomial (see Supplement 11.17-1). Solution: x

Ln–1 (λ(t – x))e–λt ftt (t) dt,

y(x) = eλx 0

where n = 1, 2, 3, . . . References: H. M. Srivastava and R. G. Buschman (1977), A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 504).





132. x

f (0) = fx (0) = 0,

Ln (λ(t – x))y(t) dt = f (x),

x > 0.

Here Ln (x) is the Laguerre polynomial (see Supplement 11.17-1). Solution: ∞

Ln+1 (λ(x – t))eλt ftt (t) dt,

y(x) = e–λx

x

where n = 1, 2, 3, . . . References: H. M. Srivastava and R. G. Buschman (1977), A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 505).

1.8-13. Kernels Containing Jacobi Theta Functions.

x

ϑ2 (0, x – t)y(t) dt = f (x),

133.

f (0) = 0.

0

Here ϑ2 (v, q) is the Jacobi theta function (see Supplement 11.15-1). Solution: 1 x y(x) = ϑ3 (0, x – t)ft (t) dt. π 0 References: H. M. Srivastava and R. G. Buschman (1977), A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 551).

1.9. EQUATIONS WHOSE KERNELS CONTAIN ARBITRARY FUNCTIONS



111

x

ϑ3 (0, x – t)y(t) dt = f (x),

134.

f (0) = 0.

0

Here ϑ3 (v, q) is the Jacobi theta function (see Supplement 11.15-1). Solution: 1 x y(x) = ϑ2 (0, x – t)ft (t) dt. π 0 References: H. M. Srivastava and R. G. Buschman (1977), A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 551).

1.8-14. Kernels Containing Other Special Functions.

x

135. s

  x y(t) dt = f (x). (x – t)c–1 F a, b, c; 1 – t

Here Φ(a, b, c; z) is the Gaussian hypergeometric function (see Supplement 11.10-1). Solution: x  n  t (x – t)n–c–1  –a d a F –a, n – b, n – c; 1 – f (t) dt , y(x) = x x dxn x s Γ(c)Γ(n – c) where 0 < c < n and n = 1, 2, . . . If the right-hand side of the equation is differentiable sufficiently many times and the conditions f (s) = fx (s) = · · · = fx(n–1) (s) = 0 are satisfied, then the solution of the integral equation can be written in the form y(x) = s

x

(x – t)n–c–1  t  (n) F –a, –b, n – c; 1 – f (t) dt. Γ(c)Γ(n – c) x t

Reference: S. G. Samko, A. A. Kilbas, and O. I. Marichev (1993).



x

136.

 √  (x – t)–(ν+1)/2 Dν λ x – t y(t) dt = f (x).

0

Here Dν (z) is the parabolic cylinder function (see Supplement 11.12-1) and –1 < Re ν < 1. Solution:     √  d 2 1 x λ2  –λ2 t/4 y(x) = + e (x – t)(ν–1)/2 eλ t/4 Dν λ t – x f (t) dt. π 0 dt 2 References: H. M. Srivastava and R. G. Buschman (1977), A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 464).

1.9. Equations Whose Kernels Contain Arbitrary Functions 1.9-1. Equations with Degenerate Kernel: K(x, t) = g1 (x)h1 (t) + g2 (x)h2 (t).

x

g(x)h(t)y(t) dt = f (x).

1. a

Solution: y =

  1 1 d f (x) g  (x) = fx (x) – 2 x f (x). h(x) dx g(x) g(x)h(x) g (x)h(x)

112

LINEAR EQUATIONS OF THE FIRST KIND WITH VARIABLE LIMIT OF INTEGRATION



x

[g(x) – g(t)]y(t) dt = f (x).

2. a

It is assumed that f (a) = fx (a) = 0 and fx /gx ≠ const. d fx (x) . Solution: y(x) = dx gx (x)

x

[g(x) – g(t) + b]y(t) dt = f (x).

3. a

For b = 0, see equation 1.9.2. Assume that b ≠ 0. Differentiation with respect to x yields an equation of the form 2.9.2: y(x) + Solution: y(x) =

1  g (x) b x



1 1  f (x) – 2 gx (x) b x b

x

y(t) dt = a



x

a

1  f (x). b x

 g(t) – g(x)  ft (t) dt. exp b

x

[Ag(x) + Bg(t)]y(t) dt = f (x).

4. a

For B = –A, see equation 1.9.2. Assume that B ≠ –A. Solution with B ≠ –A:

x – A B sign g(x) d g(t) – A+B ft (t) dt . g(x) A+B y(x) = A + B dx a

x

[Ag(x) + Bg(t) + C]y(t) dt = f (x).

5. a

For B = –A, see equation 1.9.3. Assume that B ≠ –A and (A + B)g(x) + C > 0. Solution:

x – A – B  d A+B A+B y(x) = (A + B)g(x) + C (A + B)g(t) + C ft (t) dt . dx a

x

[g(x) + h(t)]y(t) dt = f (x).

6. a

Solution:   x  ft (t) dt Φ(x) d , y(x) = dx g(x) + h(x) a Φ(t)

x

7.

 Φ(x) = exp

x a

 ht (t) dt . g(t) + h(t)

 g(x) + (x – t)h(x) y(t) dt = f (x).

a

This is a special case of equation 1.9.15 with g1 (x) = g(x) + xh(x), h1 (t) = 1, g2 (x) = h(x), and h2 (t) = –t. Solution: 

 dt h(x) x f (t) d Φ(x) , y(x) = dx g(x) a h(t) t Φ(t)

 Φ(t) = exp – a

x

 h(t) dt . g(t)

113

1.9. EQUATIONS WHOSE KERNELS CONTAIN ARBITRARY FUNCTIONS



x

8.

 g(t) + (x – t)h(t) y(t) dt = f (x).

a

This is a special case of equation 1.9.15 with g1 (x) = x, h1 (t) = h(t), g2 (x) = 1, and h2 (t) = g(t) – th(t).

x

9.

 g(x) + (Axλ + Btµ )h(x) y(t) dt = f (x).

a

This is a special case of equation 1.9.15 with g1 (x) = g(x) + Axλ h(x), h1 (t) = 1, g2 (x) = h(x), and h2 (t) = Btµ .

x

10.

 g(t) + (Axλ + Btµ )h(t) y(t) dt = f (x).

a

This is a special case of equation 1.9.15 with g1 (x) = Axλ , h1 (t) = h(t), g2 (x) = 1, and h2 (t) = g(t) + Btµ h(t).

x

f (a) = fx (a) = 0.

[g(x)h(t) – h(x)g(t)]y(t) dt = f (x),

11. a

For g = const or h = const, see equation 1.9.2. Solution:   1 d (f /h)x , where f = f (x), y(x) = h dx (g/h)x

g = g(x),

h = h(x).

Here Af + Bg + Ch ≡/ 0, with A, B, and C being some constants.

x

[Ag(x)h(t) + Bg(t)h(x)]y(t) dt = f (x).

12. a

For B = –A, see equation 1.9.11. Solution with B ≠ –A: 1 d y(x) = (A + B)h(x) dx

x

13.





h(x) g(x)



A A+B

x a

h(t) g(t)



B A+B

   d f (t) dt . dt h(t)

 1 + [g(t) – g(x)]h(x) y(t) dt = f (x).

a

This is a special case of equation 1.9.15 with g1 (x) = 1 – g(x)h(x), h1 (t) = 1, g2 (x) = h(x), and h2 (t) = g(t). Solution: y(x) =

x

14.





x dt f (t) d h(x)Φ(x) , dx h(t) t Φ(t) a

 Φ(x) = exp

x

 gt (t)h(t) dt .

a

  e–λ(x–t) + eλx g(t) – eλt g(x) h(x) y(t) dt = f (x).

a

This is a special case of equation 1.9.15 with g1 (x) = eλx h(x), h1 (t) = g(t), g2 (x) = e–λx – g(x)h(x), and h2 (t) = eλt .

114

LINEAR EQUATIONS OF THE FIRST KIND WITH VARIABLE LIMIT OF INTEGRATION



x

15. a

[g1 (x)h1 (t) + g2 (x)h2 (t)]y(t) dt = f (x).

For g2 /g1 = const or h2 /h1 = const, see equation 1.9.1. 1◦ . Solution with g1 (x)h1 (x) + g2 (x)h2 (x) ≡/ 0 and f (x) ≡/ const g2 (x): 

x dt 1 d g2 (x)h1 (x)Φ(x) f (t) , (1) y(x) = h1 (x) dx g1 (x)h1 (x) + g2 (x)h2 (x) a g2 (t) t Φ(t) where x  

g2 (t)h1 (t) dt h2 (t) Φ(x) = exp . (2) h1 (t) t g1 (t)h1 (t) + g2 (t)h2 (t) a If f (x) ≡ const g2 (x), the solution is given by formulas (1) and (2) in which the subscript 1 must be changed by 2 and vice versa. 2◦ . Solution with g1 (x)h1 (x) + g2 (x)h2 (x) ≡ 0:     1 d (f /g2 )x 1 d (f /g2 )x =– , y(x) = h1 dx (g1 /g2 )x h1 dx (h2 /h1 )x where f = f (x), g2 = g2 (x), h1 = h1 (x), and h2 = h2 (x). 1.9-2. Equations with Difference Kernel: K(x, t) = K(x – t). x 16. K(x – t)y(t) dt = f (x). a

1◦ . Let K(0) = 1 and f (a) = 0. Differentiating the equation with respect to x yields a Volterra equation of the second kind: x y(x) + Kx (x – t)y(t) dt = fx (x). a

The solution of this equation can be represented in the form x  R(x – t)ft (t) dt. y(x) = fx (x) + a

Here the resolvent R(x) is related to the kernel K(x) of the original equation by  

 1 ˜ –1 , K(p) = L K(x) , R(x) = L–1 ˜ pK(p) where L and L–1 are the operators of the direct and inverse Laplace transforms, respectively. c+i∞ ∞



 1 –1 ˜ –px ˜ dp. ˜ e K(x) dx, R(x) = L R(p) = epx R(p) K(p) = L K(x) = 2πi c–i∞ 0 2◦ . Let K(x) have an integrable power-law singularity at x = 0. Denote by w = w(x) the solution of the simpler auxiliary equation (compared with the original equation) with a = 0 and constant right-hand side f ≡ 1, x K(x – t)w(t) dt = 1. (1) 0

Then the solution of the original integral equation with arbitrary right-hand side is expressed in terms of w as follows: x x d y(x) = w(x – t)f (t) dt = f (a)w(x – a) + w(x – t)ft (t) dt. (2) dx a a Remark. The integral equation and its solution (2) form the Sonine transform pair. References: E. C. Titchmarsh (1986), A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 426), S. G. Samko, A. A. Kilbas, and O. I. Marichev (1993).

115

1.9. EQUATIONS WHOSE KERNELS CONTAIN ARBITRARY FUNCTIONS





K(x – t)y(t) dt = f (x).

17. x

Solution: y(x) = –

where



d dx



H(t – x)f (t) dt, x

x

K(t)H(x – t) dt = 1. 0



x

18.

K(x – t)y(t) dt = Axn ,

n = 0, 1, 2, . . .

–∞

This is a special case of equation 1.9.20 with λ = 0. 1◦ . Solution with n = 0: y(x) =



A , B



B=

K(z) dz. 0

2◦ . Solution with n = 1:

AC A x+ 2 , B B

y(x) =





B=

K(z) dz,

C=



zK(z) dz.

0

0

3◦ . Solution with n = 2: A 2 AC AC 2 AD x +2 2 x+2 3 – 2 , B B B B ∞ ∞ ∞ B= K(z) dz, C = zK(z) dz, D = z 2 K(z) dz. y2 (x) =

0

0

0

4◦ . Solution with n = 3, 4, . . . is given by: n  λx 

e ∂ yn (x) = A , ∂λn B(λ) λ=0

x

19.

B(λ) =



K(z)e–λz dz.

0

K(x – t)y(t) dt = Aeλx .

–∞

Solution: y(x) =

x

20.



A λx e , B

B=



K(z)e–λz dz = L{K(z), λ}.

0

K(x – t)y(t) dt = Axn eλx ,

n = 1, 2, . . .

–∞ ◦

1 . Solution with n = 1: A λx AC λx xe + 2 e , B B ∞ ∞ –λz B= K(z)e dz, C = zK(z)e–λz dz. y1 (x) =

0

0

It is convenient to calculate the coefficients B and C using tables of Laplace transforms according to the formulas B = L{K(z), λ} and C = L{zK(z), λ}.

116

LINEAR EQUATIONS OF THE FIRST KIND WITH VARIABLE LIMIT OF INTEGRATION

2◦ . Solution with n = 2:   A 2 λx AC λx AC 2 AD λx y2 (x) = x e + 2 2 xe + 2 3 – 2 e , B B B B ∞ ∞ ∞ –λz –λz B= K(z)e dz, C = zK(z)e dz, D = z 2 K(z)e–λz dz. 0

0

0

3◦ . Solution with n = 3, 4, . . . is given by: yn (x) =

 λx  e ∂ ∂n yn–1 (x) = A n , ∂λ ∂λ B(λ)

B(λ) =



K(z)e–λz dz.

0

x

K(x – t)y(t) dt = A cosh(λx).

21. –∞

Solution: y(x) =

A λx 1 A A –λx 1  A A  A  cosh(λx) + sinh(λx), e + e = + – 2B– 2B+ 2 B– B+ 2 B– B+ ∞ ∞ B– = K(z)e–λz dz, B+ = K(z)eλz dz. 0



0

x

K(x – t)y(t) dt = A sinh(λx).

22. –∞

Solution: y(x) =

A λx 1 A A –λx 1  A A  A  cosh(λx) + sinh(λx), e – e = – + 2B– 2B+ 2 B– B+ 2 B– B+ ∞ ∞ B– = K(z)e–λz dz, B+ = K(z)eλz dz. 0



0

x

K(x – t)y(t) dt = A cos(λx).

23. –∞

Solution:

 A Bc cos(λx) – Bs sin(λx) , Bc2 + Bs2 ∞ ∞ Bc = K(z) cos(λz) dz, Bs = K(z) sin(λz) dz. y(x) =

0



0

x

K(x – t)y(t) dt = A sin(λx).

24. –∞

Solution:

 A Bc sin(λx) + Bs cos(λx) , 2 + Bs ∞ ∞ Bc = K(z) cos(λz) dz, Bs = K(z) sin(λz) dz. y(x) =

0

Bc2

0

1.9. EQUATIONS WHOSE KERNELS CONTAIN ARBITRARY FUNCTIONS



x

25.

117

K(x – t)y(t) dt = Aeµx cos(λx).

–∞

Solution:

 A eµx Bc cos(λx) – Bs sin(λx) , 2 + Bs ∞ ∞ Bc = K(z)e–µz cos(λz) dz, Bs = K(z)e–µz sin(λz) dz. y(x) =

Bc2

0



x

26.

0

K(x – t)y(t) dt = Aeµx sin(λx).

–∞

Solution:

 A eµx Bc sin(λx) + Bs cos(λx) , 2 + Bs ∞ ∞ Bc = K(z)e–µz cos(λz) dz, Bs = K(z)e–µz sin(λz) dz. y(x) =

Bc2

0



0

x

K(x – t)y(t) dt = f (x).

27. –∞

1◦ . For a polynomial right-hand side of the equation, f (x) =

n 

Ak xk , the solution has the

k=0

form y(x) =

n 

Bk xk ,

k=0

where the constants Bk are found by the method of undetermined coefficients. The solution can also be obtained by the formula given in 1.9.18 (item 4◦ ). n  2◦ . For f (x) = eλx Ak xk , the solution has the form k=0

y(x) = eλx

n 

Bk xk ,

k=0

where the constants Bk are found by the method of undetermined coefficients. The solution can also be obtained by the formula given in 1.9.20 (item 3◦ ). n  3◦ . For f (x) = Ak exp(λk x), the solution has the form k=0 n  Ak y(x) = exp(λk x), Bk k=0

4◦ . For f (x) = cos(λx)

n 





Bk =

K(z) exp(–λk z) dz. 0

Ak xk , the solution has the form

k=0

y(x) = cos(λx)

n  k=0

Bk xk + sin(λx)

n 

Ck xk ,

k=0

where the constants Bk and Ck are found by the method of undetermined coefficients.

118

LINEAR EQUATIONS OF THE FIRST KIND WITH VARIABLE LIMIT OF INTEGRATION

5◦ . For f (x) = sin(λx)

n 

Ak xk , the solution has the form

k=0

y(x) = cos(λx)

n 

Bk xk + sin(λx)

k=0

n 

Ck xk ,

k=0

where the constants Bk and Ck are found by the method of undetermined coefficients. n  Ak cos(λk x), the solution has the form 6◦ . For f (x) = k=0

Bck n 

7◦ . For f (x) =

n 

 Ak Bck cos(λk x) – Bsk sin(λk x) , 2 + Bsk ∞ ∞ = K(z) cos(λk z) dz, Bsk = K(z) sin(λk z) dz.

y(x) =

2 Bck k=0

0

0

Ak sin(λk x), the solution has the form

k=0

Bck

n 

 Ak Bck sin(λk x) + Bsk cos(λk x) , 2 + Bsk ∞ ∞ = K(z) cos(λk z) dz, Bsk = K(z) sin(λk z) dz.

y(x) =

2 Bck k=0

0

0



K(x – t)y(t) dt = f (x).

28. x

Solution: y(x) = –

where



d dx



H(t – x)f (t) dt, x

x

K(t)H(x – t) dt = 1. 0

References: A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 426), S. G. Samko, A. A. Kilbas, and O. I. Marichev (1993).





29.

K(x – t)y(t) dt = Axn ,

n = 0, 1, 2, . . .

x

This is a special case of equation 1.9.31 with λ = 0. 1◦ . Solution with n = 0: y(x) = 2◦ . Solution with n = 1: AC A y(x) = x – 2 , B B 3◦ . Solution with n = 2: B=

y2 (x) = ∞

K(–z) dz, 0



A , B

B=



K(–z) dz. 0







B=

K(–z) dz,

C=

0



zK(–z) dz. 0

A 2 AC AC 2 AD x –2 2 x+2 3 – 2 , B B B B ∞ ∞ C= zK(–z) dz, D = z 2 K(–z) dz. 0

4◦ . Solution with n = 3, 4, . . . is given by n  λx 

e ∂ yn (x) = A , n ∂λ B(λ) λ=0

0

B(λ) = 0



K(–z)eλz dz.

1.9. EQUATIONS WHOSE KERNELS CONTAIN ARBITRARY FUNCTIONS





30.

119

K(x – t)y(t) dt = Aeλx .

x

Solution: y(x) =



A λx e , B



B=

K(–z)eλz dz.

0

The expression for B is the Laplace transform of the function K(–z) with parameter p = –λ and can be calculated with the aid of tables of Laplace transforms given (e.g., see Supplement 5).



31.

K(x – t)y(t) dt = Axn eλx ,

n = 1, 2, . . .

x

1◦ . Solution with n = 1: A λx AC λx xe – 2 e , B B ∞ ∞ B= K(–z)eλz dz, C = zK(–z)eλz dz. y1 (x) =

0

0

It is convenient to calculate the coefficients B and C using tables of Laplace transforms with parameter p = –λ. 2◦ . Solution with n = 2:   A 2 λx AC AC 2 AD x e – 2 2 xeλx + 2 3 – 2 eλx , B B B B ∞ ∞ ∞ B= K(–z)eλz dz, C = zK(–z)eλz dz, D = z 2 K(–z)eλz dz. y2 (x) =

0

0

0

3◦ . Solution with n = 3, 4, . . . is given by: yn (x) =

 λx  e ∂ ∂n yn–1 (x) = A n , ∂λ ∂λ B(λ)

B(λ) =



K(–z)eλz dz.

0



K(x – t)y(t) dt = A cosh(λx).

32. x

Solution: y(x) =

1 A A –λx 1  A A A A λx cosh(λx) + sinh(λx), e + e = + – 2B+ 2B– 2 B+ B– 2 B+ B– ∞ ∞ B+ = K(–z)eλz dz, B– = K(–z)e–λz dz. 0



0



K(x – t)y(t) dt = A sinh(λx).

33. x

Solution: y(x) =

1 A A –λx 1  A A A A λx cosh(λx) + sinh(λx), e – e = – + 2B+ 2B– 2 B+ B– 2 B+ B– ∞ ∞ B+ = K(–z)eλz dz, B– = K(–z)e–λz dz. 0

0

120

LINEAR EQUATIONS OF THE FIRST KIND WITH VARIABLE LIMIT OF INTEGRATION





K(x – t)y(t) dt = A cos(λx).

34. x

Solution:

 A Bc cos(λx) + Bs sin(λx) , 2 + Bs ∞ ∞ Bc = K(–z) cos(λz) dz, Bs = K(–z) sin(λz) dz. y(x) =

Bc2

0



0



K(x – t)y(t) dt = A sin(λx).

35. x

Solution:

 A Bc sin(λx) – Bs cos(λx) , 2 + Bs ∞ ∞ Bc = K(–z) cos(λz) dz, Bs = K(–z) sin(λz) dz. y(x) =

Bc2

0





36.

0

K(x – t)y(t) dt = Aeµx cos(λx).

x

Solution:

 A eµx Bc cos(λx) + Bs sin(λx) , Bc2 + Bs2 ∞ ∞ Bc = K(–z)eµz cos(λz) dz, Bs = K(–z)eµz sin(λz) dz. y(x) =

0





37.

0

K(x – t)y(t) dt = Aeµx sin(λx).

x

Solution:

 A eµx Bc sin(λx) – Bs cos(λx) , Bc2 + Bs2 ∞ ∞ µz Bc = K(–z)e cos(λz) dz, Bs = K(–z)eµz sin(λz) dz. y(x) =

0



0



K(x – t)y(t) dt = f (x).

38. x

1◦ . For a polynomial right-hand side of the equation, f (x) =

n 

Ak xk , the solution has the

k=0

form y(x) =

n 

Bk xk ,

k=0

where the constants Bk are found by the method of undetermined coefficients. The solution can also be obtained by the formula given in 1.9.29 (item 4◦ ).

1.9. EQUATIONS WHOSE KERNELS CONTAIN ARBITRARY FUNCTIONS

2◦ . For f (x) = eλx

n 

121

Ak xk , the solution has the form

k=0

y(x) = e

λx

n 

Bk xk ,

k=0

where the constants Bk are found by the method of undetermined coefficients. The solution can also be obtained by the formula given in 1.9.31 (item 3◦ ). n  Ak exp(λk x), the solution has the form 3◦ . For f (x) = k=0 n  Ak y(x) = exp(λk x), Bk k=0

4◦ . For f (x) = cos(λx)

n 





Bk =

K(–z) exp(λk z) dz. 0

Ak xk , the solution has the form

k=0

y(x) = cos(λx)

n 

Bk xk + sin(λx)

k=0

n 

Ck xk ,

k=0

where the constants Bk and Ck are found by the method of undetermined coefficients. n  Ak xk , the solution has the form 5◦ . For f (x) = sin(λx) k=0

y(x) = cos(λx)

n 

Bk xk + sin(λx)

k=0

n 

Ck xk ,

k=0

where the constants Bk and Ck are found by the method of undetermined coefficients. n  Ak cos(λk x), the solution has the form 6◦ . For f (x) = k=0

Bck

n 

 Ak Bck cos(λk x) + Bsk sin(λk x) , 2 + Bsk ∞ ∞ = K(–z) cos(λk z) dz, Bsk = K(–z) sin(λk z) dz. y(x) =

2 Bck k=0

0

7◦ . For f (x) =

n 

0

Ak sin(λk x), the solution has the form

k=0

Bck

n 

 Ak Bck sin(λk x) – Bsk cos(λk x) , 2 + Bsk ∞ ∞ = K(–z) cos(λk z) dz, Bsk = K(–z) sin(λk z) dz. y(x) =

2 Bck k=0

0

0

8◦ . For arbitrary right-hand side f = f (x), the solution of the integral equation can be calculated by the formula c+i∞ ˜ 1 f (p) px y(x) = e dp, ˜ 2πi c–i∞ k(–p) ∞ ∞ ˜ ˜ = f (x)e–px dx, k(–p) = K(–z)epz dz. f(p) 0

0

˜ To calculate f˜(p) and k(–p), it is convenient to use tables of Laplace transforms, and to determine y(x), tables of inverse Laplace transforms.

122

LINEAR EQUATIONS OF THE FIRST KIND WITH VARIABLE LIMIT OF INTEGRATION

1.9-3. Other Equations.

x

39.

g(x) – g(t)

n

y(t) dt = f (x),

n = 1, 2, . . .

a

The right-hand side of the equation is assumed to satisfy the conditions f (a) = fx (a) = · · · = fx(n) (a) = 0. n+1  1  1 d f (x). Solution: y(x) = g (x)  n! x gx (x) dx

x

40.

 g(x) – g(t) y(t) dt = f (x),

f (a) = 0.

a

Solution:



x

41. a



2 x  f (t)gt (t) dt 1 d 2  √ . y(x) = gx (x)  π gx (x) dx g(x) – g(t) a

y(t) dt g(x) – g(t)

gx > 0.

= f (x),

Solution:



1 d y(x) = π dx

x

42. a

eλ(x–t) y(t) dt = f (x), √ g(x) – g(t)

x

x

e–λt f (t)gt (t) √ dt. g(x) – g(t)

gx > 0.

d 1 y(x) = eλx π dx

f (t)gt (t) dt √ . g(x) – g(t)

a

Solution:

43.

x

[g(x) – g(t)]λ y(t) dt = f (x),

a

f (a) = 0,

0 < λ < 1.

a

Solution: y(x) =

x

44. a

kgx (x)



1 d gx (x) dx

h(t)y(t) dt = f (x), [g(x) – g(t)]λ

2 a

gx > 0,





x

K

45. 0

t x

sin(πλ) . πλ

k=

0 < λ < 1.

Solution: y(x) =

gt (t)f (t) dt , [g(x) – g(t)]λ

x

sin(πλ) d πh(x) dx



x a

f (t)gt (t) dt . [g(x) – g(t)]1–λ

 y(t) dt = Axλ + Bxµ .

Solution: y(x) =

A λ–1 B µ–1 x + x , Iλ Iµ





1

K(z)z λ–1 dz,

Iλ = 0

1

K(z)z µ–1 dz.

Iµ = 0

1.9. EQUATIONS WHOSE KERNELS CONTAIN ARBITRARY FUNCTIONS





x

K

46. 0

Solution:

t x

 y(t) dt = Pn (x),

Pn (x) = xλ

n 

123

Am xm .

m=0 n  Am m–1 x , y(x) = x I m=0 m λ



1

K(z)z λ+m–1 dz.

Im = 0

The integral I0 is supposed to converge.

x

47.



a



 g1 (x) h1 (t) – h1 (x) + g2 (x) h2 (t) – h2 (x) y(t) dt = f (x).

This is a special case of equation 1.9.52 with g3 (x) = –g1 (x)h1 (x) – g2 (x)h2 (x) and h3 (t) = 1. x

The substitution Y (x) = equation of the form 1.9.15:

a

y(t) dt followed by integration by parts leads to an integral

x



  g1 (x) h1 (t) t + g2 (x) h2 (t) t Y (t) dt = –f (x). a



x

48.



a



 g1 (x) h1 (t) – eλ(x–t) h1 (x) + g2 (x) h2 (t) – eλ(x–t) h2 (x) y(t) dt = f (x).

 This is a special case of equation 1.9.52 with g3 (x) = –eλx g1 (x)h1 (x) + g2 (x)h2 (x) , and h3 (t) = e–λt . x

e–λt y(t) dt followed by integration by parts leads to an integral The substitution Y (x) = a equation of the form 1.9.15: x



  g1 (x) eλt h1 (t) t + g2 (x) eλt h2 (t) t Y (t) dt = –f (x). a



x

49.

 Ag λ (x)g µ (t) + Bg λ+β (x)g µ–β (t) – (A + B)g λ+γ (x)g µ–γ (t) y(t) dt = f (x).

a

This is a special case of equation 1.9.52 with g1 (x) = Ag λ (x), h1 (t) = g µ (t), g2 (x) = Bg λ+β (x), h2 (t) = g µ–β (t), g3 (x) = –(A + B)g λ+γ (x), and h3 (t) = g µ–γ (t).

x

50.

a

Ag λ (x)h(x)g µ (t) + Bg λ+β (x)h(x)g µ–β (t)  – (A + B)g λ+γ (x)g µ–γ (t)h(t) y(t) dt = f (x).

This is a special case of equation 1.9.52 with g1 (x) = Ag λ (x)h(x), h1 (t) = g µ (t), g2 (x) = Bg λ+β (x)h(x), h2 (t) = g µ–β (t), g3 (x) = –(A + B)g λ+γ (x), and h3 (t) = g µ–γ (t)h(t).

x

51. a

Ag λ (x)h(x)g µ (t) + Bg λ+β (x)h(t)g µ–β (t)  – (A + B)g λ+γ (x)g µ–γ (t)h(t) y(t) dt = f (x).

This is a special case of equation 1.9.52 with g1 (x) = Ag λ (x)h(x), h1 (t) = g µ (t), g2 (x) = Bg λ+β (x), h2 (t) = g µ–β (t)h(t), g3 (x) = –(A + B)g λ+γ (x), and h3 (t) = g µ–γ (t)h(t).

124

LINEAR EQUATIONS OF THE FIRST KIND WITH VARIABLE LIMIT OF INTEGRATION



x

52.

a

 g1 (x)h1 (t) + g2 (x)h2 (t) + g3 (x)h3 (t) y(t) dt = f (x), where g1 (x)h1 (x) + g2 (x)h2 (x) + g3 (x)h3 (x) ≡ 0.

x

h3 (t)y(t) dt followed by integration by parts leads to an integral The substitution Y (x) = a equation of the form 1.9.15:  

  x h1 (t) h2 (t) g1 (x) Y (t) dt = –f (x). + g2 (x) h3 (t) t h3 (t) t a

x

53.

Q(x – t)eαt y(ξ) dt = Aepx ,

ξ = eβt g(x – t).

–∞

Solution:



A p–α y(ξ) = ξ β , q

q=



Q(z)[g(z)]

p–α β e–pz

dz.

0

1.10. Some Formulas and Transformations 1. Let the solution of the integral equation

x

K(x, t)y(t) dt = f (x)

(1)

 y(x) = F f (x) ,

(2)

a

have the form

where F is some linear integro-differential operator. Then the solution of the more complicated integral equation x K(x, t)g(x)h(t)y(t) dt = f (x) (3) a

has the form y(x) =

 f (x)  1 F . h(x) g(x)

(4)

Below are formulas for the solutions of integral equations of the form (3) for some specific functions g(x) and h(t). In all cases, it is assumed that the solution of equation (1) is known and is determined by formula (2). (a) The solution of the equation

x

K(x, t)(x/t)λy(t) dt = f (x) a

has the form

 y(x) = xλ F x–λ f (x) .

(b) The solution of the equation

x

K(x, t)eλ(x–t) y(t) dt = f (x) a

has the form

 y(x) = eλx F e–λx f (x) .

1.10. SOME FORMULAS AND TRANSFORMATIONS

125

2. Let the solution of the integral equation (1) have the form  d  d  f (x) + L2 x, y(x) = L1 x, dx dx 



x

R(x, t)f (t) dt,

(5)

a

where L1 and L2 are some linear differential operators. The solution of the more complicated integral equation

x

  K ϕ(x), ϕ(t) y(t) dt = f (x),

(6)

a

where ϕ(x) is an arbitrary monotone function (differentiable sufficiently many times, ϕx > 0), is determined by the formula   d 1 f (x) y(x) = ϕx (x)L1 ϕ(x),  ϕx (x) dx  x  (7)   1 d R ϕ(x), ϕ(t) ϕt (t)f (t) dt. + ϕx (x)L2 ϕ(x),  ϕx (x) dx a Below are formulas for the solutions of integral equations of the form (6) for some specific functions ϕ(x). In all cases, it is assumed that the solution of equation (1) is known and is determined by formula (5). (a) For ϕ(x) = xλ ,   x     d d 1 1 λ–1 λ 2 λ–1 λ f (x) + λ x L2 x , R xλ , tλ tλ–1 f (t) dt. y(x) = λx L1 x , λxλ–1 dx λxλ–1 dx a (b) For ϕ(x) = eλx ,   x     1 d 1 d 2 λx λx f (x) + λ e e L , R eλx , eλt eλt f (t) dt. y(x) = λeλx L1 eλx , 2 λx λx λe dx λe dx a (c) For ϕ(x) = ln(λx),   x    1  1 d d 1 f (x) + L2 ln(λx), x R ln(λx), ln(λt) f (t) dt. y(x) = L1 ln(λx), x x dx x dx t a (d) For ϕ(x) = cos(λx),  y(x) = –λ sin(λx)L1 cos(λx),

 –1 d f (x) λ sin(λx) dx  x    d –1 2 + λ sin(λx)L2 cos(λx), R cos(λx), cos(λt) sin(λt)f (t) dt. λ sin(λx) dx a

(e) For ϕ(x) = sin(λx),  y(x) = λ cos(λx)L1 sin(λx),

 d 1 f (x) λ cos(λx) dx  x    d 1 R sin(λx), sin(λt) cos(λt)f (t) dt. + λ2 cos(λx)L2 sin(λx), λ cos(λx) dx a

Chapter 2

Linear Equations of the Second Kind with Variable Limit of Integration  Notation: f = f (x), g = g(x), h = h(x), K = K(x), and M = M (x) are arbitrary functions (these may be composite functions of the argument depending on two variables x and t); A, B, C, D, a, b, c, α, β, γ, λ, and µ are free parameters; and m and n are nonnegative integers.

2.1. Equations Whose Kernels Contain Power-Law Functions 2.1-1. Kernels Linear in the Arguments x and t. 1.

x

y(x) – λ

y(t) dt = f (x). a



Solution:

x

eλ(x–t) f (t) dt.

y(x) = f (x) + λ a

2.

x

y(t) dt = f (x).

y(x) + λx a



Solution:

x

 x exp 12 λ(t2 – x2 ) f (t) dt.

x

 t exp 12 λ(t2 – x2 ) f (t) dt.

y(x) = f (x) – λ a

3.

x

y(x) + λ

ty(t) dt = f (x). a



Solution: y(x) = f (x) – λ

a

4.

x

(x – t)y(t) dt = f (x).

y(x) + λ a

This is a special case of equation 2.1.34 with n = 1. 1◦ . Solution with λ > 0:

x

sin[k(x – t)]f (t) dt,

y(x) = f (x) – k a

127

k=

√ λ.

128

LINEAR EQUATIONS OF THE SECOND KIND WITH VARIABLE LIMIT OF INTEGRATION

2◦ . Solution with λ < 0:



y(x) = f (x) + k

x

sinh[k(x – t)]f (t) dt,

k=



–λ.

a

5.

x

y(x) +

 A + B(x – t) y(t) dt = f (x).

a

1◦ . Solution with A2 > 4B:

x y(x) = f (x) – R(x – t)f (t) dt, a     2B – A2 R(x) = exp – 12 Ax A cosh(βx) + sinh(βx) , 2β

 β=

1 2 4A

– B.

2◦ . Solution with A2 < 4B:

x y(x) = f (x) – R(x – t)f (t) dt, a     2B – A2 R(x) = exp – 21 Ax A cos(βx) + sin(βx) , 2β

3◦ . Solution with A2 = 4B: x y(x) = f (x) – R(x – t)f (t) dt, a

6.

x

y(x) –



 β=

B – 14 A2 .

   R(x) = exp – 12 Ax A – 14 A2 x .

 Ax + Bt + C y(t) dt = f (x).

a

This is a special case of equation 2.9.6 with g(x) = –Ax and h(t) = –Bt – C. For B = –A see equation 2.1.5. x

By differentiation followed by the substitution Y (x) =

y(t) dt, the original equation a

can be reduced to the second-order linear ordinary differential equation

  Yxx – (A + B)x + C Yx – AY = fx (x) under the initial conditions Y (a) = 0,

Yx (a) = f (a).

(1) (2)

A fundamental system of solutions of the homogeneous equation (1) with f ≡ 0 has the form     Y1 (x) = Φ α, 12 ; kz 2 , Y2 (x) = Ψ α, 12 ; kz 2 , A+B C A , k= , z =x+ , α= 2(A + B) 2 A+B     where Φ α, β; x and Ψ α, β; x are degenerate hypergeometric functions. Solving the homogeneous equation (1) under conditions (2) for an arbitrary function f = f (x) and taking into account the relation y(x) = Yx (x), we thus obtain the solution of the integral equation in the form x y(x) = f (x) – R(x, t)f (t) dt, a √      2 πk C 2 ∂ 2 Y1 (x)Y2 (t) – Y2 (x)Y1 (t) , W (t) = exp k t + . R(x, t) = ∂x∂t W (t) Γ(α) A+B

129

2.1. EQUATIONS WHOSE KERNELS CONTAIN POWER-LAW FUNCTIONS

2.1-2. Kernels Quadratic in the Arguments x and t. 7.

y(x) + A

x

x2 y(t) dt = f (x).

a

This is a special case of equation 2.1.50 with λ = 2 and µ = 0. Solution: x

 y(x) = f (x) – A x2 exp 13 A(t3 – x3 ) f (t) dt. a

8.

y(x) + A

x

xty(t) dt = f (x). a

This is a special case of equation 2.1.50 with λ = 1 and µ = 1. Solution: x

 y(x) = f (x) – A xt exp 13 A(t3 – x3 ) f (t) dt. a

9.

y(x) + A

x

t2 y(t) dt = f (x).

a

This is a special case of equation 2.1.50 with λ = 0 and µ = 2. Solution: x

 y(x) = f (x) – A t2 exp 13 A(t3 – x3 ) f (t) dt. a

10.

x

y(x) + λ

(x – t)2 y(t) dt = f (x).

a

This is a special case of equation 2.1.34 with n = 2. Solution: x y(x) = f (x) – R(x – t)f (t) dt, a  √  √ √  R(x) = 23 ke–2kx – 23 kekx cos 3 kx – 3 sin 3 kx , 11.

y(x) + A

x

k=

 1 1/3 . 4λ

(x2 – t2 )y(t) dt = f (x).

a

This is a special case of equation 2.9.5 with g(x) = Ax2 . Solution: x

  1 y(x) = f (x) + u1 (x)u2 (t) – u2 (x)u1 (t) f (t) dt, W a where the primes denote differentiation with respect to the argument specified in the parentheses; u1 (x), u2 (x) is a fundamental system of solutions of the second-order linear homogeneous ordinary differential equation uxx + 2Axu = 0; and the functions u1 (x) and u2 (x) are expressed in terms of Bessel functions or modified Bessel functions, depending on the sign of the parameter A: For A > 0,     √ √ 3/2 3/2 8 8 W = 3/π, u1 (x) = x J1/3 , u2 (x) = x Y1/3 . 9Ax 9Ax For A < 0, W = – 32 , u1 (x) =

    √ √ 3/2 3/2 8 8 , u . x I1/3 |A| x (x) = x K |A| x 2 1/3 9 9

130 12.

LINEAR EQUATIONS OF THE SECOND KIND WITH VARIABLE LIMIT OF INTEGRATION

y(x) + A

x

(xt – t2 )y(t) dt = f (x).

a

This is a special case of equation 2.9.4 with g(t) = At. Solution: x

 A y(x) = f (x) + t y1 (x)y2 (t) – y2 (x)y1 (t) f (t) dt, W a where y1 (x), y2 (x) is a fundamental system of solutions of the second-order linear homo geneous ordinary differential equation yxx + Axy = 0; the functions y1 (x) and y2 (x) are expressed in terms of Bessel functions or modified Bessel functions, depending on the sign of the parameter A: For A > 0,  √   √  √ √ W = 3/π, y1 (x) = x J1/3 23 A x3/2 , y2 (x) = x Y1/3 23 A x3/2 . For A < 0,

13.

      √ √ W = – 32 , y1 (x) = x I1/3 23 |A| x3/2 , y2 (x) = x K1/3 23 |A| x3/2 . x (x2 – xt)y(t) dt = f (x). y(x) + A a

This is a special case of equation 2.9.3 with g(x) = Ax. Solution: x

 A y(x) = f (x) + x y1 (x)y2 (t) – y2 (x)y1 (t) f (t) dt, W a where y1 (x), y2 (x) is a fundamental system of solutions of the second-order linear homo geneous ordinary differential equation yxx + Axy = 0; the functions y1 (x) and y2 (x) are expressed in terms of Bessel functions or modified Bessel functions, depending on the sign of the parameter A: For A > 0,  √   √  √ √ W = 3/π, y1 (x) = x J1/3 23 A x3/2 , y2 (x) = x Y1/3 23 A x3/2 . For A < 0,

14.

      √ √ W = – 32 , y1 (x) = x I1/3 23 |A| x3/2 , y2 (x) = x K1/3 23 |A| x3/2 . x (t2 – 3x2 )y(t) dt = f (x). y(x) + A

15.

This is a special case of equation 2.1.55 with λ = 1 and µ = 2. x y(x) + A (2xt – 3x2 )y(t) dt = f (x).

a

a

16.

This is a special case of equation 2.1.55 with λ = 2 and µ = 1. x y(x) – (ABxt – ABx2 + Ax + B)y(t) dt = f (x). a

This is a special case of equation 2.9.16 with g(x) = Ax and h(x) = B. Solution: x y(x) = f (x) + R(x, t)f (t) dt, a x



 R(x, t) = (Ax + B) exp 12 A(x2 – t2 ) + B 2 exp 12 A(s 2 – t2 ) + B(x – s) ds. t

2.1. EQUATIONS WHOSE KERNELS CONTAIN POWER-LAW FUNCTIONS



x

131

  Ax2 – At2 + Bx – Ct + D y(t) dt = f (x).

17.

y(x) +

18.

This is a special case of equation 2.9.6 with g(x) = Ax2 + Bx + D and h(t) = –At2 – Ct. Solution:   x ∂ 2 Y1 (x)Y2 (t) – Y2 (x)Y1 (t) y(x) = f (x) + f (t) dt. W (t) a ∂x∂t Here Y1 (x), Y2 (x) is a fundamental system of solutions of the second-order homogeneous   ordinary differential equation Yxx + (B – C)x + D Yx + (2Ax + B)Y = 0 (see A. D. Polyanin and V. F. Zaitsev (2003) for details about this equation):     Y1 (x) = exp(–kx)Φ α, 12 ; 12 (C – B)z 2 , Y2 (x) = exp(–kx)Ψ α, 12 ; 12 (C – B)z 2 , √ 

2A 2π(C – B) W (x) = – exp 12 (C – B)z 2 – 2kx , k = , Γ(α) B–C 4A + (C – B)D 4A2 + 2AD(C – B) + B(C – B)2 , z =x– , α=– 2(C – B)3 (C – B)2     where Φ α, β; x and Ψ α, β; x are degenerate hypergeometric functions and Γ(α) is the gamma function. x

 y(x) – Ax + B + (Cx + D)(x – t) y(t) dt = f (x).

a

a

This is a special case of equation 2.9.11 with g(x) = Ax + B and h(x) = Cx + D. Solution with A ≠ 0: x

  f (t) y(x) = f (x) + dt. Y2 (x)Y1 (t) – Y1 (x)Y2 (t) W (t) a Here Y1 (x), Y2 (x) is a fundamental system of solutions of the second-order homogeneous  ordinary differential equation Yxx – (Ax + B)Yx – (Cx + D)Y = 0 (see A. D. Polyanin and V. F. Zaitsev (2003) for details about this equation):     Y1 (x) = exp(–kx)Φ α, 12 ; 12 Az 2 , Y2 (x) = exp(–kx)Ψ α, 12 ; 12 Az 2 , √

–1   W (x) = – 2πA Γ(α) exp 12 Az 2 – 2kx , k = C/A,

19.

α = 12 (A2 D – ABC – C 2 )A–3 , z = x + (AB + 2C)A–2 ,     where Φ α, β; x and Ψ α, β; x are degenerate hypergeometric functions, Γ(α) is the gamma function. x

 y(x) + At + B + (Ct + D)(t – x) y(t) dt = f (x). a

This is a special case of equation 2.9.12 with g(t) = –At – B and h(t) = –Ct – D. Solution with A ≠ 0: x

 f (t) y(x) = f (x) – dt. Y1 (x)Y2 (t) – Y1 (t)Y2 (x) W (x) a Here Y1 (x), Y2 (x) is a fundamental system of solutions of the second-order homogeneous  ordinary differential equation Yxx – (Ax + B)Yx – (Cx + D)Y = 0 (see A. D. Polyanin and V. F. Zaitsev (2003) for details about this equation):     Y1 (x) = exp(–kx)Φ α, 12 ; 12 Az 2 , Y2 (x) = exp(–kx)Ψ α, 12 ; 12 Az 2 , √

–1   W (x) = – 2πA Γ(α) exp 12 Az 2 – 2kx , k = C/A, α = 12 (A2 D – ABC – C 2 )A–3 , z = x + (AB + 2C)A–2 ,     where Φ α, β; x and Ψ α, β; x are degenerate hypergeometric functions and Γ(α) is the gamma function.

132

LINEAR EQUATIONS OF THE SECOND KIND WITH VARIABLE LIMIT OF INTEGRATION

2.1-3. Kernels Cubic in the Arguments x and t. 20.

y(x) + A

x

x3 y(t) dt = f (x).

a



Solution:

x

 x3 exp 14 A(t4 – x4 ) f (t) dt.

x

 x2 t exp 14 A(t4 – x4 ) f (t) dt.

x

 xt2 exp 14 A(t4 – x4 ) f (t) dt.

y(x) = f (x) – A a

21.

y(x) + A

x

x2 ty(t) dt = f (x).

a



Solution: y(x) = f (x) – A

a

22.

y(x) + A

x

xt2 y(t) dt = f (x).

a



Solution: y(x) = f (x) – A

a

23.

y(x) + A

x

t3 y(t) dt = f (x).

a



Solution:

x

y(x) = f (x) – A a

24.

x

y(x) + λ

 t3 exp 14 A(t4 – x4 ) f (t) dt.

(x – t)3 y(t) dt = f (x).

a

This is a special case of equation 2.1.34 with n = 3. Solution: x R(x – t)f (t) dt,

y(x) = f (x) – a

where   k cosh(kx) sin(kx) – sinh(kx) cos(kx) , R(x) =

 1 s = (–6λ)1/4 2 s sin(sx) – sinh(sx) , 25.

y(x) + A

x

k=

(x3 – t3 )y(t) dt = f (x).

a

This is a special case of equation 2.1.52 with λ = 3. 26.

y(x) – A

x

 3 3 4x – t y(t) dt = f (x).

a

This is a special case of equation 2.1.55 with λ = 1 and µ = 3. 27.

y(x) + A

x

(xt2 – t3 )y(t) dt = f (x).

a

This is a special case of equation 2.1.49 with λ = 2.

 3 1/4 2λ

for λ > 0, for λ < 0.

2.1. EQUATIONS WHOSE KERNELS CONTAIN POWER-LAW FUNCTIONS

28.

y(x) + A

x



133

 x2 t – t3 y(t) dt = f (x).

a

The transformation z = x2 , τ = t2 , y(x) = w(z) leads to an equation of the form 2.1.4: z 1 w(z) + 2 A (z – τ )w(τ ) dτ = F (z), F (z) = f (x). a2

29.

x

y(x) +

  Ax2 t + Bt3 y(t) dt = f (x).

a

The transformation z = x2 , τ = t2 , y(x) = w(z) leads to an equation of the form 2.1.6: z 1  1 w(z) + F (z) = f (x). 2 Az + 2 Bτ w(τ ) dτ = F (z), a2



x

 3  2x – xt2 y(t) dt = f (x).

30.

y(x) + B

31.

This is a special case of equation 2.1.55 with λ = 2, µ = 2, and B = –2A. x  3  4x – 3x2 t y(t) dt = f (x). y(x) – A

32.

This is a special case of equation 2.1.55 with λ = 3 and µ = 1. x   ABx3 – ABx2 t – Ax2 – B y(t) dt = f (x). y(x) +

a

a

a

This is a special case of equation 2.9.7 with g(x) = Ax2 and λ = B. Solution: x y(x) = f (x) + R(x – t)f (t) dt, a x



 R(x, t) = (Ax2 + B) exp 13 A(x3 – t3 ) + B 2 exp 13 A(s 3 – t3 ) + B(x – s) ds. 33.

t

x

y(x) +

  ABxt2 – ABt3 + At2 + B y(t) dt = f (x).

a

This is a special case of equation 2.9.8 with g(t) = At2 and λ = B. Solution: x y(x) = f (x) + R(x – t)f (t) dt, a x

1 

 2 3 3 2 R(x, t) = –(At + B) exp 3 A(t – x ) + B exp 13 A(s 3 – x3 ) + B(t – s) ds. t

2.1-4. Kernels Containing Higher-Order Polynomials in x and t. 34.

y(x) + A

x

(x – t)n y(t) dt = f (x),

n = 1, 2, . . .

a

1◦ . Differentiating the equation n + 1 times with respect to x yields an (n + 1)st-order linear ordinary differential equation with constant coefficients for y = y(x): yx(n+1) + An! y = fx(n+1) (x). This equation under the initial conditions y(a) = f (a), yx (a) = fx (a), . . . , yx(n) (a) = fx(n) (a) determines the solution of the original integral equation.

134

LINEAR EQUATIONS OF THE SECOND KIND WITH VARIABLE LIMIT OF INTEGRATION

2◦ . Solution:

x

y(x) = f (x) +

R(x – t)f (t) dt, a

R(x) =

n

 1  exp(σk x) σk cos(βk x) – βk sin(βk x) , n+1 k=0

where the coefficients σk and βk are given by

35.

 2πk   2πk  1 1 , βk = |An!| n+1 sin σk = |An!| n+1 cos n+1 n+1  2πk + π   2πk + π  1 1 σk = |An!| n+1 cos , βk = |An!| n+1 sin n+1 n+1 ∞ y(x) + A (t – x)n y(t) dt = f (x), n = 1, 2, . . .

for A < 0, for A > 0.

x

The Picard–Goursat equation. This is a special case of equation 2.9.62 with K(z) = A(–z)n . 1◦ . A solution of the homogeneous equation (f ≡ 0) is   1 λ = –An! n+1 ,

y(x) = Ce–λx ,

where C is an arbitrary constant and A < 0. This is a unique solution for n = 0, 1, 2, 3. The general solution of the homogeneous equation for any sign of A has the form y(x) =

s 

Ck exp(–λk x).

(1)

k=1

Here Ck are arbitrary constants and λk are the roots of the algebraic equation λn+1 + An! = 0 that satisfy

 the condition Re λk > 0. The number of terms in (1) is determined by the inequality s ≤ 2 n4 + 1, where [a] stands for the integral part of a number a. For more details about the solution of the homogeneous Picard–Goursat equation, see Subsection 11.11-1 (Example 1). 2◦ . For f (x) =

m 

ak exp(–βk x), where βk > 0, a solution of the equation has the form

k=1

y(x) =

m 

ak βkn+1 n+1 β + An! k=1 k

exp(–βk x),

(2)

where βkn+1 + An! ≠ 0. For A > 0, this formula can also be used for arbitrary f (x) expandable into a convergent exponential series (which corresponds to m = ∞). 3◦ . For f (x) = e–βx

m 

ak xk , where β > 0, a solution of the equation has the form

k=1

y(x) = e–βx

m 

Bk xk ,

(3)

k=0

where the constants Bk are found by the method of undetermined coefficients. The solution can also be constructed using the formulas given in item 3◦ , equation 2.9.55.

135

2.1. EQUATIONS WHOSE KERNELS CONTAIN POWER-LAW FUNCTIONS

4◦ . For f (x) = cos(βx)

m 

ak exp(–µk x), a solution of the equation has the form

k=1

y(x) = cos(βx)

m 

Bk exp(–µk x) + sin(βx)

k=1

m 

Ck exp(–µk x),

(4)

k=1

where the constants Bk and Ck are found by the method of undetermined coefficients. The solution can also be constructed using the formulas given in 2.9.60. 5◦ . For f (x) = sin(βx)

m 

ak exp(–µk x), a solution of the equation has the form

k=1

y(x) = cos(βx)

m 

Bk exp(–µk x) + sin(βx)

k=1

m 

Ck exp(–µk x),

(5)

k=1

where the constants Bk and Ck are found by the method of undetermined coefficients. The solution can also be constructed using the formulas given in 2.9.61. 6◦ . To obtain the general solution in item 2◦ –5◦ , the solution (1) of the homogeneous equation must be added to each right-hand side of (2)–(5). 36.

y(x) + A

x

(x – t)tn y(t) dt = f (x),

n = 1, 2, . . .

a

This is a special case of equation 2.1.49 with λ = n. 37.

y(x) + A

x

(xn – tn )y(t) dt = f (x),

n = 1, 2, . . .

a

This is a special case of equation 2.1.52 with λ = n. 38.

x

y(x) +

  ABxn+1 – ABxn t – Axn – B y(t) dt = f (x),

n = 1, 2, . . .

a

This is a special case of equation 2.9.7 with g(x) = Axn and λ = B. Solution: x y(x) = f (x) + R(x – t)f (t) dt, a    x A  n+1 n+1  A  n+1 n+1  n 2 x –t s –t R(x, t) = (Ax +B) exp +B +B(x– s) ds. exp n+1 n+1 t 39.

x

y(x) +

  ABxtn – ABtn+1 + Atn + B y(t) dt = f (x),

n = 1, 2, . . .

a

This is a special case of equation 2.9.8 with g(t) = Atn and λ = B. Solution: x y(x) = f (x) + R(x – t)f (t) dt, a     x A  n+1 n+1  A  n+1 n+1  t –x s –x R(x, t) = –(Atn +B) exp +B 2 +B(t–s) ds. exp n+1 n+1 t

136

LINEAR EQUATIONS OF THE SECOND KIND WITH VARIABLE LIMIT OF INTEGRATION

2.1-5. Kernels Containing Rational Functions. 40.

y(x) + x–3

x

 t 2Ax + (1 – A)t y(t) dt = f (x).

a

This equation can be obtained by differentiating the equation x

2  2 Ax t + (1 – A)xt y(t) dt = F (x), F (x) = a

41.

x

y(x) – λ 0

t3 f (t) dt, a

which has the form 1.1.17: Solution:   x 1 d –A A–1  x t ϕt (t) dt , y(x) = x dx a

x

y(t) dt x+t

1 ϕ(x) = x



x

t3 f (t) dt. a

= f (x).

Dixon’s equation. This is a special case of equation 2.1.62 with a = b = 1 and µ = 0. 1◦ . The solution of the homogeneous equation (f ≡ 0) is y(x) = Cxβ

(β > –1, λ > 0).

(1)

Here C is an arbitrary constant, and β = β(λ) is determined by the transcendental equation λI(β) = 1,

1

where I(β) = 0

z β dz . 1+z

(2)

2◦ . For a polynomial right-hand side, f (x) =

N 

An xn

n=0

the solution bounded at zero is given by ⎧ N  ⎪ An ⎪ ⎪ ⎪ xn ⎪ ⎨ 1 – (λ/λ ) n n=0 y(x) = N ⎪  ⎪ An ⎪ ⎪ xn + Cxβ ⎪ ⎩ 1 – (λ/λn ) n=0

λn =

1 , I(n)

for λ < λ0 , for λ > λ0 and λ ≠ λn ,

  n  (–1)m I(n) = (–1)n ln 2 + , m m=1

where C is an arbitrary constant, and β = β(λ) is determined by the transcendental equation (2). For special λ = λn (n = 1, 2, . . . ), the solution differs in one term and has the form y(x) =

n–1  m=0

N  Am Am λ¯ n n xm + xm – An x ln x + Cxn , 1 – (λn /λm ) 1 – (λn /λm ) λn m=n+1



n (–1) π2  + where λ¯ n = (–1)n+1 12 k=1 k 2

 k –1 .

137

2.1. EQUATIONS WHOSE KERNELS CONTAIN POWER-LAW FUNCTIONS

Remark. For arbitrary f (x), expandable into power series, the formulas of item 2◦ can be used, in which one should set N = ∞. In this case, the radius of convergence of the solution y(x) is equal to the radius of convergence of f (x).

3◦ . For logarithmic-polynomial right-hand side,   N n An x , f (x) = ln x n=0

the solution with logarithmic singularity at zero is given by ⎧ N N   ⎪ An An Dn λ ⎪ n ⎪ ⎪ ln x x + xn for λ < λ0 , ⎪ ⎨ 1 – (λ/λn ) [1 – (λ/λn )]2 n=0 n=0 y(x) = N N ⎪   ⎪ An An Dn λ ⎪ n ⎪ x ln x + xn + Cxβ for λ > λ0 and λ ≠ λn , ⎪ ⎩ 1 – (λ/λn ) [1 – (λ/λn )]2 n=0 n=0    2   n n  (–1)k (–1)k 1 n n+1 π , I(n) = (–1) ln 2 + , Dn = (–1) + . λn = I(n) k 12 k2 k=1

k=1

4◦ . For arbitrary f (x), the transformation x = 12 e2z ,

t = 12 e2τ ,

y(x) = e–z w(z),

f (x) = e–z g(z)

leads to an integral equation with difference kernel of the form 2.9.51: z w(τ ) dτ w(z) – λ = g(z). cosh(z – τ) –∞ 42.

x

y(x) – λ a

x+b t+b

y(t) dt = f (x).

This is a special case of equation 2.9.1 with g(x) = x + b. Solution: x x + b λ(x–t) e f (t) dt. y(x) = f (x) + λ a t+b 43.

y(x) =

2 (1 – λ2 )x2



x

λx

t y(t) dt. 1+t

This equation is encountered in nuclear physics and describes deceleration of neutrons in matter. 1◦ . Solution with λ = 0: y(x) =

C , (1 + x)2

where C is an arbitrary constant. 2◦ . For λ ≠ 0, the solution can be found in the series form y(x) =

∞  n=0

Reference: I. Sneddon (1995).

An xn .

138

LINEAR EQUATIONS OF THE SECOND KIND WITH VARIABLE LIMIT OF INTEGRATION

2.1-6. Kernels Containing Square Roots and Fractional Powers. 44.

y(x) + A

x

√ (x – t) t y(t) dt = f (x).

a

This is a special case of equation 2.1.49 with λ = 12 . 45.

y(x) + A

x

√  √ x – t y(t) dt = f (x).

a

This is a special case of equation 2.1.52 with λ = 12 .

46.

x

y(t) dt = f (x). √ x–t a Abel’s equation of the second kind. This equation is encountered in problems of heat and mass transfer. Solution: x 2 y(x) = F (x) + πλ exp[πλ2 (x – t)]F (t) dt, y(x) + λ

a



where

x

F (x) = f (x) – λ a

f (t) dt √ . x–t

References: H. Brakhage, K. Nickel, and P. Rieder (1965), Yu. I. Babenko (1986).



47.

x

y(t) dt = f (x), a > 0, b > 0. √ ax2 + bt2 0 1◦ . The solution of the homogeneous equation (f ≡ 0) is y(x) – λ

y(x) = Cxβ

(β > –1, λ > 0).

(1)

Here C is an arbitrary constant, and β = β(λ) is determined by the transcendental equation 1 z β dz √ . (2) λI(β) = 1, where I(β) = a + bz 2 0 2◦ . For a polynomial right-hand side, f (x) =

N 

An xn

n=0

the solution bounded at zero is given by ⎧ N  ⎪ An ⎪ ⎪ ⎪ xn for λ < λ0 , ⎪ ⎨ 1 – (λ/λn ) n=0 y(x) = N ⎪ ⎪ An ⎪ ⎪ xn + Cxβ for λ > λ0 and λ ≠ λn , ⎪ ⎩ 1 – (λ/λn ) n=0 √ 1 b z n dz 1 √ , I(n) = λ0 = .   , λn = I(n) Arsinh b/a a + bz 2 0 Here C is an arbitrary constant, and β = β(λ) is determined by the transcendental equation (2).

2.1. EQUATIONS WHOSE KERNELS CONTAIN POWER-LAW FUNCTIONS

139

3◦ . For special λ = λn (n = 1, 2, . . . ), the solution differs in one term and has the form y(x) =

n–1  m=0



N  Am Am λ¯ n n xm + xm – An x ln x + Cxn , 1 – (λn /λm ) 1 – (λn /λm ) λn

1

m=n+1

–1

z n ln z dz √ . a + bz 2 0 4◦ . For arbitrary f (x), expandable into power series, the formulas of item 2◦ can be used, in which one should set N = ∞. In this case, the radius of convergence of the solution y(x) is equal to the radius of convergence of f (x). x y(t) dt y(x) + λ = f (x). 3/4 a (x – t) This equation admits solution by quadratures (see equation 2.1.60 and Example 2 in Subsection 11.4-2). where λ¯ n =

48.

2.1-7. Kernels Containing Arbitrary Powers. 49.

y(x) + A

x

(x – t)tλ y(t) dt = f (x).

a

This is a special case of equation 2.9.4 with g(t) = Atλ . Solution: x

 A y(x) = f (x) + y1 (x)y2 (t) – y2 (x)y1 (t) tλ f (t) dt, W a where y1 (x), y2 (x) is a fundamental system of solutions of the second-order linear homo geneous ordinary differential equation yxx + Axλ y = 0; the functions y1 (x) and y2 (x) are expressed in terms of Bessel functions or modified Bessel functions, depending on the sign of A: For A > 0, √  √  √ √ 2q A q A q λ+2 W = , y1 (x) = x J 1 x , y2 (x) = x Y 1 x , q= , π q q 2 2q 2q For A < 0, √ W = –q, y1 (x) = x I

50.

y(x) + A

x

√ 1 2q

 √  √ |A| q |A| q λ+2 x , y2 (x) = x K 1 x , q= . q q 2 2q

xλ tµ y(t) dt = f (x).

a

This is a special case of equation 2.9.2 with g(x) = –Axλ and h(t) = tµ (λ and µ are arbitrary numbers). Solution: x y(x) = f (x) – R(x, t)f (t) dt, a ⎧   λ+µ+1  A ⎨ Axλ tµ exp t for λ + µ + 1 ≠ 0, – xλ+µ+1 λ+µ+1 R(x, t) = ⎩ λ–A µ+A for λ + µ + 1 = 0. Ax t

140 51.

LINEAR EQUATIONS OF THE SECOND KIND WITH VARIABLE LIMIT OF INTEGRATION

y(x) + A

x

(x – t)xλ tµ y(t) dt = f (x).

a

The substitution u(x) = x–λ y(x) leads to an equation of the form 2.1.49: x (x – t)tλ+µ u(t) dt = f (x)x–λ . u(x) + A 52.

y(x) + A

a x

(xλ – tλ )y(t) dt = f (x).

a

This is a special case of equation 2.9.5 with g(x) = Axλ . Solution: x

  1 y(x) = f (x) + u1 (x)u2 (t) – u2 (x)u1 (t) f (t) dt, W a where the primes denote differentiation with respect to the argument specified in the parentheses, and u1 (x), u2 (x) is a fundamental system of solutions of the second-order linear homogeneous ordinary differential equation uxx + Aλxλ–1 u = 0; the functions u1 (x) and u2 (x) are expressed in terms of Bessel functions or modified Bessel functions, depending on the sign of A: For Aλ > 0, √  √  √ √ 2q λ+1 Aλ q Aλ q W = , u1 (x) = x J 1 x , u2 (x) = x Y 1 x , q= , π q q 2 2q 2q For Aλ < 0, √ W = –q, u1 (x) = x I 53.

x

y(x) –



√  √  √ |Aλ| q |Aλ| q λ+1 x , u2 (x) = x λK 1 x , q= . 1 q q 2 2q 2q

 Axλ tλ–1 + Bt2λ–1 y(t) dt = f (x).

a

The transformation z = xλ ,

54.

τ = tλ ,

y(x) = Y (z)

leads to an equation of the form 2.1.6:  z B A z + τ Y (τ ) dτ = F (z), F (z) = f (x), b = aλ . Y (z) – λ λ b x   Axλ+µ tλ–µ–1 + Bxµ t2λ–µ–1 y(t) dt = f (x). y(x) – a

The substitution y(x) = xµ w(x) leads to an equation of the form 2.1.53: x  λ λ–1  Ax t + Bt2λ–1 w(t) dt = x–µ f (x). w(x) – 55.

y(x) + A

a x

 λxλ–1 tµ – (λ + µ)xλ+µ–1 y(t) dt = f (x).

a

This equation can be obtained by differentiating equation 1.1.52: x

 1 + A(xλ tµ – xλ+µ ) y(t) dt = F (x), F (x) = a

f (x) dx. a

Solution: y(x) =

x

d dx



xλ Φ(x)

a

x

 t–λ F (t) t Φ(t) dt ,

  Aµ µ+λ x . Φ(x) = exp – µ+λ

141

2.1. EQUATIONS WHOSE KERNELS CONTAIN POWER-LAW FUNCTIONS

56.

x

  ABxλ+1 – ABxλ t – Axλ – B y(t) dt = f (x).

y(x) + a

This is a special case of equation 2.9.7. Solution:

x

R(x – t)f (t) dt, a    x A  λ+1 λ+1  A  λ+1 λ+1  λ 2 x –t s –t R(x, t) = (Ax + B) exp +B + B(x – s) ds. exp λ+1 λ+1 t y(x) = f (x) +

57.

x

  ABxtλ – ABtλ+1 + Atλ + B y(t) dt = f (x).

y(x) + a

This is a special case of equation 2.9.8. Solution:

x

R(x – t)f (t) dt, a    x A  λ+1 λ+1  A  λ+1 λ+1  λ 2 t –x s –x R(x, t) = –(At +B) exp +B +B(t–s) ds. exp λ+1 λ+1 t y(x) = f (x) +

58.

x

y(x) – λ a

 x + b µ t+b

y(t) dt = f (x).

This is a special case of equation 2.9.1 with g(x) = (x + b)µ . Solution: x x + b µ λ(x–t) e f (t) dt. y(x) = f (x) + λ t+b a 59.

x

y(x) – λ a

xµ + b y(t) dt = f (x). tµ + b

This is a special case of equation 2.9.1 with g(x) = xµ + b. Solution: x µ x + b λ(x–t) e f (t) dt. y(x) = f (x) + λ µ a t +b 60.

x

y(x) – λ 0

y(t) dt = f (x), (x – t)α

0 < α < 1.

Generalized Abel equation of the second kind. 1◦ . Assume that the number α can be represented in the form α=1–

m , n

where m = 1, 2, . . . ,

n = 2, 3, . . .

(m < n).

In this case, the solution of the generalized Abel equation of the second kind can be written in closed form (in quadratures):

x

R(x – t)f (t) dt,

y(x) = f (x) + 0

142

LINEAR EQUATIONS OF THE SECOND KIND WITH VARIABLE LIMIT OF INTEGRATION

where n–1 ν ν m–1    λ Γ (m/n) (νm/n)–1 b  x + εµ exp εµ bx R(x) = Γ(νm/n) m ν=1

µ=0

 m–1  n–1   x (νm/n)–1   b  λν Γν (m/n)  εµ exp εµ bx t exp –εµ bt dt , + m ν=1 Γ(νm/n) µ=0 0  2πµi  , i2 = –1, µ = 0, 1, . . . , m – 1. b = λn/m Γn/m (m/n), εµ = exp m 2◦ . Solution with any α from 0 < α < 1:

n ∞  λΓ(1 – α)x1–α

 . where R(x) = xΓ n(1 – α) n=1

x

R(x – t)f (t) dt,

y(x) = f (x) + 0

References: H. Brakhage, K. Nickel, and P. Rieder (1965), V. I. Smirnov (1974).

61.

y(x) –

λ





x

y(t) dt (x – t)1–α

0

0 < α ≤ 1.

= f (x),

1◦ . The solution of the homogeneous equation (f ≡ 0) is y(x) = Cxβ

(β > –1, λ > 0).

(1)

Here C is an arbitrary constant, and β = β(λ) is determined by the transcendental equation λB(α, β + 1) = 1, where B(p, q) =

 1 p–1 z (1 – z)q–1 dz 0

(2)

is the beta function.

2◦ . For a polynomial right-hand side, f (x) =

N 

An xn

n=0

the solution bounded at zero is given by ⎧ N  ⎪ An ⎪ ⎪ ⎪ xn ⎪ ⎨ 1 – (λ/λ ) n n=0 y(x) = N ⎪ ⎪ An ⎪ ⎪ xn + Cxβ ⎪ ⎩ 1 – (λ/λn )

for λ < α, for λ > α and λ ≠ λn ,

n=0

λn =

(α)n+1 , n!

(α)n+1 = α(α + 1) . . . (α + n).

Here C is an arbitrary constant, and β = β(λ) is determined by the transcendental equation (2). For special λ = λn (n = 1, 2, . . . ), the solution differs in one term and has the form y(x) =

n–1  m=0

 where λ¯ n =

N  Am Am λ¯ n n xm + xm – An x ln x + Cxn , 1 – (λn /λm ) 1 – (λn /λm ) λn

–1

1

(1 – z)α–1 z n ln z dz 0

m=n+1

.

2.1. EQUATIONS WHOSE KERNELS CONTAIN POWER-LAW FUNCTIONS

143

3◦ . For arbitrary f (x), expandable into power series, the formulas of item 2◦ can be used, in which one should set N = ∞. In this case, the radius of convergence of the solution y(x) is equal to the radius of convergence of f (x). 4◦ . For f (x) = ln(kx)

N 

An xn ,

n=0

a solution has the form y(x) = ln(kx)

N 

Bn xn +

n=0

62.

N 

Dn xn ,

n=0

where the constants Bn and Dn are found by the method of undetermined coefficients. To obtain the general solution we must add the solution (1) of the homogeneous equation. In Mikhailov (1966), solvability conditions for the integral equation in question were investigated for various classes of f (x). x λ y(t) dt y(x) – µ = f (x). x 0 (ax + bt)1–µ Here a > 0, b > 0, and µ is an arbitrary number. 1◦ . The solution of the homogeneous equation (f ≡ 0) is y(x) = Cxβ

(β > –1, λ > 0).

(1)

Here C is an arbitrary constant, and β = β(λ) is determined by the transcendental equation 1 z β (a + bz)µ–1 dz. (2) λI(β) = 1, where I(β) = 0

2◦ . For a polynomial right-hand side, f (x) =

N 

An xn

n=0

the solution bounded at zero is given by ⎧ N  ⎪ An ⎪ ⎪ ⎪ xn for λ < λ0 , ⎪ ⎨ 1 – (λ/λn ) n=0 y(x) = N ⎪  ⎪ An ⎪ ⎪ xn + Cxβ for λ > λ0 and λ ≠ λn , ⎪ ⎩ 1 – (λ/λn ) n=0 1 1 , I(n) = λn = z n (a + bz)µ–1 dz. I(n) 0 Here C is an arbitrary constant, and β = β(λ) is determined by the transcendental equation (2). 3◦ . For special λ = λn (n = 1, 2, . . . ), the solution differs in one term and has the form N  Am Am λ¯ n n xm + xm – An x ln x + Cxn , 1 – (λ /λ ) 1 – (λ /λ ) λ n m n m n m=0 m=n+1  1 –1 where λ¯ n = z n (a + bz)µ–1 ln z dz .

y(x) =

n–1 

0

4◦ . For arbitrary f (x) expandable into power series, the formulas of item 2◦ can be used, in which one should set N = ∞. In this case, the radius of convergence of the solution y(x) is equal to the radius of convergence of f (x).

144

LINEAR EQUATIONS OF THE SECOND KIND WITH VARIABLE LIMIT OF INTEGRATION

2.2. Equations Whose Kernels Contain Exponential Functions 2.2-1. Kernels Containing Exponential Functions. 1.

y(x) + A

x

eλ(x–t) y(t) dt = f (x).

a



Solution:

x

e(λ–A)(x–t) f (t) dt.

y(x) = f (x) – A a

2.

y(x) + A

x

eλx+βt y(t) dt = f (x).

a

This is a special case of equation 2.9.2 with g(x) = –Aeλx and h(t) = eβt . For β = –λ, see equation 2.2.1. Solution:

x A (λ+β)t (λ+β)x  λx+βt y(x) = f (x) – e . R(x, t)f (t) dt, R(x, t) = Ae exp –e λ+β a 3.

y(x) + A

x

 eλ(x–t) – 1 y(t) dt = f (x).

a

1◦ . Solution with D ≡ λ(λ – 4A) > 0: 2Aλ y(x) = f (x) – √ D



x

R(x – t)f (t) dt, a

   √  R(x) = exp 12 λx sinh 12 D x .

2◦ . Solution with D ≡ λ(λ – 4A) < 0: 2Aλ y(x) = f (x) – √ |D|



x

R(x – t)f (t) dt, a

     R(x) = exp 12 λx sin 12 |D| x .

3◦ . Solution with λ = 4A: 2

y(x) = f (x) – 4A

x

 (x – t) exp 2A(x – t) f (t) dt.

a

4.

x

y(x) +

 Aeλ(x–t) + B y(t) dt = f (x).

a

This is a special case of equation 2.2.10 with A1 = A, A2 = B, λ1 = λ, and λ2 = 0. 1◦ . The structure of the solution depends on the sign of the discriminant D ≡ (A – B – λ)2 + 4AB

(1)

µ2 + (A + B – λ)µ – Bλ = 0.

(2)

of the square equation 2◦ . If D > 0, then equation (2) has the real different roots √ √ µ1 = 12 (λ – A – B) + 12 D, µ2 = 12 (λ – A – B) – 12 D.

145

2.2. EQUATIONS WHOSE KERNELS CONTAIN EXPONENTIAL FUNCTIONS

In this case, the original integral equation has the solution y(x) = f (x) +

x

 E1 eµ1 (x–t) + E2 eµ2 (x–t) f (t) dt,

a

where E1 = A

µ1 µ1 – λ +B , µ2 – µ1 µ2 – µ1

E2 = A

µ2 µ2 – λ +B . µ1 – µ2 µ1 – µ2

3◦ . If D < 0, then equation (2) has the complex conjugate roots µ1 = σ + iβ,

σ = 12 (λ – A – B),

µ2 = σ – iβ,

β=

1 2

√ –D.

In this case, the original integral equation has the solution x  E1 eσ(x–t) cos[β(x – t)] + E2 eσ(x–t) sin[β(x – t)] f (t) dt, y(x) = f (x) + a

where E1 = –A – B,

5.

y(x) + A

x

E2 =

1 (–Aσ – Bσ + Bλ). β

(eλx – eλt )y(t) dt = f (x).

a

This is a special case of equation 2.9.5 with g(x) = Aeλx . Solution: x

  1 y(x) = f (x) + u1 (x)u2 (t) – u2 (x)u1 (t) f (t) dt, W a where the primes denote differentiation with respect to the argument specified in the parentheses, and u1 (x), u2 (x) is a fundamental system of solutions of the second-order linear homogeneous ordinary differential equation uxx + Aλeλx u = 0; the functions u1 (x) and u2 (x) are expressed in terms of Bessel functions or modified Bessel functions, depending on the sign of A: For Aλ > 0,  √   √  λ 2 Aλ λx/2 2 Aλ λx/2 W = , u1 (x) = J0 e e , u2 (x) = Y0 , π λ λ For Aλ < 0,  √   √  2 |Aλ| λx/2 2 |Aλ| λx/2 λ e e , u2 (x) = K0 . W = – , u1 (x) = I0 2 λ λ 6.

x

y(x) +

 λx  Ae + Beλt y(t) dt = f (x).

a

This is a special case of equation 2.9.6 with g(x) = Aeλx and h(t) = Beλt . For B = –A, see equation 2.2.5. x Differentiating the original integral equation followed by substituting Y (x) = y(t) dt yields the second-order linear ordinary differential equation  Yxx + (A + B)eλx Yx + Aλeλx Y = fx (x)

a

(1)

146

LINEAR EQUATIONS OF THE SECOND KIND WITH VARIABLE LIMIT OF INTEGRATION

under the initial conditions Y (a) = 0,

Yx (a) = f (a).

(2)

A fundamental system of solutions of the homogeneous equation (1) with f ≡ 0 has the form   A A m m Y1 (x) = Φ , 1; – eλx , Y2 (x) = Ψ , 1; – eλx , m = A + B, m λ m λ     where Φ α, β; x and Ψ α, β; x are degenerate hypergeometric functions. Solving the homogeneous equation (1) under conditions (2) for an arbitrary function f = f (x) and taking into account the relation y(x) = Yx (x), we thus obtain the solution of the integral equation in the form x y(x) = f (x) – R(x, t)f (t) dt, a

 m   Γ(A/m) ∂ 2 λt R(x, t) = exp e Y1 (x)Y2 (t) – Y2 (x)Y1 (t) . λ ∂x∂t λ 7.

y(x) + A

x

 eλ(x+t) – e2λt y(t) dt = f (x).

a

The transformation z = eλx , τ = eλt leads to an equation of the form 2.1.4. 1◦ . Solution with Aλ > 0:

 eλt sin k(eλx – eλt ) f (t) dt,

k=

 eλt sinh k(eλx – eλt ) f (t) dt,

k=

x

y(x) = f (x) – λk



A/λ.

a

2◦ . Solution with Aλ < 0: y(x) = f (x) + λk

x



|A/λ|.

a

8.

y(x) + A

x

 eλx+µt – e(λ+µ)t y(t) dt = f (x).

a

The transformation z = eµx , τ = eµt , Y (z) = y(x) leads to an equation of the form 2.1.52: A z k Y (z) + (z – τ k )Y (τ ) dτ = F (z), F (z) = f (x), µ b where k = λ/µ, b = eµa . 9.

y(x) + A

x

 λeλx+µt – (λ + µ)e(λ+µ)x y(t) dt = f (x).

a

This equation can be obtained by differentiating an equation of the form 1.2.22: x x

 1 + Aeλx (eµt – eµx ) y(t) dt = F (x), F (x) = f (t) dt. a

a

Solution:

 x dt d F (t) λx y(x) = e Φ(x) , dx eλt t Φ(t) a

  Aµ (λ+µ)x Φ(x) = exp e . λ+µ

2.2. EQUATIONS WHOSE KERNELS CONTAIN EXPONENTIAL FUNCTIONS

10.

x

y(x) + a

147

 A1 eλ1 (x–t) + A2 eλ2 (x–t) y(t) dt = f (x).

1◦ . Introduce the notation I1 =



x

e

λ1 (x–t)

y(t) dt,

a

x

eλ2 (x–t) y(t) dt.

I2 = a

Differentiating the integral equation twice yields (the first line is the original equation) y + A1 I1 + A2 I2 = f ,

f = f (x),

(1)

yx + (A1 + A2 )y + A1 λ1 I1 + A2 λ2 I2 = fx ,   + (A1 + A2 )yx + (A1 λ1 + A2 λ2 )y + A1 λ21 I1 + A2 λ22 I2 = fxx . yxx

(2) (3)

Eliminating I1 and I2 , we arrive at the second-order linear ordinary differential equation with constant coefficients   yxx + (A1 + A2 – λ1 – λ2 )yx + (λ1 λ2 – A1 λ2 – A2 λ1 )y = fxx – (λ1 + λ2 )fx + λ1 λ2 f .

(4)

Substituting x = a into (1) and (2) yields the initial conditions yx (a) = fx (a) – (A1 + A2 )f (a).

y(a) = f (a),

(5)

Solving the differential equation (4) under conditions (5), we can find the solution of the integral equation. 2◦ . Consider the characteristic equation µ2 + (A1 + A2 – λ1 – λ2 )µ + λ1 λ2 – A1 λ2 – A2 λ1 = 0

(6)

which corresponds to the homogeneous differential equation (4) (with f (x) ≡ 0). The structure of the solution of the integral equation depends on the sign of the discriminant D ≡ (A1 – A2 – λ1 + λ2 )2 + 4A1 A2 of the quadratic equation (6). If D > 0, the quadratic equation (6) has the real different roots √ √ µ1 = 12 (λ1 + λ2 – A1 – A2 ) + 12 D, µ2 = 12 (λ1 + λ2 – A1 – A2 ) – 12 D. In this case, the solution of the original integral equation has the form x

 y(x) = f (x) + B1 eµ1 (x–t) + B2 eµ2 (x–t) f (t) dt, a

where

µ1 – λ2 µ1 – λ1 µ2 – λ2 µ2 – λ1 + A2 , B2 = A1 + A2 . µ2 – µ1 µ2 – µ1 µ1 – µ2 µ1 – µ2 If D < 0, the quadratic equation (6) has the complex conjugate roots √ µ1 = σ + iβ, µ2 = σ – iβ, σ = 12 (λ1 + λ2 – A1 – A2 ), β = 12 –D. B1 = A1

In this case, the solution of the original integral equation has the form x   y(x) = f (x) + B1 eσ(x–t) cos[β(x – t)] + B2 eσ(x–t) sin[β(x – t)] f (t) dt, a

where B1 = –A1 – A2 ,

B2 =

 1 A1 (λ2 – σ) + A2 (λ1 – σ) . β

148

LINEAR EQUATIONS OF THE SECOND KIND WITH VARIABLE LIMIT OF INTEGRATION

11.

x

y(x) +

 Aeλ(x+t) – Ae2λt + Beλt y(t) dt = f (x).

a

The transformation z = eλx , τ = eλt , Y (z) = y(x) leads to an equation of the form 2.1.5:

z

Y (z) +

 B1 (z – τ ) + A1 Y (τ ) dτ = F (z),

F (z) = f (x),

b

where A1 = B/λ, B1 = A/λ, b = eλa . 12.

x

y(x) +

 Aeλ(x+t) + Be2λt + Ceλt y(t) dt = f (x).

a

The transformation z = eλx , τ = eλt , Y (z) = y(x) leads to an equation of the form 2.1.6:

z

Y (z) –

(A1 z + B1 τ + C1 )Y (τ ) dτ = F (z),

F (z) = f (x),

b

where A1 = –A/λ, B1 = –B/λ, C1 = –C/λ, b = eλa . 13.

x

y(x) +

  λeλ(x–t) + A µeµx+λt – λeλx+µt y(t) dt = f (x).

a

This is a special case of equation 2.9.23 with h(t) = A. Solution:

 x F (t) e2λt 1 d Φ(x) dt , eλx dx eλt t Φ(t) a   x λ – µ (λ+µ)x e , F (x) = Φ(x) = exp A f (t) dt. λ+µ a y(x) =

14.

x

y(x) –

  λe–λ(x–t) + A µeλx+µt – λeµx+λt y(t) dt = f (x).

a

This is a special case of equation 2.9.24 with h(x) = A. Assume that f (a) = 0. Solution:

x

w(t) dt,

y(x) = a

15.

x

y(x) +



 d e2λx x f (t) w(x) = e–λx Φ(t) dt , dx Φ(x) a eλt t   λ – µ (λ+µ)x e . Φ(x) = exp A λ+µ

  λeλ(x–t) + Aeβt µeµx+λt – λeλx+µt y(t) dt = f (x).

a

This is a special case of equation 2.9.23 with h(t) = Aeβt . Solution:

 x d F (t) e(2λ+β)t y(x) = e Φ(x) dt , dx eλt t Φ(t) a   x λ – µ (λ+µ+β)x e , F (x) = Φ(x) = exp A f (t) dt. λ+µ+β a –(λ+β)x

2.2. EQUATIONS WHOSE KERNELS CONTAIN EXPONENTIAL FUNCTIONS

16.

x

y(x) –

149

  λe–λ(x–t) + Aeβx µeλx+µt – λeµx+λt y(t) dt = f (x).

a

This is a special case of equation 2.9.24 with h(x) = Aeβx . Assume that f (a) = 0. Solution:

x

y(x) =

w(t) dt, a

17.

x

y(x) +



 d e(2λ+β)x x f (t) w(x) = e–λx Φ(t) dt , dx Φ(x) a e(λ+β)t t   λ – µ (λ+µ+β)x e . Φ(x) = exp A λ+µ+β

 ABe(λ+1)x+t – ABeλx+2t – Aeλx+t – Bet y(t) dt = f (x).

a

The transformation z = ex , τ = et , Y (z) = y(x) leads to an equation of the form 2.1.56: z   ABz λ+1 – ABz λ τ – Az λ – B Y (τ ) dτ = F (z), Y (z) + b

where F (z) = f (x) and b = ea . 18.

x

y(x) +

 ABex+λt – ABe(λ+1)t + Aeλt + Bet y(t) dt = f (x).

a

The transformation z = ex , τ = et , Y (z) = y(x) leads to an equation of the form 2.1.57 (in which λ is substituted by λ – 1): z   Y (z) + ABzτ λ–1 – ABτ λ + Aτ λ–1 + B Y (τ ) dτ = F (z), b

where F (z) = f (x) and b = ea . 19.

x

y(x) + a

 n

 Ak eλk (x–t) y(t) dt = f (x).

k=1



1 . This integral equation can be reduced to an nth-order linear nonhomogeneous ordinary differential equation with constant coefficients. Set x Ik (x) = eλk (x–t) y(t) dt. (1) a

Differentiating (1) with respect to x yields Ik = y(x) + λk



x

eλk (x–t) y(t) dt,

(2)

a

where the prime stands for differentiation with respect to x. From the comparison of (1) with (2) we see that Ik = y(x) + λk Ik , Ik = Ik (x). (3) The integral equation can be written in terms of Ik (x) as follows: y(x) +

n  k=1

Ak Ik = f (x).

(4)

150

LINEAR EQUATIONS OF THE SECOND KIND WITH VARIABLE LIMIT OF INTEGRATION

Differentiating (4) with respect to x and taking account of (3), we obtain yx (x) + σn y(x) +

n 

Ak λk Ik = fx (x),

σn =

k=1

n 

Ak .

(5)

k=1

Eliminating the integral In from (4) and (5), we find that n–1   Ak (λk – λn )Ik = fx (x) – λn f (x). yx (x) + σn – λn )y(x) +

(6)

k=1

Differentiating (6) with respect to x and eliminating In–1 from the resulting equation with the aid of (6), we obtain a similar equation whose left-hand side is a second-order linear n–2  1 differential operator (acting on y) with constant coefficients plus the sum Ak Ik . If we k=1

proceed with successively eliminating In–2 , In–3 , . . . , I1 with the aid of differentiation and formula (3), then we will finally arrive at an nth-order linear nonhomogeneous ordinary differential equation with constant coefficients. The initial conditions for y(x) can be obtained by setting x = a in the integral equation and all its derivative equations. 2◦ . The solution of the equation can be represented in the form y(x) = f (x) +

x  n a

 Bk e

µk (x–t)

f (t) dt.

(7)

k=1

The unknown constants µk are the roots of the algebraic equation n  Ak + 1 = 0, z – λk

(8)

k=1

which is reduced (by separating the numerator) to the problem of finding the roots of an nth-order characteristic polynomial. After the µk have been calculated, the coefficients Bk can be found from the following linear system of algebraic equations: n  k=1

Bk + 1 = 0, λm – µk

m = 1, . . . , n.

(9)

Another way of determining the Bk is presented in item 3◦ below. If all the roots µk of equation (8) are real and different, then the solution of the original integral equation can be calculated by formula (7). To a pair of complex conjugate roots µk,k+1 = α ± iβ of the characteristic polynomial (8) there corresponds a pair of complex conjugate coefficients Bk,k+1 in equation (9). In this case, µk (x–t) the corresponding terms + Bk+1 eµk+1 (x–t)

 Bk e α(x–t)  in solution (7) can be written in the form α(x–t) cos β(x – t) + B k+1 e sin β(x – t) , where B k and B k+1 are real coefficients. Bk e 3◦ . For a = 0, the solution of the original integral equation is given by y(x) = f (x) –

x

R(x – t)f (t) dt, 0

 R(x) = L–1 R(p) ,

(10)

151

2.2. EQUATIONS WHOSE KERNELS CONTAIN EXPONENTIAL FUNCTIONS

 where L–1 R(p) is the inverse Laplace transform of the function K(p) , R(p) = 1 + K(p)

n  Ak K(p) = . p – λk

(11)

k=1

The transform R(p) of the resolvent R(x) can be represented as a regular fractional function: Q(p) , P (p) = (p – µ1 )(p – µ2 ) . . . (p – µn ), R(p) = P (p) where Q(p) is a polynomial in p of degree < n. The roots µk of the polynomial P (p) coincide with the roots of equation (8). If all µk are real and different, then the resolvent can be determined by the formula R(x) =

n 

Bk eµk x ,

k=1

Bk =

Q(µk ) , P  (µk )

where the prime stands for differentiation. 2.2-2. Kernels Containing Power-Law and Exponential Functions. 20.

y(x) + A

x

xeλ(x–t) y(t) dt = f (x).

a



Solution:

x

y(x) = f (x) – A a

21.

y(x) + A

x

 x exp 12 A(t2 – x2 ) + λ(x – t) f (t) dt.

teλ(x–t) y(t) dt = f (x).

a



Solution:

x

y(x) = f (x) – A a

22.

y(x) + A

x

 t exp 12 A(t2 – x2 ) + λ(x – t) f (t) dt.

(x – t)eλt y(t) dt = f (x).

a

This is a special case of equation 2.9.4 with g(t) = Aeλt . Solution: x

 A y(x) = f (x) + u1 (x)u2 (t) – u2 (x)u1 (t) eλt f (t) dt, W a where u1 (x), u2 (x) is a fundamental system of solutions of the second-order linear homogeneous ordinary differential equation uxx + Aeλx u = 0; the functions u1 (x) and u2 (x) are expressed in terms of Bessel functions or modified Bessel functions, depending on sign A: λ W = , π λ W =– , 2

 √   √  2 A λx/2 2 A λx/2 u1 (x) = J0 e e , u2 (x) = Y0 λ λ √ √     2 |A| λx/2 2 |A| λx/2 u1 (x) = I0 e e , u2 (x) = K0 λ λ

for A > 0, for A < 0.

152 23.

LINEAR EQUATIONS OF THE SECOND KIND WITH VARIABLE LIMIT OF INTEGRATION

y(x) + A

x

(x – t)eλ(x–t) y(t) dt = f (x).

a

1◦ . Solution with A > 0:



eλ(x–t) sin[k(x – t)]f (t) dt,

k=

√ A.

eλ(x–t) sinh[k(x – t)]f (t) dt,

k=

√ –A.

x

y(x) = f (x) – k a ◦

2 . Solution with A < 0:



x

y(x) = f (x) + k a

24.

y(x) + A

x

(x – t)eλx+µt y(t) dt = f (x).

a

The substitution u(x) = e–λx y(x) leads to an equation of the form 2.2.22: x u(x) + A (x – t)e(λ+µ)t u(t) dt = f (x)e–λx . a

25.

x

y(x) –

(Ax + Bt + C)eλ(x–t) y(t) dt = f (x).

a

The substitution u(x) = e–λx y(x) leads to an equation of the form 2.1.6: x (Ax + Bt + C)u(t) dt = f (x)e–λx . u(x) – a

26.

y(x) + A

x

x2 eλ(x–t) y(t) dt = f (x).

a



Solution:

x

y(x) = f (x) – A a

27.

y(x) + A

x

 x2 exp 13 A(t3 – x3 ) + λ(x – t) f (t) dt.

xteλ(x–t) y(t) dt = f (x).

a



Solution:

x

y(x) = f (x) – A a

28.

y(x) + A

x

 xt exp 13 A(t3 – x3 ) + λ(x – t) f (t) dt.

t2 eλ(x–t) y(t) dt = f (x).

a



Solution:

x

y(x) = f (x) – A a

29.

y(x) + A

x

 t2 exp 13 A(t3 – x3 ) + λ(x – t) f (t) dt.

(x – t)2 eλ(x–t) y(t) dt = f (x).

a

Solution:



x

y(x) = f (x) –

R(x – t)f (t) dt, a

√  √ √  R(x) = 23 ke(λ–2k)x – 23 ke(λ+k)x cos 3 kx – 3 sin 3 kx ,

k=

 1 1/3 . 4A

2.2. EQUATIONS WHOSE KERNELS CONTAIN EXPONENTIAL FUNCTIONS

30.

y(x) + A

x

(x2 – t2 )eλ(x–t) y(t) dt = f (x).

0

The substitution u(x) = e–λx y(x) leads to an equation of the form 2.1.11: x u(x) + A (x2 – t2 )u(t) dt = f (x)e–λx . 0

31.

y(x) + A

x

(x – t)n eλ(x–t) y(t) dt = f (x),

n = 1, 2, . . .

a

Solution:



x

y(x) = f (x) +

R(x – t)f (t) dt, a

 1 λx  e exp(σk x) σk cos(βk x) – βk sin(βk x) , n+1 n

R(x) =

k=0

where

32.

 2πk   2πk  1 1 σk = |An!| n+1 cos , βk = |An!| n+1 sin for A < 0, n+1 n+1     1 1 2πk + π 2πk + π σk = |An!| n+1 cos , βk = |An!| n+1 sin for A > 0. n+1 n+1 x exp[λ(x – t)] y(x) + b y(t) dt = f (x). √ x–t a Solution:

x y(x) = eλx F (x) + πb2 exp[πb2 (x – t)]F (t) dt , a

where

F (x) = e

–λx

f (x) – b a

33.

y(x) + A

x

x

e–λt f (t) √ dt. x–t

(x – t)tk eλ(x–t) y(t) dt = f (x).

a

The substitution u(x) = e–λx y(x) leads to an equation of the form 2.1.49: x u(x) + A (x – t)tk u(t) dt = f (x)e–λx . a

34.

y(x) + A

x

(xk – tk )eλ(x–t) y(t) dt = f (x).

a

The substitution u(x) = e–λx y(x) leads to an equation of the form 2.1.52: x u(x) + A (xk – tk )u(t) dt = f (x)e–λx . a

35.

y(x) – λ 0

x

eµ(x–t) (x – t)α

Solution:



y(t) dt = f (x),

0 < α < 1.

x

y(x) = f (x) +

R(x – t)f (t) dt, 0

where R(x) = e

µx

n ∞  λΓ(1 – α)x1–α

 . xΓ n(1 – α) n=1

153

154 36.

LINEAR EQUATIONS OF THE SECOND KIND WITH VARIABLE LIMIT OF INTEGRATION

y(x) + A

x

 exp λ(x2 – t2 ) y(t) dt = f (x).

a

Solution:



x

y(x) = f (x) – A

 exp λ(x2 – t2 ) – A(x – t) f (t) dt.

a

37.

y(x) + A

x

  exp λx2 + βt2 y(t) dt = f (x).

a

In the case β = –λ, see equation 2.2.36. This is a special case of equation 2.9.2 with  g(x) = –A exp λx2 ) and h(t) = exp βt2 . 38.

y(x) + A



 √  exp –λ t – x y(t) dt = f (x).

x

 √  This is a special case of equation 2.9.62 with K(x) = A exp –λ –x . 39.

y(x) + A

x

 exp λ(xµ – tµ ) y(t) dt = f (x),

µ > 0.

a

    This is a special case of equation 2.9.2 with g(x) = –A exp λxµ and h(t) = exp –λtµ . Solution: x

 y(x) = f (x) – A exp λ(xµ – tµ ) – A(x – t) f (t) dt. a

40.

x

y(x) + k 0

 t 1 exp –λ y(t) dt = g(x). x x

This is a special case of equation 2.9.71 with f (z) = ke–λz . N  For a polynomial right-hand side, g(x) = An xn , a solution is given by n=0

y(x) =

N  n=0

An xn , 1 + kBn

 n! 1 n! –λ – e . λn+1 k! λn–k+1 n

Bn =

k=0

2.3. Equations Whose Kernels Contain Hyperbolic Functions 2.3-1. Kernels Containing Hyperbolic Cosine.

1.

y(x) – A

x

cosh(λx)y(t) dt = f (x).

a

This is a special case of equation 2.9.2 with g(x) = A cosh(λx) and h(t) = 1. Solution:

x

y(x) = f (x) + A a

A  sinh(λx) – sinh(λt) f (t) dt. cosh(λx) exp λ

2.3. EQUATIONS WHOSE KERNELS CONTAIN HYPERBOLIC FUNCTIONS

2.

y(x) – A

155

x

cosh(λt)y(t) dt = f (x).

a

This is a special case of equation 2.9.2 with g(x) = A and h(t) = cosh(λt). Solution: x A  y(x) = f (x) + A sinh(λx) – sinh(λt) f (t) dt. cosh(λt) exp λ a 3.

y(x) + A

x

cosh[λ(x – t)] y(t) dt = f (x).

a

This is a special case of equation 2.9.28 with g(t) = A. Therefore, solving the original integral equation is reduced to solving the second-order linear nonhomogeneous ordinary differential equation with constant coefficients   yxx + Ayx – λ2 y = fxx – λ2 f ,

f = f (x),

under the initial conditions yx (a) = fx (a) – Af (a).

y(a) = f (a), Solution:

x y(x) = f (x) + R(x – t)f (t) dt, a      A2 R(x) = exp – 21 Ax sinh(kx) – A cosh(kx) , k = λ2 + 14 A2 . 2k 4.

x

y(x) +

 n

a

Ak cosh[λk (x – t)] y(t) dt = f (x).

k=1

This equation  can be  reduced to an equation of the form 2.2.19 by using the identity cosh z ≡ 12 ez + e–z . Therefore, the integral equation in question can be reduced to a linear nonhomogeneous ordinary differential equation of order 2n with constant coefficients. 5.

y(x) – A

x

a

cosh(λx) cosh(λt)

y(t) dt = f (x).

Solution:

x

y(x) = f (x) + A

eA(x–t)

cosh(λx) f (t) dt. cosh(λt)

eA(x–t)

cosh(λt) f (t) dt. cosh(λx)

a

6.

y(x) – A

x

a

cosh(λt) cosh(λx)

y(t) dt = f (x).

Solution:

x

y(x) = f (x) + A a

7.

y(x) – A

x

coshk (λx) coshm (µt)y(t) dt = f (x).

a

This is a special case of equation 2.9.2 with g(x) = A coshk (λx) and h(t) = coshm (µt).

156 8.

LINEAR EQUATIONS OF THE SECOND KIND WITH VARIABLE LIMIT OF INTEGRATION

y(x) + A

x

t cosh[λ(x – t)] y(t) dt = f (x).

a

This is a special case of equation 2.9.28 with g(t) = At. 9.

y(x) + A

x

tk coshm (λx)y(t) dt = f (x).

a

This is a special case of equation 2.9.2 with g(x) = –A coshm (λx) and h(t) = tk . 10.

y(x) + A

x

xk coshm (λt)y(t) dt = f (x).

a

This is a special case of equation 2.9.2 with g(x) = –Axk and h(t) = coshm (λt). 11.

x

y(x) –

 A cosh(kx) + B – AB(x – t) cosh(kx) y(t) dt = f (x).

a

This is a special case of equation 2.9.7 with λ = B and g(x) = A cosh(kx). Solution: x y(x) = f (x) + R(x, t)f (t) dt, a   x G(x) B2 A R(x, t) = [A cosh(kx) + B] + sinh(kx) . eB(x–s) G(s) ds, G(x) = exp G(t) G(t) t k 12.

x

y(x) +

 A cosh(kt) + B + AB(x – t) cosh(kt) y(t) dt = f (x).

a

This is a special case of equation 2.9.8 with λ = B and g(t) = A cosh(kt). Solution: x y(x) = f (x) + R(x, t)f (t) dt, a   x G(t) B2 A B(t–s) R(x, t) = –[A cosh(kt) + B] + sinh(kx) . e G(s) ds, G(x) = exp G(x) G(x) t k 13.

y(x) + A



 √  cosh λ t – x y(t) dt = f (x).

x

 √  This is a special case of equation 2.9.62 with K(x) = A cosh λ –x .

2.3-2. Kernels Containing Hyperbolic Sine. 14.

y(x) – A

x

sinh(λx)y(t) dt = f (x).

a

This is a special case of equation 2.9.2 with g(x) = A sinh(λx) and h(t) = 1. Solution:

x  A cosh(λx) – cosh(λt) f (t) dt. sinh(λx) exp y(x) = f (x) + A λ a

157

2.3. EQUATIONS WHOSE KERNELS CONTAIN HYPERBOLIC FUNCTIONS

15.

y(x) – A

x

sinh(λt)y(t) dt = f (x).

a

This is a special case of equation 2.9.2 with g(x) = A and h(t) = sinh(λt). Solution:

x  A cosh(λx) – cosh(λt) f (t) dt. sinh(λt) exp y(x) = f (x) + A λ a 16.

y(x) + A

x

sinh[λ(x – t)] y(t) dt = f (x). a

This is a special case of equation 2.9.30 with g(x) = A. 1◦ . Solution with λ(A – λ) > 0: Aλ x y(x) = f (x) – sin[k(x – t)]f (t) dt, k a

where k =

2◦ . Solution with λ(A – λ) < 0: Aλ x y(x) = f (x) – sinh[k(x – t)]f (t) dt, k a

where k =



λ(A – λ).



λ(λ – A).

3◦ . Solution with A = λ: 2

x

(x – t)f (t) dt.

y(x) = f (x) – λ

a

17.

y(x) + A

x

sinh3 [λ(x – t)] y(t) dt = f (x).

a

Using the formula sinh3 β = y(x) + a

18.

x

y(x) + a



x

1 4

sinh 3β –

1 4 A sinh

3 4

sinh β, we arrive at an equation of the form 2.3.18:

  3λ(x – t) – 34 A sinh[λ(x – t)] y(t) dt = f (x).

 A1 sinh[λ1 (x – t)] + A2 sinh[λ2 (x – t)] y(t) dt = f (x).



1 . Introduce the notation x x sinh[λ1 (x – t)] y(t) dt, I2 = sinh[λ2 (x – t)] y(t) dt, I1 = a a x x J1 = cosh[λ1 (x – t)] y(t) dt, J2 = cosh[λ2 (x – t)] y(t) dt. a

a

Successively differentiating the integral equation four times yields (the first line is the original equation) y + A1 I1 + A2 I2 = f ,

f = f (x),

yx + A1 λ1 J1 + A2 λ2 J2 = fx ,   + (A1 λ1 + A2 λ2 )y + A1 λ21 I1 + A2 λ22 I2 = fxx , yxx   3 3  , yxxx + (A1 λ1 + A2 λ2 )yx + A1 λ1 J1 + A2 λ2 J2 = fxxx   3 3 yxxxx + (A1 λ1 + A2 λ2 )yxx + (A1 λ1 + A2 λ2 )y + A1 λ41 I1

(1) (2) (3)  + A2 λ42 I2 = fxxxx .

(4) (5)

158

LINEAR EQUATIONS OF THE SECOND KIND WITH VARIABLE LIMIT OF INTEGRATION

Eliminating I1 and I2 from (1), (3), and (5), we arrive at a fourth-order linear ordinary differential equation with constant coefficients:   yxxxx – (λ21 + λ22 – A1 λ1 – A2 λ2 )yxx + (λ21 λ22 – A1 λ1 λ22 – A2 λ21 λ2 )y =   fxxxx – (λ21 + λ22 )fxx + λ21 λ22 f .

(6)

The initial conditions can be obtained by setting x = a in (1)–(4): y(a) = f (a), yx (a) = fx (a),   (a) = fxx (a) – (A1 λ1 + A2 λ2 )f (a), yxx

(7)

  (a) = fxxx (a) – (A1 λ1 + A2 λ2 )fx (a). yxxx

On solving the differential equation (6) under conditions (7), we thus find the solution of the integral equation. 2◦ . Consider the characteristic equation z 2 – (λ21 + λ22 – A1 λ1 – A2 λ2 )z + λ21 λ22 – A1 λ1 λ22 – A2 λ21 λ2 = 0,

(8)

whose roots, z1 and z2 , determine the solution structure of the integral equation. Assume that the discriminant of equation (8) is positive: D ≡ (A1 λ1 – A2 λ2 – λ21 + λ22 )2 + 4A1 A2 λ1 λ2 > 0. In this case, the quadratic equation (8) has the real (different) roots √ √ z1 = 12 (λ21 + λ22 – A1 λ1 – A2 λ2 ) + 12 D, z2 = 12 (λ21 + λ22 – A1 λ1 – A2 λ2 ) – 12 D. Depending on the signs of z1 and z2 the following three cases are possible. Case 1. If z1 > 0 and z2 > 0, then the solution of the integral equation has the form (i = 1, 2): x

 √ y(x) = f (x) + {B1 sinh[µ1 (x – t)] + B2 sinh µ2 (x – t) f (t) dt, µi = zi , a

where B1 = A1

λ1 (µ21 – λ22 ) λ2 (µ21 – λ21 ) + A , 2 µ1 (µ22 – µ21 ) µ1 (µ22 – µ21 )

B2 = A1

λ1 (µ22 – λ22 ) λ2 (µ22 – λ21 ) + A . 2 µ2 (µ21 – µ22 ) µ2 (µ21 – µ22 )

Case 2. If z1 < 0 and z2 < 0, then the solution of the integral equation has the form x 

 y(x) = f (x) + {B1 sin[µ1 (x – t)] + B2 sin µ2 (x – t) f (t) dt, µi = |zi |, a

where the coefficients B1 and B2 are found by solving the following system of linear algebraic equations: B2 µ2 B1 µ1 B2 µ2 B1 µ1 + 2 + 1 = 0, + 2 + 1 = 0. 2 2 2 2 2 λ1 + µ1 λ1 + µ2 λ2 + µ1 λ2 + µ22 Case 3. If z1 > 0 and z2 < 0, then the solution of the integral equation has the form x 

 y(x) = f (x) + {B1 sinh[µ1 (x – t)] + B2 sin µ2 (x – t) f (t) dt, µi = |zi |, a

where B1 and B2 are determined from the following system of linear algebraic equations: B1 µ1 B2 µ2 + 2 + 1 = 0, 2 2 λ1 – µ1 λ1 + µ22

B1 µ1 B2 µ2 + 2 + 1 = 0. 2 2 λ2 – µ1 λ2 + µ22

159

2.3. EQUATIONS WHOSE KERNELS CONTAIN HYPERBOLIC FUNCTIONS

19.

x

y(x) +

 n

a

Ak sinh[λk (x – t)] y(t) dt = f (x).

k=1



1 . This equation can   be reduced to an equation of the form 2.2.19 with the aid of the formula sinh z = 12 ez – e–z . Therefore, the original integral equation can be reduced to a linear nonhomogeneous ordinary differential equation of order 2n with constant coefficients. 2◦ . Let us find the roots zk of the algebraic equation n  λk Ak + 1 = 0. z – λ2k

(1)

k=1

By reducing it to a common denominator, we arrive at the problem of determining the roots of an nth-degree characteristic polynomial. Assume that all zk are real, different, and nonzero. Let us divide the roots into two groups z1 > 0, zs+1 < 0,

z2 > 0, zs+2 < 0,

..., ...,

zs > 0 zn < 0

(positive roots); (negative roots).

Then the solution of the integral equation can be written in the form y(x) = f (x)+

x  s a

n

 

 Bk sinh µk (x–t) + Ck sin µk (x–t) f (t) dt,

k=1

µk =

 |zk |. (2)

k=s+1

The coefficients Bk and Ck are determined from the following system of linear algebraic equations: s n   Bk µk Ck µk + + 1 = 0, 2 – µ2 2 + µ2 λ λ m k k k=0 k=s+1 m

µk =

 |zk |,

m = 1, . . . , n.

(3)

In the case of a nonzero root zs = 0, we can introduce the new constant D = Bs µs and proceed to the limit  µs → 0. As a result, the term D(x – t) appears in solution (2) instead of Bs sinh µs (x – t) and the corresponding terms Dλ–2 m appear in system (3). 20.

y(x) – A

x

a

sinh(λx) sinh(λt)

y(t) dt = f (x).

Solution:

x

y(x) = f (x) + A

eA(x–t)

sinh(λx) f (t) dt. sinh(λt)

eA(x–t)

sinh(λt) f (t) dt. sinh(λx)

a

21.

y(x) – A

x a

sinh(λt) y(t) dt = f (x). sinh(λx)

Solution:

x

y(x) = f (x) + A a

22.

y(x) – A

x

sinhk (λx) sinhm (µt)y(t) dt = f (x).

a

This is a special case of equation 2.9.2 with g(x) = A sinhk (λx) and h(t) = sinhm (µt).

160 23.

LINEAR EQUATIONS OF THE SECOND KIND WITH VARIABLE LIMIT OF INTEGRATION

y(x) + A

x

t sinh[λ(x – t)] y(t) dt = f (x).

a

This is a special case of equation 2.9.30 with g(t) = At. Solution:  Aλ x y(x) = f (x) + t u1 (x)u2 (t) – u2 (x)u1 (t) f (t) dt, W a where u1 (x), u2 (x) is a fundamental system of solutions of the second-order linear ordinary differential equation uxx + λ(Ax – λ)u = 0, and W is the Wronskian. The functions u1 (x) and u2 (x) are expressed in terms of Bessel functions or modified Bessel functions, depending on the sign of Aλ, as follows: if Aλ > 0, then  √   √  u1 (x) = ξ 1/2 J1/3 23 Aλ ξ 3/2 , u2 (x) = ξ 1/2 Y1/3 23 Aλ ξ 3/2 , W = 3/π, ξ = x – (λ/A); if Aλ < 0, then u1 (x) = ξ 1/2 I1/3

2√  3/2 , 3 –Aλ ξ W = – 23 ,

24.

y(x) + A

u2 (x) = ξ 1/2 K1/3

2√  3/2 , 3 –Aλ ξ

ξ = x – (λ/A).

x

x sinh[λ(x – t)] y(t) dt = f (x).

a

This is a special case of equation 2.9.31 with g(x) = Ax and h(t) = 1. Solution:  Aλ x y(x) = f (x) + x u1 (x)u2 (t) – u2 (x)u1 (t) f (t) dt, W a

25.

where u1 (x), u2 (x) is a fundamental system of solutions of the second-order linear ordinary differential equation uxx + λ(Ax – λ)u = 0, and W is the Wronskian. The functions u1 (x), u2 (x), and W are specified in 2.3.23. x tk sinhm (λx)y(t) dt = f (x). y(x) + A

26.

This is a special case of equation 2.9.2 with g(x) = –A sinhm (λx) and h(t) = tk . x y(x) + A xk sinhm (λt)y(t) dt = f (x).

27.

This is a special case of equation 2.9.2 with g(x) = –Axk and h(t) = sinhm (λt). x

 A sinh(kx) + B – AB(x – t) sinh(kx) y(t) dt = f (x). y(x) –

a

a

a

This is a special case of equation 2.9.7 with λ = B and g(x) = A sinh(kx). Solution: x y(x) = f (x) + R(x, t)f (t) dt, a   x G(x) B2 A R(x, t) = [A sinh(kx) + B] + cosh(kx) . eB(x–s) G(s) ds, G(x) = exp G(t) G(t) t k

161

2.3. EQUATIONS WHOSE KERNELS CONTAIN HYPERBOLIC FUNCTIONS

28.

x

y(x) +

 A sinh(kt) + B + AB(x – t) sinh(kt) y(t) dt = f (x).

a

This is a special case of equation 2.9.8 with λ = B and g(t) = A sinh(kt). Solution: x y(x) = f (x) + R(x, t)f (t) dt, a   x G(t) B2 A R(x, t) = –[sinh(kt) + B] + cosh(kx) . eB(t–s) G(s) ds, G(x) = exp G(x) G(x) t k 29.

y(x) + A



 √  sinh λ t – x y(t) dt = f (x).

x

 √  This is a special case of equation 2.9.62 with K(x) = A sinh λ –x .

2.3-3. Kernels Containing Hyperbolic Tangent. 30.

y(x) – A

x

tanh(λx)y(t) dt = f (x).

a

This is a special case of equation 2.9.2 with g(x) = A tanh(λx) and h(t) = 1. Solution: A/λ  x cosh(λx) y(x) = f (x) + A tanh(λx) f (t) dt. cosh(λt) a 31.

y(x) – A

x

tanh(λt)y(t) dt = f (x). a

This is a special case of equation 2.9.2 with g(x) = A and h(t) = tanh(λt). Solution: A/λ  x cosh(λx) y(x) = f (x) + A tanh(λt) f (t) dt. cosh(λt) a 32.

y(x) + A

x

 tanh(λx) – tanh(λt) y(t) dt = f (x).

a

This is a special case of equation 2.9.5 with g(x) = A tanh(λx). Solution: x

  1 Y1 (x)Y2 (t) – Y2 (x)Y1 (t) f (t) dt, y(x) = f (x) + W a where Y1 (x), Y2 (x) is a fundamental system of solutions of the second-order linear ordinary  differential equation cosh2 (λx)Yxx + AλY = 0, W is the Wronskian, and the primes stand for the differentiation with respect to the argument specified in the parentheses. As shown in A. D. Polyanin and V. F. Zaitsev (2003), the functions Y1 (x) and Y2 (x) can be represented in the form  Y1 (x) = F α, β, 1;

eλx  , 1 + eλx

Y2 (x) = Y1 (x) a

x

dξ , Y12 (ξ)

W = 1,

where F (α, β, γ; z) is the hypergeometric function, in which α and β are determined from the algebraic system α + β = 1, αβ = –A/λ.

162 33.

LINEAR EQUATIONS OF THE SECOND KIND WITH VARIABLE LIMIT OF INTEGRATION

y(x) – A

x

a

tanh(λx) y(t) dt = f (x). tanh(λt)

Solution:

x

y(x) = f (x) + A

eA(x–t)

tanh(λx) f (t) dt. tanh(λt)

eA(x–t)

tanh(λt) f (t) dt. tanh(λx)

a

34.

y(x) – A

x

tanh(λt) tanh(λx)

a

y(t) dt = f (x).

Solution:

x

y(x) = f (x) + A a

35.

y(x) – A

x

tanhk (λx) tanhm (µt)y(t) dt = f (x).

a

36.

This is a special case of equation 2.9.2 with g(x) = A tanhk (λx) and h(t) = tanhm (µt). x y(x) + A tk tanhm (λx)y(t) dt = f (x). a

37.

This is a special case of equation 2.9.2 with g(x) = –A tanhm (λx) and h(t) = tk . x y(x) + A xk tanhm (λt)y(t) dt = f (x).

38.

This is a special case of equation 2.9.2 with g(x) = –Axk and h(t) = tanhm (λt). ∞ y(x) + A tanh[λ(t – x)] y(t) dt = f (x).

39.

This is a special case of equation 2.9.62 with K(z) = A tanh(–λz). ∞  √  y(x) + A tanh λ t – x y(t) dt = f (x).

a

x

x

 √  This is a special case of equation 2.9.62 with K(z) = A tanh λ –z .

x

 A tanh(kx) + B – AB(x – t) tanh(kx) y(t) dt = f (x).

40.

y(x) –

41.

This is a special case of equation 2.9.7 with λ = B and g(x) = A tanh(kx). x

 y(x) + A tanh(kt) + B + AB(x – t) tanh(kt) y(t) dt = f (x).

a

a

This is a special case of equation 2.9.8 with λ = B and g(t) = A tanh(kt). 2.3-4. Kernels Containing Hyperbolic Cotangent. 42.

y(x) – A

x

coth(λx)y(t) dt = f (x).

a

This is a special case of equation 2.9.2 with g(x) = A coth(λx) and h(t) = 1. Solution: x  sinh(λx) A/λ y(x) = f (x) + A coth(λx) f (t) dt. sinh(λt) a

2.3. EQUATIONS WHOSE KERNELS CONTAIN HYPERBOLIC FUNCTIONS

43.

y(x) – A

x

coth(λt)y(t) dt = f (x).

a

This is a special case of equation 2.9.2 with g(x) = A and h(t) = coth(λt). Solution: x  sinh(λx) A/λ y(x) = f (x) + A coth(λt) f (t) dt. sinh(λt) a 44.

y(x) – A

x

a

coth(λt) y(t) dt = f (x). coth(λx)

Solution:

x

y(x) = f (x) + A

eA(x–t)

coth(λt) f (t) dt. coth(λx)

eA(x–t)

coth(λx) f (t) dt. coth(λt)

a

45.

y(x) – A

x

coth(λx) coth(λt)

a

y(t) dt = f (x).

Solution:

x

y(x) = f (x) + A a

46.

y(x) – A

x

cothk (λx) cothm (µt)y(t) dt = f (x).

a

This is a special case of equation 2.9.2 with g(x) = A cothk (λx) and h(t) = cothm (µt). 47.

y(x) + A

x

tk cothm (λx)y(t) dt = f (x).

a

This is a special case of equation 2.9.2 with g(x) = –A cothm (λx) and h(t) = tk . 48.

y(x) + A

x

xk cothm (λt)y(t) dt = f (x).

a

This is a special case of equation 2.9.2 with g(x) = –Axk and h(t) = cothm (λt). 49.

y(x) + A



coth[λ(t – x)] y(t) dt = f (x). x

This is a special case of equation 2.9.62 with K(z) = A coth(–λz). 50.

y(x) + A



 √  coth λ t – x y(t) dt = f (x).

x

 √  This is a special case of equation 2.9.62 with K(z) = A coth λ –z . 51.

x

y(x) –

 A coth(kx) + B – AB(x – t) coth(kx) y(t) dt = f (x).

a

This is a special case of equation 2.9.7 with λ = B and g(x) = A coth(kx). 52.

x

y(x) +

 A coth(kt) + B + AB(x – t) coth(kt) y(t) dt = f (x).

a

This is a special case of equation 2.9.8 with λ = B and g(t) = A coth(kt).

163

164

LINEAR EQUATIONS OF THE SECOND KIND WITH VARIABLE LIMIT OF INTEGRATION

2.3-5. Kernels Containing Combinations of Hyperbolic Functions. 53.

y(x) – A

x

coshk (λx) sinhm (µt)y(t) dt = f (x).

a

54.

This is a special case of equation 2.9.2 with g(x) = A coshk (λx) and h(t) = sinhm (µt). x   A + B cosh(λx) + B(x – t)[λ sinh(λx) – A cosh(λx)] y(t) dt = f (x). y(x) –

55.

This is a special case of equation 2.9.32 with b = B and g(x) = A. x   A + B sinh(λx) + B(x – t)[λ cosh(λx) – A sinh(λx)] y(t) dt = f (x). y(x) –

56.

This is a special case of equation 2.9.33 with b = B and g(x) = A. x tanhk (λx) cothm (µt)y(t) dt = f (x). y(x) – A

a

a

a

This is a special case of equation 2.9.2 with g(x) = A tanhk (λx) and h(t) = cothm (µt).

2.4. Equations Whose Kernels Contain Logarithmic Functions 2.4-1. Kernels Containing Logarithmic Functions. 1.

y(x) – A

x

ln(λx)y(t) dt = f (x).

a

2.

This is a special case of equation 2.9.2 with g(x) = A ln(λx) and h(t) = 1. Solution: x (λx)Ax y(x) = f (x) + A ln(λx)e–A(x–t) f (t) dt. (λt)At a x y(x) – A ln(λt)y(t) dt = f (x).

3.

This is a special case of equation 2.9.2 with g(x) = A and h(t) = ln(λt). Solution: x (λx)Ax y(x) = f (x) + A ln(λt)e–A(x–t) f (t) dt. (λt)At a x y(x) + A (ln x – ln t)y(t) dt = f (x).

a

a

This is a special case of equation 2.9.5 with g(x) = A ln x. Solution: x

  1 y(x) = f (x) + u1 (x)u2 (t) – u2 (x)u1 (t) f (t) dt, W a where the primes denote differentiation with respect to the argument specified in the parentheses; and u1 (x), u2 (x) is a fundamental system of solutions of the second-order linear homogeneous ordinary differential equation uxx + Ax–1 u = 0, with u1 (x) and u2 (x) expressed in terms of Bessel functions or modified Bessel functions, depending on the sign of A:  √   √  √ √ W = π1 , u1 (x) = x J1 2 Ax , u2 (x) = x Y1 2 Ax for A > 0, √ √     √ √ W = – 12 , u1 (x) = x I1 2 –Ax , u2 (x) = x K1 2 –Ax for A < 0.

2.4. EQUATIONS WHOSE KERNELS CONTAIN LOGARITHMIC FUNCTIONS

4.

y(x) – A

x

a

ln(λx) y(t) dt = f (x). ln(λt)

Solution:

x

y(x) = f (x) + A

eA(x–t)

ln(λx) f (t) dt. ln(λt)

eA(x–t)

ln(λt) f (t) dt. ln(λx)

a

5.

y(x) – A

x

ln(λt) ln(λx)

a

y(t) dt = f (x).

Solution:

x

y(x) = f (x) + A a

6.

y(x) – A

x

lnk (λx) lnm (µt)y(t) dt = f (x).

a

This is a special case of equation 2.9.2 with g(x) = A lnk (λx) and h(t) = lnm (µt). 7.



ln(t – x)y(t) dt = f (x).

y(x) + a x

This is a special case of equation 2.9.62 with K(x) = a ln(–x). m  Ak exp(–λk x), where λk > 0, a solution of the equation has the form For f (x) = k=1

y(x) =

m  Ak exp(–λk x), Bk

Bk = 1 –

k=1

8.

a (ln λk + C), λk

where C = 0.5772 . . . is the Euler constant. ∞ y(x) + a ln2 (t – x)y(t) dt = f (x). x

This is a special case of equation 2.9.62 with K(x) = a ln2 (–x). m  For f (x) = Ak exp(–λk x), where λk > 0, a solution of the equation has the form k=1

y(x) =

m  Ak exp(–λk x), Bk k=1

Bk = 1 +

 a 1 2 π + (ln λk + C)2 , λk 6

where C = 0.5772 . . . is the Euler constant. 2.4-2. Kernels Containing Power-Law and Logarithmic Functions. 9.

y(x) – A

x

xk lnm (λt)y(t) dt = f (x).

a

This is a special case of equation 2.9.2 with g(x) = Axk and h(t) = lnm (λt). 10.

y(x) – A

x

tk lnm (λx)y(t) dt = f (x).

a

This is a special case of equation 2.9.2 with g(x) = A lnm (λx) and h(t) = tk .

165

166

LINEAR EQUATIONS OF THE SECOND KIND WITH VARIABLE LIMIT OF INTEGRATION



x

 A ln(kx) + B – AB(x – t) ln(kx) y(t) dt = f (x).

11.

y(x) –

12.

This is a special case of equation 2.9.7 with λ = B and g(x) = A ln(kx). x

 y(x) + A ln(kt) + B + AB(x – t) ln(kt) y(t) dt = f (x).

a

a

13.

This is a special case of equation 2.9.8 with λ = B and g(t) = A ln(kt). ∞ y(x) + a (t – x)n ln(t – x)y(t) dt = f (x), n = 1, 2, . . . x m 

For f (x) =

Ak exp(–λk x), where λk > 0, a solution of the equation has the form

k=1 m  Ak exp(–λk x), Bk

y(x) =

Bk = 1 +

k=1

14.

an!  1+ λn+1 k

1 2

+

1 3

+ ···+

1 n

 – ln λk – C ,

where C = 0.5772 . . . is the Euler constant. ∞ ln(t – x) y(x) + a y(t) dt = f (x). √ t–x x This is a special case of equation 2.9.62 with K(–x) = ax–1/2 ln x. m  For f (x) = Ak exp(–λk x), where λk > 0, a solution of the equation has the form k=1

y(x) =

m  Ak exp(–λk x), Bk k=1

 Bk = 1 – a

 π ln(4λk ) + C , λk

where C = 0.5772 . . . is the Euler constant.

2.5. Equations Whose Kernels Contain Trigonometric Functions 2.5-1. Kernels Containing Cosine. 1.

y(x) – A

x

cos(λx)y(t) dt = f (x).

a

2.

This is a special case of equation 2.9.2 with g(x) = A cos(λx) and h(t) = 1. Solution: x A  y(x) = f (x) + A sin(λx) – sin(λt) f (t) dt. cos(λx) exp λ a x y(x) – A cos(λt)y(t) dt = f (x). a

This is a special case of equation 2.9.2 with g(x) = A and h(t) = cos(λt). Solution: x A  y(x) = f (x) + A sin(λx) – sin(λt) f (t) dt. cos(λt) exp λ a

2.5. EQUATIONS WHOSE KERNELS CONTAIN TRIGONOMETRIC FUNCTIONS

3.

y(x) + A

167

x

cos[λ(x – t)] y(t) dt = f (x).

a

This is a special case of equation 2.9.34 with g(t) = A. Therefore, solving this integral equation is reduced to solving the following second-order linear nonhomogeneous ordinary differential equation with constant coefficients:   yxx + Ayx + λ2 y = fxx + λ2 f ,

f = f (x),

with the initial conditions yx (a) = fx (a) – Af (a).

y(a) = f (a), 1◦ . Solution with |A| > 2|λ|:

x y(x) = f (x) + R(x – t)f (t) dt, a      A2 R(x) = exp – 12 Ax sinh(kx) – A cosh(kx) , k = 14 A2 – λ2 . 2k 2◦ . Solution with |A| < 2|λ|:

x

y(x) = f (x) +

R(x – t)f (t) dt, a

   A2 R(x) = exp – 12 Ax sin(kx) – A cos(kx) , 2k

 k=

λ2 – 14 A2 .

3◦ . Solution with λ = ± 12 A:

   R(x) = exp – 21 Ax 12 A2 x – A .

x

R(x – t)f (t) dt,

y(x) = f (x) + a

4.

x

y(x) + a

 n

Ak cos[λk (x – t)] y(t) dt = f (x).

k=1

This integral equation is reduced to a linear nonhomogeneous ordinary differential equation of order 2n with constant coefficients. Set x Ik (x) = cos[λk (x – t)] y(t) dt. (1) a

Differentiating (1) with respect to x twice yields Ik = y(x) – λk



x

sin[λk (x – t)] y(t) dt, a

Ik = yx (x) – λ2k

(2)

x

cos[λk (x – t)] y(t) dt, a

where the primes stand for differentiation with respect to x. Comparing (1) and (2), we see that Ik = yx (x) – λ2k Ik , Ik = Ik (x). (3)

168

LINEAR EQUATIONS OF THE SECOND KIND WITH VARIABLE LIMIT OF INTEGRATION

With the aid of (1), the integral equation can be rewritten in the form y(x) +

n 

Ak Ik = f (x).

(4)

k=1

Differentiating (4) with respect to x twice taking into account (3) yields  yxx (x) + σn yx (x) –

n 

 Ak λ2k Ik = fxx (x),

σn =

k=1

n 

Ak .

(5)

k=1

Eliminating the integral In from (4) and (5), we obtain  (x) + σn yx (x) + λ2n y(x) + yxx

n–1 

 Ak (λ2n – λ2k )Ik = fxx (x) + λ2n f (x).

(6)

k=1

Differentiating (6) with respect to x twice followed by eliminating In–1 from the resulting expression with the aid of (6) yields a similar equation whose left-hand side is a fourthn–2  order differential operator (acting on y) with constant coefficients plus the sum Bk Ik . k=1

Successively eliminating the terms In–2 , In–3 , . . . using double differentiation and formula (3), we finally arrive at a linear nonhomogeneous ordinary differential equation of order 2n with constant coefficients. The initial conditions for y(x) can be obtained by setting x = a in the integral equation and all its derivative equations. 5.

y(x) – A

x

a

cos(λx) y(t) dt = f (x). cos(λt)

Solution:

x

y(x) = f (x) + A

eA(x–t)

cos(λx) f (t) dt. cos(λt)

eA(x–t)

cos(λt) f (t) dt. cos(λx)

a

6.

y(x) – A

x a

cos(λt) y(t) dt = f (x). cos(λx)

Solution:

x

y(x) = f (x) + A a

7.

y(x) – A

x

cosk (λx) cosm (µt)y(t) dt = f (x).

a

This is a special case of equation 2.9.2 with g(x) = A cosk (λx) and h(t) = cosm (µt). 8.

y(x) + A

x

t cos[λ(x – t)] y(t) dt = f (x).

a

This is a special case of equation 2.9.34 with g(t) = At. 9.

y(x) + A

x

tk cosm (λx)y(t) dt = f (x).

a

This is a special case of equation 2.9.2 with g(x) = –A cosm (λx) and h(t) = tk .

2.5. EQUATIONS WHOSE KERNELS CONTAIN TRIGONOMETRIC FUNCTIONS

10.

y(x) + A

x

169

xk cosm (λt)y(t) dt = f (x).

a

This is a special case of equation 2.9.2 with g(x) = –Axk and h(t) = cosm (λt). 11.

x

y(x) –

 A cos(kx) + B – AB(x – t) cos(kx) y(t) dt = f (x).

a

This is a special case of equation 2.9.7 with λ = B and g(x) = A cos(kx). Solution: x R(x, t)f (t) dt, y(x) = f (x) + a   x G(x) B2 A R(x, t) = [A cos(kx) + B] + sin(kx) . eB(x–s) G(s) ds, G(x) = exp G(t) G(t) t k 12.

x

y(x) +

 A cos(kt) + B + AB(x – t) cos(kt) y(t) dt = f (x).

a

This is a special case of equation 2.9.8 with λ = B and g(t) = A cos(kt). Solution: x R(x, t)f (t) dt, y(x) = f (x) + a x B2 G(t) + eB(t–s) G(s) ds, R(x, t) = –[A cos(kt) + B] G(x) G(x) t 13.

y(x) + A



  A sin(kx) . G(x) = exp k

 √  cos λ t – x y(t) dt = f (x).

x

 √  This is a special case of equation 2.9.62 with K(x) = A cos λ –x .

2.5-2. Kernels Containing Sine. 14.

y(x) – A

x

sin(λx)y(t) dt = f (x).

a

This is a special case of equation 2.9.2 with g(x) = A sin(λx) and h(t) = 1. Solution:

x

y(x) = f (x) + A a

15.

y(x) – A

A  cos(λt) – cos(λx) f (t) dt. sin(λx) exp λ

x

sin(λt)y(t) dt = f (x).

a

This is a special case of equation 2.9.2 with g(x) = A and h(t) = sin(λt). Solution:

x

y(x) = f (x) + A a

A  cos(λt) – cos(λx) f (t) dt. sin(λt) exp λ

170 16.

LINEAR EQUATIONS OF THE SECOND KIND WITH VARIABLE LIMIT OF INTEGRATION

y(x) + A

x

sin[λ(x – t)] y(t) dt = f (x).

a

This is a special case of equation 2.9.36 with g(t) = A. 1◦ . Solution with λ(A + λ) > 0: Aλ x y(x) = f (x) – sin[k(x – t)]f (t) dt, k a 2◦ . Solution with λ(A + λ) < 0: Aλ x y(x) = f (x) – sinh[k(x – t)]f (t) dt, k a 3◦ . Solution with A = –λ:

2

y(x) = f (x) + λ

where k =

where k =





λ(A + λ).

–λ(λ + A).

x

(x – t)f (t) dt. a

17.

y(x) + A

x

sin3 [λ(x – t)] y(t) dt = f (x).

a

Using the formula sin3 β = – 14 sin 3β + 34 sin β, we arrive at an equation of the form 2.5.18: x  1  – 4 A sin[3λ(x – t)] + 34 A sin[λ(x – t)] y(t) dt = f (x). y(x) + a

18.

x

y(x) + a



 A1 sin[λ1 (x – t)] + A2 sin[λ2 (x – t)] y(t) dt = f (x).

This equation can be solved by the same method as equation 2.3.18, by reducing it to a fourth-order linear ordinary differential equation with constant coefficients. Consider the characteristic equation z 2 + (λ21 + λ22 + A1 λ1 + A2 λ2 )z + λ21 λ22 + A1 λ1 λ22 + A2 λ21 λ2 = 0,

(1)

whose roots, z1 and z2 , determine the solution structure of the integral equation. Assume that the discriminant of equation (1) is positive: D ≡ (A1 λ1 – A2 λ2 + λ21 – λ22 )2 + 4A1 A2 λ1 λ2 > 0. In this case, the quadratic equation (1) has the real (different) roots √ √ z1 = – 12 (λ21 + λ22 + A1 λ1 + A2 λ2 ) + 12 D, z2 = – 12 (λ21 + λ22 + A1 λ1 + A2 λ2 ) – 12 D. Depending on the signs of z1 and z2 the following three cases are possible. Case 1. If z1 > 0 and z2 > 0, then the solution of the integral equation has the form (i = 1, 2): x

 √ y(x) = f (x) + {B1 sinh[µ1 (x – t)] + B2 sinh µ2 (x – t) f (t) dt, µi = zi , a

where the coefficients B1 and B2 are determined from the following system of linear algebraic equations: B1 µ1 B2 µ2 B1 µ1 B2 µ2 + 2 – 1 = 0, + 2 – 1 = 0. 2 2 2 2 2 λ1 + µ1 λ1 + µ2 λ2 + µ1 λ2 + µ22

171

2.5. EQUATIONS WHOSE KERNELS CONTAIN TRIGONOMETRIC FUNCTIONS

Case 2. If z1 < 0 and z2 < 0, then the solution of the integral equation has the form

x

y(x) = f (x) +

 {B1 sin[µ1 (x – t)] + B2 sin µ2 (x – t) f (t) dt,

µi =

 |zi |,

a

where B1 and B2 are determined from the system B2 µ2 B1 µ1 + 2 – 1 = 0, 2 2 λ1 – µ1 λ1 – µ22

B1 µ1 B2 µ2 + 2 – 1 = 0. 2 2 λ2 – µ1 λ2 – µ22

Case 3. If z1 > 0 and z2 < 0, then the solution of the integral equation has the form y(x) = f (x) +

x

 {B1 sinh[µ1 (x – t)] + B2 sin µ2 (x – t) f (t) dt,

µi =

 |zi |,

a

where B1 and B2 are determined from the system B1 µ1 B2 µ2 + 2 – 1 = 0, 2 2 λ1 + µ1 λ1 – µ22

B1 µ1 B2 µ2 + 2 – 1 = 0. 2 2 λ2 + µ1 λ2 – µ22

Remark. The solution of the original integral equation can be obtained from the solution of equation 2.3.18 by performing the following change of parameters:

λk → iλk , 19.

x

y(x) +

 n

a

µk → iµk ,

Ak → –iAk ,

Bk → –iBk ,

i2 = –1 (k = 1, 2).

Ak sin[λk (x – t)] y(t) dt = f (x).

k=1



1 . This integral equation can be reduced to a linear nonhomogeneous ordinary differential equation of order 2n with constant coefficients. Set

x

Ik (x) =

sin[λk (x – t)] y(t) dt.

(1)

a

Differentiating (1) with respect to x twice yields Ik

= λk

x

Ik

cos[λk (x – t)] y(t) dt,

= λk y(x) –

x

λ2k

a

sin[λk (x – t)] y(t) dt,

(2)

a

where the primes stand for differentiation with respect to x. Comparing (1) and (2), we see that Ik = λk y(x) – λ2k Ik , Ik = Ik (x). (3) With aid of (1), the integral equation can be rewritten in the form y(x) +

n 

Ak Ik = f (x).

(4)

k=1

Differentiating (4) with respect to x twice taking into account (3) yields  (x) + σn y(x) – yxx

n  k=1

 Ak λ2k Ik = fxx (x),

σn =

n  k=1

Ak λk .

(5)

172

LINEAR EQUATIONS OF THE SECOND KIND WITH VARIABLE LIMIT OF INTEGRATION

Eliminating the integral In from (4) and (5), we obtain  (x) + (σn + λ2n )y(x) + yxx

n–1 

 Ak (λ2n – λ2k )Ik = fxx (x) + λ2n f (x).

(6)

k=1

Differentiating (6) with respect to x twice followed by eliminating In–1 from the resulting expression with the aid of (6) yields a similar equation whose left-hand side is a fourthn–2  order differential operator (acting on y) with constant coefficients plus the sum Bk Ik . k=1

Successively eliminating the terms In–2 , In–3 , . . . using double differentiation and formula (3), we finally arrive at a linear nonhomogeneous ordinary differential equation of order 2n with constant coefficients. The initial conditions for y(x) can be obtained by setting x = a in the integral equation and all its derivative equations. 2◦ . Let us find the roots zk of the algebraic equation n  λk Ak + 1 = 0. z + λ2k k=1

(7)

By reducing it to a common denominator, we arrive at the problem of determining the roots of an nth-degree characteristic polynomial. Assume that all zk are real, different, and nonzero. Let us divide the roots into two groups z1 > 0,

z2 > 0,

...,

zs > 0

(positive roots);

zs+1 < 0,

zs+2 < 0,

...,

zn < 0

(negative roots).

Then the solution of the integral equation can be written in the form

x  s n

 

 y(x) = f (x)+ Bk sinh µk (x–t) + Ck sin µk (x–t) f (t) dt, a

k=1

µk =

 |zk |. (8)

k=s+1

The coefficients Bk and Ck are determined from the following system of linear algebraic equations: s n   Bk µk Ck µk + – 1 = 0, 2 + µ2 2 – µ2 λ λ m k k k=0 k=s+1 m

µk =

 |zk |

m = 1, 2, . . . , n.

(9)

In the case of a nonzero root zs = 0, we can introduce the new constant D = Bs µs and proceed to the limit  µs → 0. As a result, the term D(x – t) appears in solution (8) instead of Bs sinh µs (x – t) and the corresponding terms Dλ–2 m appear in system (9). 20.

y(x) – A

x

a

sin(λx) sin(λt)

y(t) dt = f (x).

Solution:

x

y(x) = f (x) + A

eA(x–t)

sin(λx) f (t) dt. sin(λt)

eA(x–t)

sin(λt) f (t) dt. sin(λx)

a

21.

y(x) – A

x a

sin(λt) sin(λx)

y(t) dt = f (x).

Solution:

x

y(x) = f (x) + A a

2.5. EQUATIONS WHOSE KERNELS CONTAIN TRIGONOMETRIC FUNCTIONS

22.

y(x) – A

x

173

sink (λx) sinm (µt)y(t) dt = f (x).

a

23.

This is a special case of equation 2.9.2 with g(x) = A sink (λx) and h(t) = sinm (µt). x t sin[λ(x – t)] y(t) dt = f (x). y(x) + A a

This is a special case of equation 2.9.36 with g(t) = At. Solution:  Aλ x y(x) = f (x) + t u1 (x)u2 (t) – u2 (x)u1 (t) f (t) dt, W a where u1 (x), u2 (x) is a fundamental system of solutions of the second-order linear ordinary differential equation uxx + λ(Ax + λ)u = 0, and W is the Wronskian. Depending on the sign of Aλ, the functions u1 (x) and u2 (x) are expressed in terms of Bessel functions or modified Bessel functions as follows: if Aλ > 0, then  √   √  u1 (x) = ξ 1/2 J1/3 23 Aλ ξ 3/2 , u2 (x) = ξ 1/2 Y1/3 23 Aλ ξ 3/2 , W = 3/π, if Aλ < 0, then u1 (x) = ξ 1/2 I1/3

24.

y(x) + A

ξ = x + (λ/A);

2√  3/2 , 3 –Aλ ξ W = – 32 ,

u2 (x) = ξ 1/2 K1/3

2√  3/2 , 3 –Aλ ξ

ξ = x + (λ/A).

x

x sin[λ(x – t)] y(t) dt = f (x).

a

25.

This is a special case of equation 2.9.37 with g(x) = Ax and h(t) = 1. Solution:  Aλ x y(x) = f (x) + x u1 (x)u2 (t) – u2 (x)u1 (t) f (t) dt, W a where u1 (x), u2 (x) is a fundamental system of solutions of the second-order linear ordinary differential equation uxx + λ(Ax + λ)u = 0, and W is the Wronskian. The functions u1 (x), u2 (x), and W are specified in 2.5.23. x tk sinm (λx)y(t) dt = f (x). y(x) + A a

26.

This is a special case of equation 2.9.2 with g(x) = –A sinm (λx) and h(t) = tk . x xk sinm (λt)y(t) dt = f (x). y(x) + A

27.

This is a special case of equation 2.9.2 with g(x) = –Axk and h(t) = sinm (λt). x

 A sin(kx) + B – AB(x – t) sin(kx) y(t) dt = f (x). y(x) –

a

a

This is a special case of equation 2.9.7 with λ = B and g(x) = A sin(kx). Solution: x y(x) = f (x) + R(x, t)f (t) dt, a   x G(x) B2 A R(x, t) = [A sin(kx) + B] + eB(x–s) G(s) ds, G(x) = exp – cos(kx) . G(t) G(t) t k

174

LINEAR EQUATIONS OF THE SECOND KIND WITH VARIABLE LIMIT OF INTEGRATION

28.

x

y(x) +

 A sin(kt) + B + AB(x – t) sin(kt) y(t) dt = f (x).

a

This is a special case of equation 2.9.8 with λ = B and g(t) = A sin(kt). Solution: x y(x) = f (x) + R(x, t)f (t) dt, a   x G(t) B2 A R(x, t) = –[A sin(kt) + B] + eB(t–s) G(s) ds, G(x) = exp – cos(kx) . G(x) G(x) t k 29.

y(x) + A



 √  sin λ t – x y(t) dt = f (x).

x

 √  This is a special case of equation 2.9.62 with K(x) = A sin λ –x .

2.5-3. Kernels Containing Tangent. 30.

y(x) – A

x

tan(λx)y(t) dt = f (x).

a

This is a special case of equation 2.9.2 with g(x) = A tan(λx) and h(t) = 1. Solution: x cos(λt) A/λ y(x) = f (x) + A tan(λx) f (t) dt. cos(λx) a 31.

y(x) – A

x

tan(λt)y(t) dt = f (x).

a

This is a special case of equation 2.9.2 with g(x) = A and h(t) = tan(λt). Solution: x cos(λt) A/λ y(x) = f (x) + A tanh(λt) f (t) dt. cos(λx) a 32.

y(x) + A

x

 tan(λx) – tan(λt) y(t) dt = f (x).

a

This is a special case of equation 2.9.5 with g(x) = A tan(λx). Solution: x

  1 y(x) = f (x) + Y1 (x)Y2 (t) – Y2 (x)Y1 (t) f (t) dt, W a where Y1 (x), Y2 (x) is a fundamental system of solutions of the second-order linear ordinary  differential equation cos2 (λx)Yxx + AλY = 0, W is the Wronskian, and the primes stand for the differentiation with respect to the argument specified in the parentheses. As shown in A. D. Polyanin and V. F. Zaitsev (2003), the functions Y1 (x) and Y2 (x) can be expressed via the hypergeometric function. 33.

y(x) – A

x

a

tan(λx) tan(λt)

y(t) dt = f (x).

Solution:

x

eA(x–t)

y(x) = f (x) + A a

tan(λx) f (t) dt. tan(λt)

2.5. EQUATIONS WHOSE KERNELS CONTAIN TRIGONOMETRIC FUNCTIONS

34.

y(x) – A

x

a

tan(λt) y(t) dt = f (x). tan(λx)

Solution:

x

eA(x–t)

y(x) = f (x) + A a

35.

y(x) – A

x

tan(λt) f (t) dt. tan(λx)

tank (λx) tanm (µt)y(t) dt = f (x).

a

This is a special case of equation 2.9.2 with g(x) = A tank (λx) and h(t) = tanm (µt). 36.

y(x) + A

x

tk tanm (λx)y(t) dt = f (x).

a

This is a special case of equation 2.9.2 with g(x) = –A tanm (λx) and h(t) = tk . 37.

y(x) + A

x

xk tanm (λt)y(t) dt = f (x).

a

This is a special case of equation 2.9.2 with g(x) = –Axk and h(t) = tanm (λt). 38.

x

y(x) –

 A tan(kx) + B – AB(x – t) tan(kx) y(t) dt = f (x).

a

This is a special case of equation 2.9.7 with λ = B and g(x) = A tan(kx). 39.

x

y(x) +

 A tan(kt) + B + AB(x – t) tan(kt) y(t) dt = f (x).

a

This is a special case of equation 2.9.8 with λ = B and g(t) = A tan(kt). 2.5-4. Kernels Containing Cotangent. 40.

y(x) – A

x

cot(λx)y(t) dt = f (x).

a

This is a special case of equation 2.9.2 with g(x) = A cot(λx) and h(t) = 1. Solution: x sin(λx) A/λ cot(λx) f (t) dt. y(x) = f (x) + A sin(λt) a 41.

y(x) – A

x

cot(λt)y(t) dt = f (x).

a

This is a special case of equation 2.9.2 with g(x) = A and h(t) = cot(λt). Solution: x sin(λx) A/λ coth(λt) f (t) dt. y(x) = f (x) + A sin(λt) a 42.

y(x) – A

x a

cot(λx) cot(λt)

y(t) dt = f (x).

Solution:

x

eA(x–t)

y(x) = f (x) + A a

cot(λx) f (t) dt. cot(λt)

175

176 43.

LINEAR EQUATIONS OF THE SECOND KIND WITH VARIABLE LIMIT OF INTEGRATION

y(x) – A

x

a

cot(λt) y(t) dt = f (x). cot(λx)

Solution:

x

eA(x–t)

y(x) = f (x) + A a

cot(λt) f (t) dt. cot(λx)

44.

y(x) + A

45.

This is a special case of equation 2.9.2 with g(x) = –A cotm (λx) and h(t) = tk . x y(x) + A xk cotm (λt)y(t) dt = f (x).

x

tk cotm (λx)y(t) dt = f (x).

a

a

46.

This is a special case of equation 2.9.2 with g(x) = –Axk and h(t) = cotm (λt). x

 y(x) – A cot(kx) + B – AB(x – t) cot(kx) y(t) dt = f (x).

47.

This is a special case of equation 2.9.7 with λ = B and g(x) = A cot(kx). x

 y(x) + A cot(kt) + B + AB(x – t) cot(kt) y(t) dt = f (x).

a

a

This is a special case of equation 2.9.8 with λ = B and g(t) = A cot(kt). 2.5-5. Kernels Containing Combinations of Trigonometric Functions. 48.

y(x) – A

49.

This is a special case of equation 2.9.2 with g(x) = A cosk (λx) and h(t) = sinm (µt). x   y(x) – A + B cos(λx) – B(x – t)[λ sin(λx) + A cos(λx)] y(t) dt = f (x).

50.

This is a special case of equation 2.9.38 with b = B and g(x) = A. x   y(x) – A + B sin(λx) + B(x – t)[λ cos(λx) – A sin(λx)] y(t) dt = f (x).

51.

This is a special case of equation 2.9.39 with b = B and g(x) = A. x y(x) – A tank (λx) cotm (µt)y(t) dt = f (x).

x

cosk (λx) sinm (µt)y(t) dt = f (x).

a

a

a

a

This is a special case of equation 2.9.2 with g(x) = A tank (λx) and h(t) = cotm (µt).

2.6. Equations Whose Kernels Contain Inverse Trigonometric Functions 2.6-1. Kernels Containing Arccosine. 1.

y(x) – A

x

arccos(λx)y(t) dt = f (x).

a

This is a special case of equation 2.9.2 with g(x) = A arccos(λx) and h(t) = 1.

2.6. EQUATIONS WHOSE KERNELS CONTAIN INVERSE TRIGONOMETRIC FUNCTIONS

2.

y(x) – A

x

arccos(λt)y(t) dt = f (x).

a

This is a special case of equation 2.9.2 with g(x) = A and h(t) = arccos(λt). 3.

y(x) – A

x a

arccos(λx) y(t) dt = f (x). arccos(λt)

Solution:



x

y(x) = f (x) + A

eA(x–t)

arccos(λx) f (t) dt. arccos(λt)

eA(x–t)

arccos(λt) f (t) dt. arccos(λx)

a

4.

y(x) – A

x a

arccos(λt) y(t) dt = f (x). arccos(λx)

Solution:



x

y(x) = f (x) + A a

5.

x

y(x) –

 A arccos(kx) + B – AB(x – t) arccos(kx) y(t) dt = f (x).

a

This is a special case of equation 2.9.7 with λ = B and g(x) = A arccos(kx). 6.

x

y(x) +

 A arccos(kt) + B + AB(x – t) arccos(kt) y(t) dt = f (x).

a

This is a special case of equation 2.9.8 with λ = B and g(t) = A arccos(kt).

2.6-2. Kernels Containing Arcsine. 7.

y(x) – A

x

arcsin(λx)y(t) dt = f (x).

a

This is a special case of equation 2.9.2 with g(x) = A arcsin(λx) and h(t) = 1. 8.

y(x) – A

x

arcsin(λt)y(t) dt = f (x). a

This is a special case of equation 2.9.2 with g(x) = A and h(t) = arcsin(λt). 9.

y(x) – A

x a

arcsin(λx) arcsin(λt)

y(t) dt = f (x).

Solution:



x

y(x) = f (x) + A

eA(x–t)

arcsin(λx) f (t) dt. arcsin(λt)

eA(x–t)

arcsin(λt) f (t) dt. arcsin(λx)

a

10.

y(x) – A

x a

arcsin(λt) arcsin(λx)

y(t) dt = f (x).

Solution:



x

y(x) = f (x) + A a

177

178

LINEAR EQUATIONS OF THE SECOND KIND WITH VARIABLE LIMIT OF INTEGRATION



x

 A arcsin(kx) + B – AB(x – t) arcsin(kx) y(t) dt = f (x).

11.

y(x) –

12.

This is a special case of equation 2.9.7 with λ = B and g(x) = A arcsin(kx). x

 y(x) + A arcsin(kt) + B + AB(x – t) arcsin(kt) y(t) dt = f (x).

a

a

This is a special case of equation 2.9.8 with λ = B and g(t) = A arcsin(kt). 2.6-3. Kernels Containing Arctangent. 13.

y(x) – A

x

arctan(λx)y(t) dt = f (x).

a

14.

This is a special case of equation 2.9.2 with g(x) = A arctan(λx) and h(t) = 1. x y(x) – A arctan(λt)y(t) dt = f (x). a

15.

16.

This is a special case of equation 2.9.2 with g(x) = A and h(t) = arctan(λt). x arctan(λx) y(t) dt = f (x). y(x) – A a arctan(λt) Solution: x arctan(λx) f (t) dt. y(x) = f (x) + A eA(x–t) arctan(λt) a y(x) – A

x a

arctan(λt) y(t) dt = f (x). arctan(λx)

Solution:

x

eA(x–t)

y(x) = f (x) + A a

17.

y(x) + A

arctan(λt) f (t) dt. arctan(λx)



arctan[λ(t – x)] y(t) dt = f (x). x

18.

This is a special case of equation 2.9.62 with K(x) = A arctan(–λx). x

 y(x) – A arctan(kx) + B – AB(x – t) arctan(kx) y(t) dt = f (x).

19.

This is a special case of equation 2.9.7 with λ = B and g(x) = A arctan(kx). x

 y(x) + A arctan(kt) + B + AB(x – t) arctan(kt) y(t) dt = f (x).

a

a

This is a special case of equation 2.9.8 with λ = B and g(t) = A arctan(kt). 2.6-4. Kernels Containing Arccotangent. 20.

y(x) – A

x

arccot(λx)y(t) dt = f (x).

a

This is a special case of equation 2.9.2 with g(x) = A arccot(λx) and h(t) = 1.

2.7. EQUATIONS WHOSE KERNELS CONTAIN COMBINATIONS OF ELEMENTARY FUNCTIONS

21.

y(x) – A

x

arccot(λt)y(t) dt = f (x).

a

This is a special case of equation 2.9.2 with g(x) = A and h(t) = arccot(λt).

22.

y(x) – A

x

arccot(λx) arccot(λt)

a

y(t) dt = f (x).

Solution:



x

y(x) = f (x) + A

eA(x–t)

arccot(λx) f (t) dt. arccot(λt)

eA(x–t)

arccot(λt) f (t) dt. arccot(λx)

a

23.

y(x) – A

x

arccot(λt) arccot(λx)

a

y(t) dt = f (x).

Solution:



x

y(x) = f (x) + A a

24.

y(x) + A



arccot[λ(t – x)] y(t) dt = f (x). x

This is a special case of equation 2.9.62 with K(x) = A arccot(–λx). 25.

x

y(x) –

 A arccot(kx) + B – AB(x – t) arccot(kx) y(t) dt = f (x).

a

This is a special case of equation 2.9.7 with λ = B and g(x) = A arccot(kx). 26.

x

y(x) +

 A arccot(kt) + B + AB(x – t) arccot(kt) y(t) dt = f (x).

a

This is a special case of equation 2.9.8 with λ = B and g(t) = A arccot(kt).

2.7. Equations Whose Kernels Contain Combinations of Elementary Functions 2.7-1. Kernels Containing Exponential and Hyperbolic Functions.

1.

y(x) + A

x

eµ(x–t) cosh[λ(x – t)] y(t) dt = f (x).

a

Solution: x R(x – t)f (t) dt, y(x) = f (x) + a  

 A2 R(x) = exp (µ – 12 A)x sinh(kx) – A cosh(kx) , 2k

 k=

λ2 + 14 A2 .

179

180 2.

LINEAR EQUATIONS OF THE SECOND KIND WITH VARIABLE LIMIT OF INTEGRATION

y(x) + A

x

eµ(x–t) sinh[λ(x – t)] y(t) dt = f (x).

a

1◦ . Solution with λ(A – λ) > 0: y(x) = f (x) –

Aλ k



x

eµ(x–t) sin[k(x – t)]f (t) dt,

where k =



λ(A – λ).

a

2◦ . Solution with λ(A – λ) < 0: Aλ k

y(x) = f (x) –



x

eµ(x–t) sinh[k(x – t)]f (t) dt,

where k =



λ(λ – A).

a

3◦ . Solution with A = λ:

x

(x – t)eµ(x–t) f (t) dt.

y(x) = f (x) – λ2 a

3.

x

y(x) + a

  eµ(x–t) A1 sinh[λ1 (x – t)] + A2 sinh[λ2 (x – t)] y(t) dt = f (x).

The substitution w(x) = e–µx y(x) leads to an equation of the form 2.3.18:

x

 A1 sinh[λ1 (x – t)] + A2 sinh[λ2 (x – t)] w(t) dt = e–µx f (x).

w(x) + a

4.

y(x) + A

x

teµ(x–t) sinh[λ(x – t)] y(t) dt = f (x).

a

The substitution w(x) = e–µx y(x) leads to an equation of the form 2.3.23:

x

t sinh[λ(x – t)]w(t) dt = e–µx f (x).

w(x) + A a

2.7-2. Kernels Containing Exponential and Logarithmic Functions. 5.

y(x) – A

x

eµt ln(λx)y(t) dt = f (x).

a

This is a special case of equation 2.9.2 with g(x) = A ln(λx) and h(t) = eµt . 6.

y(x) – A

x

eµx ln(λt)y(t) dt = f (x).

a

This is a special case of equation 2.9.2 with g(x) = Aeµx and h(t) = ln(λt). 7.

y(x) – A

x

eµ(x–t) ln(λx)y(t) dt = f (x).

a

Solution:



x

e(µ–A)(x–t) ln(λx)

y(x) = f (x) + A a

(λx)Ax f (t) dt. (λt)At

2.7. EQUATIONS WHOSE KERNELS CONTAIN COMBINATIONS OF ELEMENTARY FUNCTIONS

8.

y(x) – A

x

181

eµ(x–t) ln(λt)y(t) dt = f (x).

a

Solution:



x

e(µ–A)(x–t) ln(λt)

y(x) = f (x) + A 9.

a

y(x) + A

x

(λx)Ax f (t) dt. (λt)At

eµ(x–t) (ln x – ln t)y(t) dt = f (x).

a

x

 1 eµ(x–t) u1 (x)u2 (t) – u2 (x)u1 (t) f (t) dt, W a where the primes stand for the differentiation with respect to the argument specified in the parentheses, and u1 (x), u2 (x) is a fundamental system of solutions of the second-order linear homogeneous ordinary differential equation uxx + Ax–1 u = 0, with u1 (x) and u2 (x) expressed in terms of Bessel functions or modified Bessel functions, depending on the sign of A:  √   √  √ √ W = π1 , u1 (x) = x J1 2 Ax , u2 (x) = x Y1 2 Ax for A > 0, √ √     √ √ W = – 21 , u1 (x) = x I1 2 –Ax , u2 (x) = x K1 2 –Ax for A < 0. ∞ y(x) + a eλ(x–t) ln(t – x)y(t) dt = f (x).

Solution:

y(x) = f (x) +

10.

x

This is a special case of equation 2.9.62 with K(x) = aeλx ln(–x). 2.7-3. Kernels Containing Exponential and Trigonometric Functions. 11.

y(x) – A

x

eµt cos(λx)y(t) dt = f (x).

a

12.

This is a special case of equation 2.9.2 with g(x) = A cos(λx) and h(t) = eµt . x eµx cos(λt)y(t) dt = f (x). y(x) – A

13.

This is a special case of equation 2.9.2 with g(x) = Aeµx and h(t) = cos(λt). x eµ(x–t) cos[λ(x – t)] y(t) dt = f (x). y(x) + A

a

a

1◦ . Solution with |A| > 2|λ|:

x R(x – t)f (t) dt, y(x) = f (x) + a   

 A2 R(x) = exp (µ – 12 A)x sinh(kx) – A cosh(kx) , k = 41 A2 – λ2 . 2k ◦ 2 . Solution with |A| < 2|λ|: x R(x – t)f (t) dt, y(x) = f (x) + a   2   A

1 sin(kx) – A cos(kx) , k = λ2 – 14 A2 . R(x) = exp (µ – 2 A)x 2k 3◦ . Solution with λ = ± 12 A: x R(x – t)f (t) dt, y(x) = f (x) + a

R(x) =

1

2 2A x



   – A exp µ – 12 A x .

182

LINEAR EQUATIONS OF THE SECOND KIND WITH VARIABLE LIMIT OF INTEGRATION

14.

x

y(x) –

 eµ(x–t) A cos(kx) + B – AB(x – t) cos(kx) y(t) dt = f (x).

a

Solution:

x

eµ(x–t) M (x, t)f (t) dt,

y(x) = f (x) + a

B2 G(x) + M (x, t) = [A cos(kx) + B] G(t) G(t) 15.

x

y(x) +



x

e

B(x–s)

G(s) ds,

t

  A sin(kx) . G(x) = exp k

 eµ(x–t) A cos(kt) + B + AB(x – t) cos(kt) y(t) dt = f (x).

a

Solution:

x

eµ(x–t) M (x, t)f (t) dt,   x G(t) B A B(t–s) M (x, t) = –[A cos(kt) + B] + sin(kx) . e G(s) ds, G(x) = exp G(x) G(x) t k y(x) = f (x) +

a 2

16.

y(x) – A

x

eµt sin(λx)y(t) dt = f (x).

a

This is a special case of equation 2.9.2 with g(x) = A sin(λx) and h(t) = eµt . 17.

y(x) – A

x

eµx sin(λt)y(t) dt = f (x).

a

This is a special case of equation 2.9.2 with g(x) = Aeµx and h(t) = sin(λt). 18.

y(x) + A

x

eµ(x–t) sin[λ(x – t)] y(t) dt = f (x).

a

1◦ . Solution with λ(A + λ) > 0: y(x) = f (x) –

Aλ k



x

eµ(x–t) sin[k(x – t)]f (t) dt,

where k =



λ(A + λ).

a

2◦ . Solution with λ(A + λ) < 0: y(x) = f (x) –

Aλ k



x

eµ(x–t) sinh[k(x – t)]f (t) dt,

where k =

a

3◦ . Solution with A = –λ:

x

(x – t)eµ(x–t) f (t) dt.

y(x) = f (x) + λ2 a

19.

y(x) + A

x

eµ(x–t) sin3 [λ(x – t)] y(t) dt = f (x).

a

The substitution w(x) = e–µx y(x) leads to an equation of the form 2.5.17:

x

sin3 [λ(x – t)]w(t) dt = e–µx f (x).

w(x) + A a



–λ(λ + A).

2.7. EQUATIONS WHOSE KERNELS CONTAIN COMBINATIONS OF ELEMENTARY FUNCTIONS

20.

x

y(x) + a

183

  eµ(x–t) A1 sin[λ1 (x – t)] + A2 sin[λ2 (x – t)] y(t) dt = f (x).

The substitution w(x) = e–µx y(x) leads to an equation of the form 2.5.18: w(x) +

x

 A1 sin[λ1 (x – t)] + A2 sin[λ2 (x – t)] w(t) dt = e–µx f (x).

a

21.

x

y(x) +

eµ(x–t)

a

 n

Ak sin[λk (x – t)] y(t) dt = f (x).

k=1

The substitution w(x) = e–µx y(x) leads to an equation of the form 2.5.19: w(x) +

x  n a

22.

y(x) + A

x

Ak sin[λk (x – t)] w(t) dt = e–µx f (x).

k=1

teµ(x–t) sin[λ(x – t)] y(t) dt = f (x).

a

Solution: Aλ y(x) = f (x) + W



x

 teµ(x–t) u1 (x)u2 (t) – u2 (x)u1 (t) f (t) dt,

a

where u1 (x), u2 (x) is a fundamental system of solutions of the second-order linear ordinary differential equation uxx + λ(Ax + λ)u = 0, and W is the Wronskian. Depending on the sign of Aλ, the functions u1 (x) and u2 (x) are expressed in terms of Bessel functions or modified Bessel functions as follows: if Aλ > 0, then u1 (x) = ξ 1/2 J1/3

2√  3/2 , 3 Aλ ξ W = 3/π,

u2 (x) = ξ 1/2 Y1/3

2√  3/2 , 3 Aλ ξ

ξ = x + (λ/A);

if Aλ < 0, then u1 (x) = ξ 1/2 I1/3

2√  3/2 , 3 –Aλ ξ W = – 32 ,

23.

y(x) + A

x

u2 (x) = ξ 1/2 K1/3

2√  3/2 , 3 –Aλ ξ

ξ = x + (λ/A).

xeµ(x–t) sin[λ(x – t)] y(t) dt = f (x).

a

Solution: y(x) = f (x) +

Aλ W



x

 xeµ(x–t) u1 (x)u2 (t) – u2 (x)u1 (t) f (t) dt,

a

where u1 (x), u2 (x) is a fundamental system of solutions of the second-order linear ordinary differential equation uxx + λ(Ax + λ)u = 0, and W is the Wronskian. The functions u1 (x), u2 (x), and W are specified in 2.7.22. 24.

y(x) + A

∞ x

 √  eµ(t–x) sin λ t – x y(t) dt = f (x).

 √  This is a special case of equation 2.9.62 with K(x) = Ae–µx sin λ –x .

184

LINEAR EQUATIONS OF THE SECOND KIND WITH VARIABLE LIMIT OF INTEGRATION

25.

x

y(x) –

 eµ(x–t) A sin(kx) + B – AB(x – t) sin(kx) y(t) dt = f (x).

a

Solution:

x

eµ(x–t) M (x, t)f (t) dt,

y(x) = f (x) + a

M (x, t) = [A sin(kx) + B] 26.

x

y(x) +



G(x) B2 + G(t) G(t)

x

eB(x–s) G(s) ds, t

  A G(x) = exp – cos(kx) . k

 eµ(x–t) A sin(kt) + B + AB(x – t) sin(kt) y(t) dt = f (x).

a

Solution:

x

eµ(x–t) M (x, t)f (t) dt,

y(x) = f (x) + a

G(t) B2 M (x, t) = –[A sin(kt) + B] + G(x) G(x) 27.

y(x) – A

x



x

e

B(t–s)

G(s) ds,

t

  A G(x) = exp – cos(kx) . k

eµt tan(λx)y(t) dt = f (x).

a

This is a special case of equation 2.9.2 with g(x) = A tan(λx) and h(t) = eµt . 28.

y(x) – A

x

eµx tan(λt)y(t) dt = f (x).

a

This is a special case of equation 2.9.2 with g(x) = Aeµx and h(t) = tan(λt). 29.

y(x) + A

x

 eµ(x–t) tan(λx) – tan(λt) y(t) dt = f (x).

a

The substitution w(x) = e–µx y(x) leads to an equation of the form 2.5.32:

x

w(x) + A

 tan(λx) – tan(λt) w(t) dt = e–µx f (x).

a

30.

x

y(x) –

 eµ(x–t) A tan(kx) + B – AB(x – t) tan(kx) y(t) dt = f (x).

a

The substitution w(x) = e–µx y(x) leads to an equation of the form 2.9.7 with λ = B and g(x) = A tan(kx):

x

 A tan(kx) + B – AB(x – t) tan(kx) w(t) dt = e–µx f (x).

w(x) – a

31.

x

y(x) +

 eµ(x–t) A tan(kt) + B + AB(x – t) tan(kt) y(t) dt = f (x).

a

The substitution w(x) = e–µx y(x) leads to an equation of the form 2.9.8 with λ = B and g(t) = A tan(kt):

x

w(x) + a

 A tan(kt) + B + AB(x – t) tan(kt) w(t) dt = e–µx f (x).

2.7. EQUATIONS WHOSE KERNELS CONTAIN COMBINATIONS OF ELEMENTARY FUNCTIONS

32.

y(x) – A

x

eµt cot(λx)y(t) dt = f (x).

a

This is a special case of equation 2.9.2 with g(x) = A cot(λx) and h(t) = eµt . 33.

y(x) – A

x

eµx cot(λt)y(t) dt = f (x).

a

This is a special case of equation 2.9.2 with g(x) = Aeµx and h(t) = cot(λt).

2.7-4. Kernels Containing Hyperbolic and Logarithmic Functions.

34.

y(x) – A

x

coshk (λx) lnm (µt)y(t) dt = f (x).

a

This is a special case of equation 2.9.2 with g(x) = A coshk (λx) and h(t) = lnm (µt). 35.

y(x) – A

x

coshk (λt) lnm (µx)y(t) dt = f (x).

a

This is a special case of equation 2.9.2 with g(x) = A lnm (µx) and h(t) = coshk (λt). 36.

y(x) – A

x

sinhk (λx) lnm (µt)y(t) dt = f (x).

a

This is a special case of equation 2.9.2 with g(x) = A sinhk (λx) and h(t) = lnm (µt). 37.

y(x) – A

x

sinhk (λt) lnm (µx)y(t) dt = f (x).

a

This is a special case of equation 2.9.2 with g(x) = A lnm (µx) and h(t) = sinhk (λt). 38.

y(x) – A

x

tanhk (λx) lnm (µt)y(t) dt = f (x).

a

This is a special case of equation 2.9.2 with g(x) = A tanhk (λx) and h(t) = lnm (µt). 39.

y(x) – A

x

tanhk (λt) lnm (µx)y(t) dt = f (x).

a

This is a special case of equation 2.9.2 with g(x) = A lnm (µx) and h(t) = tanhk (λt). 40.

y(x) – A

x

cothk (λx) lnm (µt)y(t) dt = f (x).

a

This is a special case of equation 2.9.2 with g(x) = A cothk (λx) and h(t) = lnm (µt). 41.

y(x) – A

x

cothk (λt) lnm (µx)y(t) dt = f (x).

a

This is a special case of equation 2.9.2 with g(x) = A lnm (µx) and h(t) = cothk (λt).

185

186

LINEAR EQUATIONS OF THE SECOND KIND WITH VARIABLE LIMIT OF INTEGRATION

2.7-5. Kernels Containing Hyperbolic and Trigonometric Functions. 42.

y(x) – A

x

coshk (λx) cosm (µt)y(t) dt = f (x).

a

43.

This is a special case of equation 2.9.2 with g(x) = A coshk (λx) and h(t) = cosm (µt). x coshk (λt) cosm (µx)y(t) dt = f (x). y(x) – A a

44.

This is a special case of equation 2.9.2 with g(x) = A cosm (µx) and h(t) = coshk (λt). x coshk (λx) sinm (µt)y(t) dt = f (x). y(x) – A

45.

This is a special case of equation 2.9.2 with g(x) = A coshk (λx) and h(t) = sinm (µt). x coshk (λt) sinm (µx)y(t) dt = f (x). y(x) – A

a

a

46.

This is a special case of equation 2.9.2 with g(x) = A sinm (µx) and h(t) = coshk (λt). x sinhk (λx) cosm (µt)y(t) dt = f (x). y(x) – A

47.

This is a special case of equation 2.9.2 with g(x) = A sinhk (λx) and h(t) = cosm (µt). x sinhk (λt) cosm (µx)y(t) dt = f (x). y(x) – A

a

a

48.

This is a special case of equation 2.9.2 with g(x) = A cosm (µx) and h(t) = sinhk (λt). x sinhk (λx) sinm (µt)y(t) dt = f (x). y(x) – A

49.

This is a special case of equation 2.9.2 with g(x) = A sinhk (λx) and h(t) = sinm (µt). x sinhk (λt) sinm (µx)y(t) dt = f (x). y(x) – A

a

a

50.

This is a special case of equation 2.9.2 with g(x) = A sinm (µx) and h(t) = sinhk (λt). x tanhk (λx) cosm (µt)y(t) dt = f (x). y(x) – A

51.

This is a special case of equation 2.9.2 with g(x) = A tanhk (λx) and h(t) = cosm (µt). x tanhk (λt) cosm (µx)y(t) dt = f (x). y(x) – A

a

a

52.

This is a special case of equation 2.9.2 with g(x) = A cosm (µx) and h(t) = tanhk (λt). x tanhk (λx) sinm (µt)y(t) dt = f (x). y(x) – A a

53.

This is a special case of equation 2.9.2 with g(x) = A tanhk (λx) and h(t) = sinm (µt). x tanhk (λt) sinm (µx)y(t) dt = f (x). y(x) – A a

This is a special case of equation 2.9.2 with g(x) = A sinm (µx) and h(t) = tanhk (λt).

2.8. EQUATIONS WHOSE KERNELS CONTAIN SPECIAL FUNCTIONS

187

2.7-6. Kernels Containing Logarithmic and Trigonometric Functions. 54.

y(x) – A

x

cosk (λx) lnm (µt)y(t) dt = f (x).

a

55.

This is a special case of equation 2.9.2 with g(x) = A cosk (λx) and h(t) = lnm (µt). x cosk (λt) lnm (µx)y(t) dt = f (x). y(x) – A a

56.

This is a special case of equation 2.9.2 with g(x) = A lnm (µx) and h(t) = cosk (λt). x sink (λx) lnm (µt)y(t) dt = f (x). y(x) – A

57.

This is a special case of equation 2.9.2 with g(x) = A sink (λx) and h(t) = lnm (µt). x sink (λt) lnm (µx)y(t) dt = f (x). y(x) – A

a

a

58.

This is a special case of equation 2.9.2 with g(x) = A lnm (µx) and h(t) = sink (λt). x tank (λx) lnm (µt)y(t) dt = f (x). y(x) – A

59.

This is a special case of equation 2.9.2 with g(x) = A tank (λx) and h(t) = lnm (µt). x tank (λt) lnm (µx)y(t) dt = f (x). y(x) – A

60.

This is a special case of equation 2.9.2 with g(x) = A lnm (µx) and h(t) = tank (λt). x cotk (λx) lnm (µt)y(t) dt = f (x). y(x) – A

a

a

a

61.

This is a special case of equation 2.9.2 with g(x) = A cotk (λx) and h(t) = lnm (µt). x cotk (λt) lnm (µx)y(t) dt = f (x). y(x) – A a

This is a special case of equation 2.9.2 with g(x) = A lnm (µx) and h(t) = cotk (λt).

2.8. Equations Whose Kernels Contain Special Functions 2.8-1. Kernels Containing Bessel Functions. 1.

y(x) – λ

x

J0 (x – t)y(t) dt = f (x). 0



Solution:

x

R(x – t)f (t) dt,

y(x) = f (x) + 0

where

x  √ 

√  J1(t) √ λ2 λ dt. sin 1 – λ2 x + √ sin 1 – λ2 (x – t) R(x) = λ cos 1 – λ2 x + √ t 1 – λ2 1 – λ2 0 Reference: V. I. Smirnov (1974).

188 2.

LINEAR EQUATIONS OF THE SECOND KIND WITH VARIABLE LIMIT OF INTEGRATION

y(x) – A

x

Jν (λx)y(t) dt = f (x).

a

This is a special case of equation 2.9.2 with g(x) = AJν (λx) and h(t) = 1. 3.

y(x) – A

x

Jν (λt)y(t) dt = f (x).

a

This is a special case of equation 2.9.2 with g(x) = A and h(t) = Jν (λt). 4.

y(x) – A

x

Jν (λx) Jν (λt)

a

y(t) dt = f (x).

Solution:

x

y(x) = f (x) + A

eA(x–t)

Jν (λx) f (t) dt. Jν (λt)

eA(x–t)

Jν (λt) f (t) dt. Jν (λx)

a

5.

y(x) – A

x

Jν (λt) Jν (λx)

a

y(t) dt = f (x).

Solution:



x

y(x) = f (x) + A a

6.

y(x) + A



Jν (λ(t – x)) y(t) dt = f (x).

x

This is a special case of equation 2.9.62 with K(x) = AJν (–λx). 7.

x

y(x) –

a

 AJν (kx) + B – AB(x – t)Jν (kx) y(t) dt = f (x).

This is a special case of equation 2.9.7 with λ = B and g(x) = AJν (kx). 8.

x

y(x) +

a

 AJν (kt) + B + AB(x – t)Jν (kt) y(t) dt = f (x).

This is a special case of equation 2.9.8 with λ = B and g(t) = AJν (kt). 9.

x

y(x) – λ

eµ(x–t) J0 (x – t)y(t) dt = f (x).

0



Solution:

x

R(x – t)f (t) dt,

y(x) = f (x) + 0

where

10.

√  √  λ2 R(x) = eµx λ cos 1 – λ2 x + √ sin 1 – λ2 x + 1 – λ2

x

√  J1 (t) λ √ dt . sin 1 – λ2 (x – t) t 1 – λ2 0

y(x) – A

x

a

Yν (λx)y(t) dt = f (x).

This is a special case of equation 2.9.2 with g(x) = AYν (λx) and h(t) = 1.

2.8. EQUATIONS WHOSE KERNELS CONTAIN SPECIAL FUNCTIONS

11.

y(x) – A

x

a

Yν (λt)y(t) dt = f (x).

14.

This is a special case of equation 2.9.2 with g(x) = A and h(t) = Yν (λt). x Yν (λx) y(x) – A y(t) dt = f (x). a Yν (λt) Solution: x Yν (λx) f (t) dt. y(x) = f (x) + A eA(x–t) Yν (λt) a x Yν (λt) y(x) – A y(t) dt = f (x). a Yν (λx) Solution: x Yν (λt) f (t) dt. y(x) = f (x) + A eA(x–t) Y ν (λx) a ∞ y(x) + A Yν (λ(t – x)) y(t) dt = f (x).

15.

This is a special case of equation 2.9.62 with K(x) = AYν (–λx). x

 y(x) – AYν (kx) + B – AB(x – t)Yν (kx) y(t) dt = f (x).

16.

This is a special case of equation 2.9.7 with λ = B and g(x) = AYν (kx). x

 y(x) + AYν (kt) + B + AB(x – t)Yν (kt) y(t) dt = f (x).

12.

13.

x

a

a

This is a special case of equation 2.9.8 with λ = B and g(t) = AYν (kt). 2.8-2. Kernels Containing Modified Bessel Functions. 17.

y(x) – A

x

a

18.

Iν (λx)y(t) dt = f (x).

This is a special case of equation 2.9.2 with g(x) = AIν (λx) and h(t) = 1. x y(x) – A Iν (λt)y(t) dt = f (x). a

19.

20.

This is a special case of equation 2.9.2 with g(x) = A and h(t) = Iν (λt). x Iν (λx) y(x) – A y(t) dt = f (x). a Iν (λt) Solution: x Iν (λx) f (t) dt. y(x) = f (x) + A eA(x–t) Iν (λt) a x Iν (λt) y(x) – A y(t) dt = f (x). a Iν (λx) Solution: x Iν (λt) f (t) dt. y(x) = f (x) + A eA(x–t) Iν (λx) a

189

190 21.

LINEAR EQUATIONS OF THE SECOND KIND WITH VARIABLE LIMIT OF INTEGRATION

y(x) + A



Iν (λ(t – x)) y(t) dt = f (x).

x

This is a special case of equation 2.9.62 with K(x) = AIν (–λx). 22.

x

y(x) –

a

 AIν (kx) + B – AB(x – t)Iν (kx) y(t) dt = f (x).

This is a special case of equation 2.9.7 with λ = B and g(x) = AIν (kx). 23.

x

y(x) +

a

 AIν (kt) + B + AB(x – t)Iν (kt) y(t) dt = f (x).

This is a special case of equation 2.9.8 with λ = B and g(t) = AIν (kt). 24.

y(x) – A

x

Kν (λx)y(t) dt = f (x).

a

This is a special case of equation 2.9.2 with g(x) = AKν (λx) and h(t) = 1. 25.

y(x) – A

x

Kν (λt)y(t) dt = f (x).

a

This is a special case of equation 2.9.2 with g(x) = A and h(t) = Kν (λt). 26.

y(x) – A

x

Kν (λx) Kν (λt)

a

y(t) dt = f (x).

Solution:



x

y(x) = f (x) + A

eA(x–t)

Kν (λx) f (t) dt. Kν (λt)

eA(x–t)

Kν (λt) f (t) dt. Kν (λx)

a

27.

y(x) – A

x

Kν (λt) Kν (λx)

a

y(t) dt = f (x).

Solution:



x

y(x) = f (x) + A a

28.

y(x) + A



Kν (λ(t – x)) y(t) dt = f (x).

x

This is a special case of equation 2.9.62 with K(x) = AKν (–λx). 29.

x

y(x) – a

 AKν (kx) + B – AB(x – t)Kν (kx) y(t) dt = f (x).

This is a special case of equation 2.9.7 with λ = B and g(x) = AKν (kx). 30.

x

y(x) + a

 AKν (kt) + B + AB(x – t)Kν (kt) y(t) dt = f (x).

This is a special case of equation 2.9.8 with λ = B and g(t) = AKν (kt).

191

2.9. EQUATIONS WHOSE KERNELS CONTAIN ARBITRARY FUNCTIONS

2.9. Equations Whose Kernels Contain Arbitrary Functions 2.9-1. Equations with Degenerate Kernel: K(x, t) = g1 (x)h1 (t) + · · · + gn (x)hn (t). 1.

x

y(x) – λ

g(x) g(t)

a

y(t) dt = f (x).

Solution:

x

eλ(x–t)

y(x) = f (x) + λ a

2.

g(x) f (t) dt. g(t)

x

y(x) –

g(x)h(t)y(t) dt = f (x). a

Solution:



R(x, t)f (t) dt,

y(x) = f (x) + 3.

 where R(x, t) = g(x)h(t) exp

x a

x

 g(s)h(s) ds .

t

x

(x – t)g(x)y(t) dt = f (x).

y(x) + a

This is a special case of equation 2.9.11. 1◦ . Solution: 1 y(x) = f (x) + W



x

 Y1 (x)Y2 (t) – Y2 (x)Y1 (t) g(x)f (t) dt,

(1)

a

where Y1 = Y1 (x) and Y2 = Y2 (x) are two linearly independent solutions (Y1 /Y2 ≡/ const) of  the second-order linear homogeneous differential equation Yxx + g(x)Y = 0. In this case, the   Wronskian is a constant: W = Y1 (Y2 )x – Y2 (Y1 )x ≡ const.

4.

2◦ . Given only one nontrivial solution Y1 = Y1 (x) of the linear homogeneous differential  equation Yxx + g(x)Y = 0, one can obtain the solution of the integral equation by formula (1) with x dξ , W = 1, Y2 (x) = Y1 (x) 2 b Y1 (ξ) where b is an arbitrary number. x y(x) + (x – t)g(t)y(t) dt = f (x). a

This is a special case of equation 2.9.12. 1◦ . Solution: 1 y(x) = f (x) + W



x

 Y1 (x)Y2 (t) – Y2 (x)Y1 (t) g(t)f (t) dt,

(1)

a

where Y1 = Y1 (x) and Y2 = Y2 (x) are two linearly independent solutions (Y1 /Y2 ≡/ const) of  the second-order linear homogeneous differential equation Yxx + g(x)Y = 0. In this case, the   Wronskian is a constant: W = Y1 (Y2 )x – Y2 (Y1 )x ≡ const. 2◦ . Given only one nontrivial solution Y1 = Y1 (x) of the linear homogeneous differential  equation Yxx + g(x)Y = 0, one can obtain the solution of the integral equation by formula (1) with x dξ , W = 1, Y2 (x) = Y1 (x) 2 b Y1 (ξ) where b is an arbitrary number.

192

LINEAR EQUATIONS OF THE SECOND KIND WITH VARIABLE LIMIT OF INTEGRATION

5.

x

y(x) +

 g(x) – g(t) y(t) dt = f (x).

a

1◦ . Differentiating the equation with respect to x yields yx (x)

+

gx (x)



x

y(t) dt = fx (x).

(1)

a

x

y(t) dt, we obtain the second-order linear ordinary Introducing the new variable Y (x) = a differential equation  Yxx + gx (x)Y = fx (x), (2) which must be supplemented by the initial conditions Y (a) = 0,

Yx (a) = f (a).

(3)

Conditions (3) follow from the original equation and the definition of Y (x). For exact solutions of second-order linear ordinary differential equations (2) with various f (x), see E. Kamke (1977), G. M. Murphy (1960), and A. D. Polyanin and V. F. Zaitsev (2003). 2◦ . Let Y1 = Y1 (x) and Y2 = Y2 (x) be two linearly independent solutions (Y1 /Y2 ≡/ const) of  the second-order linear homogeneous differential equation Yxx + gx (x)Y = 0, which follows from (2) for f (x) ≡ 0. In this case, the Wronskian is a constant: W = Y1 (Y2 )x – Y2 (Y1 )x ≡ const . Solving the nonhomogeneous equation (2) under the initial conditions (3) with arbitrary f = f (x) and taking into account y(x) = Yx (x), we obtain the solution of the original integral equation in the form y(x) = f (x) +

1 W



x

 Y1 (x)Y2 (t) – Y2 (x)Y1 (t) f (t) dt,

(4)

a

where the primes stand for the differentiation with respect to the argument specified in the parentheses. 3◦ . Given only one nontrivial solution Y1 = Y1 (x) of the linear homogeneous differential  equation Yxx + gx (x)Y = 0, one can obtain the solution of the nonhomogeneous equation (2) under the initial conditions (3) by formula (4) with W = 1,

x

Y2 (x) = Y1 (x) b

dξ , Y12 (ξ)

where b is an arbitrary number. 6.

x

y(x) +

 g(x) + h(t) y(t) dt = f (x).

a ◦

1 . Differentiating the equation with respect to x yields

 yx (x) + g(x) + h(x) y(x) + gx (x)

a

x

y(t) dt = fx (x).

2.9. EQUATIONS WHOSE KERNELS CONTAIN ARBITRARY FUNCTIONS

193

x

y(t) dt, we obtain the second-order linear ordinary Introducing the new variable Y (x) = a differential equation

  Yxx + g(x) + h(x) Yx + gx (x)Y = fx (x), (1) which must be supplemented by the initial conditions Yx (a) = f (a).

Y (a) = 0,

(2)

Conditions (3) follow from the original equation and the definition of Y (x). For exact solutions of second-order linear ordinary differential equations (1) with various f (x), see E. Kamke (1977), G. M. Murphy (1960), and A. D. Polyanin and V. F. Zaitsev (2003). 2◦ . Let Y1 = Y1 (x) and Y2 = Y2 (x) be two linearly independent solutions (Y1/Y2 ≡/ const) of the

 second-order linear homogeneous differential equation Yxx + g(x) + h(x) Yx + gx (x)Y = 0, which follows from (1) for f (x) ≡ 0. Solving the nonhomogeneous equation (1) under the initial conditions (2) with arbitrary f = f (x) and taking into account y(x) = Yx (x), we obtain the solution of the original integral equation in the form x y(x) = f (x) + R(x, t)f (t) dt, a   ∂ 2 Y1 (x)Y2 (t) – Y2 (x)Y1 (t) R(x, t) = , W (x) = Y1 (x)Y2 (x) – Y2 (x)Y1 (x), ∂x∂t W (t) where W (x) is the Wronskian and the primes stand for the differentiation with respect to the argument specified in the parentheses. 7.

x

y(x) –

 g(x) + λ – λ(x – t)g(x) y(t) dt = f (x).

a

This is a special case of equation 2.9.16 with h(x) = λ. Solution: x R(x, t)f (t) dt, y(x) = f (x) + a  x  x G(x) λ2 λ(x–s) R(x, t) = [g(x) + λ] + e G(s) ds, G(x) = exp g(s) ds . G(t) G(t) t a 8.

x

y(x) +

 g(t) + λ + λ(x – t)g(t) y(t) dt = f (x).

a

Solution:

x

y(x) = f (x) + R(x, t)f (t) dt, a  x  x G(t) λ2 λ(t–s) e G(s) ds, G(x) = exp g(s) ds . R(x, t) = –[g(t) + λ] + G(x) G(x) t a 9.

x

y(x) – a

 g1 (x) + g2 (x)t y(t) dt = f (x).

This equation can be rewritten in the form of equation 2.9.11 with g1 (x) = g(x) + xh(x) and g2 (x) = –h(x).

194

LINEAR EQUATIONS OF THE SECOND KIND WITH VARIABLE LIMIT OF INTEGRATION



x

 g1 (t) + g2 (t)x y(t) dt = f (x).

10.

y(x) –

11.

This equation can be rewritten in the form of equation 2.9.12 with g1 (t) = g(t) + th(t) and g2 (t) = –h(t). x

 g(x) + h(x)(x – t) y(t) dt = f (x). y(x) –

a

a  1◦ . The solution of the integral equation can be represented in the form y(x) = Yxx , where Y = Y (x) is the solution of the second-order linear nonhomogeneous ordinary differential equation  Yxx – g(x)Yx – h(x)Y = f (x), (1)

under the initial conditions

Y (a) = Yx (a) = 0.

(2)



2 . Let Y1 = Y1 (x) and Y2 = Y2 (x) be two nontrivial linearly independent solutions of the  second-order linear homogeneous differential equation Yxx –g(x)Yx –h(x)Y = 0, which follows from (1) for f (x) ≡ 0. Then the solution of the nonhomogeneous differential equation (1) under conditions (2) is given by x

 f (t) Y (x) = dt, W (t) = Y1 (t)Y2 (t) – Y2 (t)Y1 (t), (3) Y2 (x)Y1 (t) – Y1 (x)Y2 (t) W (t) a where W (t) is the Wronskian and the primes denote the derivatives. Substituting (3) into (1), we obtain the solution of the original integral equation in the form x 1 y(x) = f (x) + [Y  (x)Y1 (t) – Y1 (x)Y2 (t)]. R(x, t)f (t) dt, R(x, t) = (4) W (t) 2 a 3◦ . Let Y1 = Y1 (x) be a nontrivial particular solution of the homogeneous differential equation (1) (with f ≡ 0) satisfying the initial condition Y1 (a) ≠ 0. Then the function  x  x W (t) dt, W (x) = exp g(s) ds (5) Y2 (x) = Y1 (x) 2 a [Y1 (t)] a is another nontrivial solution of the homogeneous equation. Substituting (5) into (4) yields the solution of the original integral equation in the form x R(x, t)f (t) dt, y(x) = f (x) + a W (x) Y1 (t) Y1 (t) x W (s)  + [g(x)Y1 (x) + h(x)Y1 (x)] R(x, t) = g(x) ds, Y1 (x) W (t) W (t) t [Y1 (s)]2  x  where W (x) = exp g(s) ds . a

12.

x

y(x) –

 g(t) + h(t)(t – x) y(t) dt = f (x).

a

Solution:



x

R(x, t)f (t) dt,

y(x) = f (x) + a

R(x, t) = g(t)

Y (x)W (x) + Y (x)W (x)[g(t)Yt (t) + h(t)Y (t)] Y (t)W (t)   t g(t) dt , W (t) = exp b



t

x

ds , W (s)[Y (s)]2

195

2.9. EQUATIONS WHOSE KERNELS CONTAIN ARBITRARY FUNCTIONS

where Y = Y (x) is an arbitrary nontrivial solution of the second-order homogeneous differential equation  Yxx + g(x)Yx + h(x)Y = 0 satisfying the condition Y (a) ≠ 0. 13.

x

(x – t)g(x)h(t)y(t) dt = f (x).

y(x) + a

The substitution y(x) = g(x)u(x) leads to an equation of the form 2.9.4:

x

(x – t)g(t)h(t)u(t) dt = f (x)/g(x).

u(x) + a

14.

x

y(x) –



 g(x) + λxn + λ(x – t)xn–1 [n – xg(x)] y(t) dt = f (x).

a

This is a special case of equation 2.9.16 with h(x) = λxn . Solution: x y(x) = f (x) + R(x, t)f (t) dt,

a

G(x) H(x) + λ(λx2n + nxn–1 ) R(x, t) = [g(x) + λx ] G(t) G(t)

x

n

 where G(x) = exp

x

a

15.

x

y(x) –



a

t

G(s) ds, H(s)



  λ xn+1 . g(s) ds and H(x) = exp n+1

 g(x) + λ + (x – t)[gx (x) – λg(x)] y(t) dt = f (x).

This is a special case of equation 2.9.16. Solution:

x

R(x, t)f (t) dt,

y(x) = f (x) + a

R(x, t) = [g(x) + λ]e

λ(x–t)

  + [g(x)]2 + gx (x) G(x)



x

eλ(s–t) ds, G(s)

t

 where G(x) = exp

 x g(s) ds .

a

16.

x

y(x) – a



 g(x) + h(x) + (x – t)[hx (x) – g(x)h(x)] y(t) dt = f (x).

Solution:

x

R(x, t)f (t) dt,

y(x) = f (x) + a

R(x, t) = [g(x) + h(x)]  where G(x) = exp a

x

G(x) H(x) + {[h(x)]2 + hx (x)} G(t) G(t)

  g(s) ds and H(x) = exp a

x

 h(s) ds .

t

x

G(s) ds, H(s)

196

LINEAR EQUATIONS OF THE SECOND KIND WITH VARIABLE LIMIT OF INTEGRATION

17.

x



y(x) + a

 ϕx (x)   + ϕ(t)gx (x) – ϕx (x)g(t) h(t) y(t) dt = f (x). ϕ(t)

1◦ . This equation is equivalent to the equation x a

x  ϕ(x) + ϕ(t)g(x) – ϕ(x)g(t) h(t) y(t) dt = F (x), F (x) = f (x) dx, ϕ(t) a

(1)

obtained by differentiating the original equation with respect to x. Equation (1) is a special case of equation 1.9.15 with g1 (x) = g(x),

h1 (t) = ϕ(t)h(t),

g2 (x) = ϕ(x),

h2 (t) =

1 – g(t)h(t). ϕ(t)

2◦ . Solution: 

x d F (t) ϕ2 (t)h(t) 1 Ξ(x) dt , y(x) = ϕ(x)h(x) dx ϕ(t) t Ξ(t) a  x

x g(t) F (x) = f (x) dx, Ξ(x) = exp – ϕ2 (t)h(t) dt . ϕ(t) t a a 18.

x

y(x) – a



ϕt (t) ϕ(x)



   + ϕ(x)gt (t) – ϕt (t)g(x) h(x) y(t) dt = f (x).

1◦ . Let f (a) = 0. The change



x

w(t) dt

y(x) =

(1)

a

followed by the integration by parts leads to the equation x a

 ϕ(t) + ϕ(x)g(t) – ϕ(t)g(x) h(x) w(t) dt = f (x), ϕ(x)

(2)

which is a special case of equation 1.9.15 with g1 (x) =

1 – g(x)h(x), ϕ(x)

h1 (t) = ϕ(t),

g2 (x) = ϕ(x)h(x),

h2 (t) = g(t).

The solution of equation (2) is given by

y(x) =



x dt f (t) 1 d ϕ2 (x)h(x)Φ(x) , ϕ(x) dx ϕ(t)h(t) t Φ(t) a x  

g(t) 2 Φ(x) = exp ϕ (t)h(t) dt . ϕ(t) t a

2◦ . Let f (a) ≠ 0. The substitution y(x) = y(x) ¯ + f (a) leads to the integral equation y(x) ¯ with ¯ satisfying the condition f¯(a) = 0. Thus we obtain case 1◦ . the right-hand side f(x)

2.9. EQUATIONS WHOSE KERNELS CONTAIN ARBITRARY FUNCTIONS

19.

x

y(x) – a

 n

197

 gk (x)(x – t)k–1 y(t) dt = f (x).

k=1

The solution can be represented in the form

x

y(x) = f (x) +

R(x, t)f (t) dt.

(1)

dn w , dxn

(2)

a

Here the resolvent R(x, t) is given by R(x, t) = wx(n) ,

wx(n) =

where w is the solution of the nth-order linear homogeneous ordinary differential equation wx(n) – g1 (x)wx(n–1) – g2 (x)wx(n–2) – 2g3 (x)wx(n–3) – · · · – (n – 1)! gn (x)w = 0 satisfying the following initial conditions at x = t: w = wx = · · · = wx(n–2) x=t

x=t

x=t

= 0,

wx(n–1) x=t = 1.

(3)

(4)

Note that the differential equation (3) implicitly depends on t via the initial conditions (4). References: E. Goursat (1923), A. F. Verlan’ and V. S. Sizikov (1987).

20.

x

y(x) – a

 n

gk (t)(t – x)

k–1

 y(t) dt = f (x).

k=1

The solution can be represented in the form

x

y(x) = f (x) +

R(x, t)f (t) dt.

(1)

dn u , dtn

(2)

a

Here the resolvent R(x, t) is given by R(x, t) = –u(n) t ,

u(n) t =

where u is the solution of the nth-order linear homogeneous ordinary differential equation (n–1) + g2 (t)u(n–2) + 2g3 (t)u(n–3) + · · · + (n – 1)! gn (t)u = 0, u(n) t + g1 (t)ut t t

satisfying the following initial conditions at t = x: = 0, u t=x = ut t=x = · · · = u(n–2) t t=x

= 1. u(n–1) t t=x

(3)

(4)

Note that the differential equation (3) implicitly depends on x via the initial conditions (4). References: E. Goursat (1923), A. F. Verlan’ and V. S. Sizikov (1987).

21.

x

y(x) +

 λx+µt  e – eµx+λt g(t)y(t) dt = f (x).

a

Let us differentiate the equation twice and then eliminate the integral terms from the resulting relations and the original equation. As a result, we arrive at the second-order linear ordinary differential equation

   yxx – (λ + µ)yx + (λ – µ)e(λ+µ)x g(x) + λµ y = fxx (x) – (λ + µ)fx (x) + λµf (x), which must be supplemented by the initial conditions y(a) = f (a), yx (a) = fx (a).

198

LINEAR EQUATIONS OF THE SECOND KIND WITH VARIABLE LIMIT OF INTEGRATION

22.

x

y(x) +

 eλx g(t) + eµx h(t) y(t) dt = f (x).

a

Let us differentiate the equation twice and then eliminate the integral terms from the resulting relations and the original equation. As a result, we arrive at the second-order linear ordinary differential equation



 yxx + eλx g(x) + eµx h(x) – λ – µ yx + eλx gx (x) + eµx hx (x)   + (λ – µ)eλx g(x) + (µ – λ)eµx h(x) + λµ y = fxx (x) – (λ + µ)fx (x) + λµf (x), which must be supplemented by the initial conditions

 yx (a) = fx (a) – eλa g(a) + eµa h(a) f (a).

y(a) = f (a), Example. The Arutyunyan equation, y(x) –

x a

ϕ(t)

∂ ∂t



 

1 + ψ(t) 1 – e–λ(x–t) y(t) dt = f (x), ϕ(t)

can be reduced to the above equation. The former is encountered in the theory of viscoelasticity for aging solids. The solution of the Arutyunyan equation is given by y(x) = f (x) –

x a

  x 1 ∂ ϕ(t) – λψ(t)ϕ2 (t)eη(t) e–η(s) ds f (t) dt, ϕ(t) ∂t t

where

x η(x) = a

 ϕ (t) λ 1 + ψ(t)ϕ(t) – ϕ(t)

dt.

Reference: N. Kh. Arutyunyan (1966).

23.

x

y(x) +

   λeλ(x–t) + µeµx+λt – λeλx+µt h(t) y(t) dt = f (x).

a

This is a special case of equation 2.9.17 with ϕ(x) = eλx and g(x) = eµx . Solution:

 x d F (t) e2λt h(t) 1 Φ(x) dt , eλx h(x) dx eλt t Φ(t) a   x x (λ+µ)t F (x) = f (t) dt, Φ(x) = exp (λ – µ) e h(t) dt . y(x) =

a

24.

x

y(x) –

a

   λe–λ(x–t) + µeλx+µt – λeµx+λt h(x) y(t) dt = f (x).

a

This is a special case of equation 2.9.18 with ϕ(x) = eλx and g(x) = eµx . Assume that f (a) = 0. Solution:

x



 e2λx h(x) x f (t) w(t) dt, w(x) = e Φ(t) dt , Φ(x) eλt h(t) t a   x e(λ+µ)t h(t) dt . Φ(x) = exp (λ – µ) –λx

y(x) = a

a

d dx



199

2.9. EQUATIONS WHOSE KERNELS CONTAIN ARBITRARY FUNCTIONS

25.

x

y(x) –



 g(x) + beλx + b(x – t)eλx [λ – g(x)] y(t) dt = f (x).

a

This is a special case of equation 2.9.16 with h(x) = beλx . Solution: x R(x, t)f (t) dt, y(x) = f (x) + a

G(x) H(x) + (b2 e2λx + bλeλx ) R(x, t) = [g(x) + beλx ] G(t) G(t)  where G(x) = exp a

26.

x

y(x) +



a

x



x

t

G(s) ds, H(s)

   b λx . g(s) ds and H(x) = exp e λ

  λeλ(x–t) + eλt gx (x) – λeλx g(t) h(t) y(t) dt = f (x).

This is a special case of equation 2.9.17 with ϕ(x) = eλx . 27.

x

y(x) – a



  λe–λ(x–t) + eλx gt (t) – λeλt g(x) h(x) y(t) dt = f (x).

This is a special case of equation 2.9.18 with ϕ(x) = eλx . 28.

x

y(x) +

cosh[λ(x – t)]g(t)y(t) dt = f (x). a

Differentiating the equation with respect to x twice yields yx (x)



x

sinh[λ(x – t)]g(t)y(t) dt = fx (x), x

  2  yxx (x) + g(x)y(x) x + λ cosh[λ(x – t)]g(t)y(t) dt = fxx (x). + g(x)y(x) + λ

(1)

a

(2)

a

Eliminating the integral term from (2) with the aid of the original equation, we arrive at the second-order linear ordinary differential equation

   yxx + g(x)y x – λ2 y = fxx (x) – λ2 f (x).

(3)

By setting x = a in the original equation and (1), we obtain the initial conditions for y = y(x): y(a) = f (a),

yx (a) = fx (a) – f (a)g(a).

(4)

Equation (3) under conditions (4) determines the solution of the original integral equation. 29.

x

y(x) +

cosh[λ(x – t)]g(x)h(t)y(t) dt = f (x). a

The substitution y(x) = g(x)u(x) leads to an equation of the form 2.9.28:

x

u(x) +

cosh[λ(x – t)]g(t)h(t)u(t) dt = f (x)/g(x). a

200

LINEAR EQUATIONS OF THE SECOND KIND WITH VARIABLE LIMIT OF INTEGRATION

30.

x

y(x) +

sinh[λ(x – t)]g(t)y(t) dt = f (x). a

1◦ . Differentiating the equation with respect to x twice yields yx (x) + λ



x

cosh[λ(x – t)]g(t)y(t) dt = fx (x), x   yxx (x) + λg(x)y(x) + λ2 sinh[λ(x – t)]g(t)y(t) dt = fxx (x).

(1)

a

(2)

a

Eliminating the integral term from (2) with the aid of the original equation, we arrive at the second-order linear ordinary differential equation

   yxx + λ g(x) – λ y = fxx (x) – λ2 f (x).

(3)

By setting x = a in the original equation and (1), we obtain the initial conditions for y = y(x): y(a) = f (a),

yx (a) = fx (a).

(4)

For exact solutions of second-order linear ordinary differential equations (3) with various g(x), see E. Kamke (1977), G. M. Murphy (1960), and A. D. Polyanin and V. F. Zaitsev (2003). 2◦ . Let y1 = y1 (x) and y2 = y2 (x) be two linearly solutions (y1 /y2 ≡/ const) of

independent   the homogeneous differential equation yxx + λ g(x) – λ y = 0, which follows from (3) for f (x) ≡ 0. In this case, the Wronskian is a constant: W = y1 (y2 )x – y2 (y1 )x ≡ const . The solution of the nonhomogeneous equation (3) under conditions (4) with arbitrary f = f (x) has the form x

 λ y(x) = f (x) + y1 (x)y2 (t) – y2 (x)y1 (t) g(t)f (t) dt (5) W a and determines the solution of the original integral equation. 3◦ . Given only one nontrivial solution y1 = y1 (x) of the linear homogeneous differential   equation yxx +λ g(x)–λ y = 0, one can obtain the solution of the nonhomogeneous equation (3) under the initial conditions (4) by formula (5) with W = 1,

y2 (x) = y1 (x) b

x

dξ , y12 (ξ)

where b is an arbitrary number. 31.

x

sinh[λ(x – t)]g(x)h(t)y(t) dt = f (x).

y(x) + a

The substitution y(x) = g(x)u(x) leads to an equation of the form 2.9.30:

x

u(x) +

sinh[λ(x – t)]g(t)h(t)u(t) dt = f (x)/g(x). a

201

2.9. EQUATIONS WHOSE KERNELS CONTAIN ARBITRARY FUNCTIONS

32.

x

y(x) –



 g(x) + b cosh(λx) + b(x – t)[λ sinh(λx) – cosh(λx)g(x)] y(t) dt = f (x).

a

This is a special case of equation 2.9.16 with h(x) = b cosh(λx). Solution: x R(x, t)f (t) dt, y(x) = f (x) + a

 H(x) G(x) 2 + b cosh2 (λx) + bλ sinh(λx) R(x, t) = [g(x) + b cosh(λx)] G(t) G(t)  x    b where G(x) = exp sinh(λx) . g(s) ds and H(x) = exp λ a 33.

x

y(x) –





x t

G(s) ds, H(s)

 g(x) + b sinh(λx) + b(x – t)[λ cosh(λx) – sinh(λx)g(x)] y(t) dt = f (x).

a

This is a special case of equation 2.9.16 with h(x) = b sinh(λx). Solution: x R(x, t)f (t) dt, y(x) = f (x) + a

 H(x) G(x) 2 + b sinh2 (λx) + bλ cosh(λx) R(x, t) = [g(x) + b sinh(λx)] G(t) G(t)  x    b where G(x) = exp cosh(λx) . g(s) ds and H(x) = exp λ a 34.



x

t

G(s) ds, H(s)

x

cos[λ(x – t)]g(t)y(t) dt = f (x).

y(x) + a

Differentiating the equation with respect to x twice yields x sin[λ(x – t)]g(t)y(t) dt = fx (x), yx (x) + g(x)y(x) – λ a x

  2  yxx (x) + g(x)y(x) x – λ cos[λ(x – t)]g(t)y(t) dt = fxx (x).

(1) (2)

a

Eliminating the integral term from (2) with the aid of the original equation, we arrive at the second-order linear ordinary differential equation

   yxx + g(x)y x + λ2 y = fxx (x) + λ2 f (x).

(3)

By setting x = a in the original equation and (1), we obtain the initial conditions for y = y(x): y(a) = f (a), 35.

yx (a) = fx (a) – f (a)g(a).

x

cos[λ(x – t)]g(x)h(t)y(t) dt = f (x).

y(x) + a

The substitution y(x) = g(x)u(x) leads to an equation of the form 2.9.34: x cos[λ(x – t)]g(t)h(t)u(t) dt = f (x)/g(x). u(x) + a

(4)

202

LINEAR EQUATIONS OF THE SECOND KIND WITH VARIABLE LIMIT OF INTEGRATION

36.

x

y(x) +

sin[λ(x – t)]g(t)y(t) dt = f (x). a

1◦ . Differentiating the equation with respect to x twice yields yx (x)



x

cos[λ(x – t)]g(t)y(t) dt = fx (x), a x   yxx (x) + λg(x)y(x) – λ2 sin[λ(x – t)]g(t)y(t) dt = fxx (x). +λ

(1) (2)

a

Eliminating the integral term from (2) with the aid of the original equation, we arrive at the second-order linear ordinary differential equation

   + λ g(x) + λ y = fxx (x) + λ2 f (x). yxx

(3)

By setting x = a in the original equation and (1), we obtain the initial conditions for y = y(x): y(a) = f (a),

yx (a) = fx (a).

(4)

For exact solutions of second-order linear ordinary differential equations (3) with various f (x), see E. Kamke (1977) and A. D. Polyanin and V. F. Zaitsev (2003). 2◦ . Let y1 = y1 (x) and y2 = y2 (x) be two linearly solutions (y1 /y2 ≡/ const) of

independent   the homogeneous differential equation yxx + λ g(x) – λ y = 0, which follows from (3) for f (x) ≡ 0. In this case, the Wronskian is a constant: W = y1 (y2 )x – y2 (y1 )x ≡ const . The solution of the nonhomogeneous equation (3) under conditions (4) with arbitrary f = f (x) has the form x

 λ y(x) = f (x) + y1 (x)y2 (t) – y2 (x)y1 (t) g(t)f (t) dt (5) W a and determines the solution of the original integral equation. solution y1 = y1 (x) of the linear homogeneous differential equa3◦ . Given only

one nontrivial   tion yxx + λ g(x) + λ y = 0, one can obtain the solution of the nonhomogeneous equation (3) under the initial conditions (4) by formula (5) with W = 1,

y2 (x) = y1 (x) b

x

dξ , y12 (ξ)

where b is an arbitrary number. 37.

x

y(x) +

sin[λ(x – t)]g(x)h(t)y(t) dt = f (x). a

The substitution y(x) = g(x)u(x) leads to an equation of the form 2.9.36:

x

sin[λ(x – t)]g(t)h(t)u(t) dt = f (x)/g(x).

u(x) + a

203

2.9. EQUATIONS WHOSE KERNELS CONTAIN ARBITRARY FUNCTIONS

38.

x

y(x) –



 g(x) + b cos(λx) – b(x – t)[λ sin(λx) + cos(λx)g(x)] y(t) dt = f (x).

a

This is a special case of equation 2.9.16 with h(x) = b cos(λx). Solution: x y(x) = f (x) + R(x, t)f (t) dt,

39.

a  H(x) x G(s) G(x) 2 + b cos2 (λx) – bλ sin(λx) ds, R(x, t) = [g(x) + b cos(λx)] G(t) G(t) t H(s)  x    b where G(x) = exp sin(λx) . g(s) ds and H(x) = exp λ a x   y(x) – g(x) + b sin(λx) + b(x – t)[λ cos(λx) – sin(λx)g(x)] y(t) dt = f (x). a

This is a special case of equation 2.9.16 with h(x) = b sin(λx). Solution: x

y(x) = f (x) +

R(x, t)f (t) dt, a

 H(x) G(x) 2 2 + b sin (λx) + bλ cos(λx) R(x, t) = [g(x) + b sin(λx)] G(t) G(t)  x    b where G(x) = exp g(s) ds and H(x) = exp – cos(λx) . λ a

t

x

G(s) ds, H(s)

2.9-2. Equations with Difference Kernel: K(x, t) = K(x – t). x 40. y(x) + K(x – t)y(t) dt = f (x). a

Renewal equation. 1◦ . To solve this integral equation, direct and inverse Laplace transforms are used. The solution can be represented in the form x y(x) = f (x) – R(x – t)f (t) dt.

(1)

a

Here the resolvent R(x) is expressed via the kernel K(x) of the original equation as follows: c+i∞ 1 px ˜ dp, R(x) = R(p)e 2πi c–i∞ ∞ ˜ K(p) ˜ ˜ , K(p) = K(x)e–px dx. R(p) = ˜ 1 + K(p) 0 References: R. Bellman and K. L. Cooke (1963), M. L. Krasnov, A. I. Kisilev, and G. I. Makarenko (1971), V. I. Smirnov (1974).

2◦ . Let w = w(x) be the solution of the simpler auxiliary equation with a = 0 and f ≡ 1: x K(x – t)w(t) dt = 1. (2) w(x) + 0

Then the solution of the original integral equation with arbitrary f = f (x) is expressed via the solution of the auxiliary equation (2) as x x d w(x – t)f (t) dt = f (a)w(x – a) + w(x – t)ft (t) dt. y(x) = dx a a Reference: R. Bellman and K. L. Cooke (1963).

204

LINEAR EQUATIONS OF THE SECOND KIND WITH VARIABLE LIMIT OF INTEGRATION

41.

x

y(x) +

K(x – t)y(t) dt = 0. –∞

Eigenfunctions of this integral equation are determined by the roots of the following transcendental (algebraic) equation for the parameter λ: ∞ K(z)e–λz dz = –1. (1) 0

The left-hand side of this equation is the Laplace transform of the kernel of the integral equation. 1◦ . For a real simple root λk of equation (1) there is a corresponding eigenfunction yk (x) = exp(λk x). ◦

2 . For a real root λk of multiplicity r there are corresponding r eigenfunctions yk1 (x) = exp(λk x),

yk2 (x) = x exp(λk x),

...,

ykr (x) = xr–1 exp(λk x).

3◦ . For a complex simple root λk = αk + iβk of equation (1) there is a corresponding eigenfunction pair yk(1) (x) = exp(αk x) cos(βk x),

yk(2) (x) = exp(αk x) sin(βk x).

4◦ . For a complex root λk = αk +iβk of multiplicity r there are corresponding r eigenfunction pairs (1) (2) yk1 (x) = exp(αk x) cos(βk x), (x) = exp(αk x) sin(βk x), yk1 (1) yk2 (x) = x exp(αk x) cos(βk x),

(2) yk2 (x) = x exp(αk x) sin(βk x),

⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ (1) (x) ykr

⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅

(2) = x exp(αk x) cos(βk x), ykr (x) = xr–1 exp(αk x) sin(βk x). The general solution is the combination (with arbitrary constants) of the eigenfunctions of the homogeneous integral equation. r–1

 For equations 2.9.42–2.9.51, only particular solutions are given. To obtain the general solution, one must add the general solution of the corresponding homogeneous equation 2.9.41 to the particular solution. x 42. y(x) + K(x – t)y(t) dt = Axn , n = 0, 1, 2, . . . –∞

This is a special case of equation 2.9.44 with λ = 0. 1◦ . A solution with n = 0: y(x) = 2◦ . A solution with n = 1: AC A y(x) = x + 2 , B B 3◦ . A solution with n = 2:

A , B

B =1+



K(z) dz. 0

B =1+





K(z) dz,

C=



zK(z) dz.

0

0

A 2 AC AC 2 AD x +2 2 x+2 3 – 2 , B B B ∞ B ∞ ∞ B =1+ K(z) dz, C = zK(z) dz, D = z 2 K(z) dz. y2 (x) =

0

0

4◦ . A solution with n = 3, 4, . . . is given by: n  λx 

e ∂ yn (x) = A , n ∂λ B(λ) λ=0

0

B(λ) = 1 + 0



K(z)e–λz dz.

205

2.9. EQUATIONS WHOSE KERNELS CONTAIN ARBITRARY FUNCTIONS

43.

x

y(x) +

K(x – t)y(t) dt = Aeλx .

–∞

A solution:



A y(x) = eλx , B



B =1+

K(z)e–λz dz.

0

The integral term in the expression for B is the Laplace transform of K(z), which may be calculated using tables of Laplace transforms (e.g., see Supplement 5). 44.

x

y(x) +

K(x – t)y(t) dt = Axn eλx ,

n = 1, 2, . . .

–∞

1◦ . A solution with n = 1: A λx AC λx xe + 2 e , B B ∞ ∞ B =1+ K(z)e–λz dz, C = zK(z)e–λz dz. y1 (x) =

0

0

It is convenient to calculate B and C using tables of Laplace transforms. 2◦ . A solution with n = 2:   A 2 λx AC λx AC 2 AD λx y2 (x) = x e + 2 2 xe + 2 3 – 2 e , B B B B ∞ ∞ ∞ B =1+ K(z)e–λz dz, C = zK(z)e–λz dz, D = z 2 K(z)e–λz dz. 0

0

0

3◦ . A solution with n = 3, 4, . . . is given by:  λx  e ∂ ∂n yn–1 (x) = A n , yn (x) = ∂λ ∂λ B(λ) 45.





B(λ) = 1 +

K(z)e–λz dz.

0

x

y(x) +

K(x – t)y(t) dt = A cosh(λx). –∞

A solution: y(x) =

A λx 1 A A –λx 1  A A  A  cosh(λx) + sinh(λx), e + e = + – 2B– 2B+ 2 B– B+ 2 B– B+ ∞ ∞ B– = 1 + K(z)e–λz dz, B+ = 1 + K(z)eλz dz. 0

46.

0

x

y(x) +

K(x – t)y(t) dt = A sinh(λx). –∞

A solution: y(x) =

A λx 1 A A –λx 1  A A  A  cosh(λx) + sinh(λx), e – e = – + 2B– 2B+ 2 B– B+ 2 B– B+ ∞ ∞ B– = 1 + K(z)e–λz dz, B+ = 1 + K(z)eλz dz. 0

0

206

LINEAR EQUATIONS OF THE SECOND KIND WITH VARIABLE LIMIT OF INTEGRATION

47.

x

y(x) +

K(x – t)y(t) dt = A cos(λx). –∞

A solution:

 A Bc cos(λx) – Bs sin(λx) , y(x) = 2 2 Bc + Bs ∞ ∞ Bc = 1 + K(z) cos(λz) dz, Bs = K(z) sin(λz) dz. 0

48.

0

x

y(x) +

K(x – t)y(t) dt = A sin(λx). –∞

A solution:

 A Bc sin(λx) + Bs cos(λx) , y(x) = 2 Bc + Bs2 ∞ ∞ Bc = 1 + K(z) cos(λz) dz, Bs = K(z) sin(λz) dz. 0

49.

x

y(x) +

0

K(x – t)y(t) dt = Aeµx cos(λx).

–∞

A solution:

 A eµx Bc cos(λx) – Bs sin(λx) , 2 + Bs ∞ ∞ Bc = 1 + K(z)e–µz cos(λz) dz, Bs = K(z)e–µz sin(λz) dz. y(x) =

Bc2

0

50.

x

y(x) +

0

K(x – t)y(t) dt = Aeµx sin(λx).

–∞

A solution:

 A eµx Bc sin(λx) + Bs cos(λx) , 2 + Bs ∞ ∞ Bc = 1 + K(z)e–µz cos(λz) dz, Bs = K(z)e–µz sin(λz) dz. y(x) =

Bc2

0

51.

0

x

y(x) +

K(x – t)y(t) dt = f (x). –∞

1◦ . For a polynomial right-hand side, f (x) =

n 

Ak xk , a solution has the form

k=0

y(x) =

n 

Bk xk ,

k=0

where the constants Bk are found by the method of undetermined coefficients. One can also make use of the formula given in item 4◦ of equation 2.9.42 to construct the solution.

2.9. EQUATIONS WHOSE KERNELS CONTAIN ARBITRARY FUNCTIONS

2◦ . For f (x) = eλx

n 

207

Ak xk , a solution of the equation has the form

k=0

y(x) = eλx

n 

Bk xk ,

k=0

where the Bk are found by the method of undetermined coefficients. One can also make use of the formula given in item 3◦ of equation 2.9.44 to construct the solution. n  3◦ . For f (x) = Ak exp(λk x), a solution of the equation has the form k=0

y(x) =

n  Ak exp(λk x), Bk k=0

4◦ . For f (x) = cos(λx)

n 





Bk = 1 +

K(z) exp(–λk z) dz. 0

Ak xk , a solution of the equation has the form

k=0

y(x) = cos(λx)

n 

Bk xk + sin(λx)

k=0

n 

Ck xk ,

k=0

where the constants Bk and Ck are found by the method of undetermined coefficients. n  5◦ . For f (x) = sin(λx) Ak xk , a solution of the equation has the form k=0

y(x) = cos(λx)

n 

Bk xk + sin(λx)

k=0

n 

Ck xk ,

k=0

where the constants Bk and Ck are found by the method of undetermined coefficients. n  6◦ . For f (x) = Ak cos(λk x), the solution of a equation has the form k=0

y(x) =

n 

 Ak Bck cos(λk x) – Bsk sin(λk x) , 2 + Bsk ∞ K(z) cos(λk z) dz, Bsk = K(z) sin(λk z) dz.

2 Bck k=0 ∞

Bck = 1 + 0

7◦ . For f (x) =

n 

0

Ak sin(λk x), a solution of the equation has the form

k=0

y(x) =

n 

Bck = 1 + 0

8◦ . For f (x) = cos(λx)

 Ak Bck sin(λk x) + Bsk cos(λk x) , 2 + Bsk ∞ K(z) cos(λk z) dz, Bsk = K(z) sin(λk z) dz.

2 Bck k=0 ∞

n 

0

Ak exp(µk x), a solution of the equation has the form

k=0 n n   Ak Bck Ak Bsk exp(µ x) – sin(λx) exp(µk x), k 2 + B2 2 2 B B ck sk ck + Bsk k=0 k=0 ∞ ∞ =1+ K(z) exp(–µk z) cos(λz) dz, Bsk = K(z) exp(–µk z) sin(λz) dz.

y(x) = cos(λx) Bck

0

0

208

LINEAR EQUATIONS OF THE SECOND KIND WITH VARIABLE LIMIT OF INTEGRATION

9◦ . For f (x) = sin(λx)

n 

Ak exp(µk x), a solution of the equation has the form

k=0 n n   Ak Bck Ak Bsk exp(µ x) + cos(λx) exp(µk x), k 2 + B2 2 2 Bck B sk ck + Bsk k=0 k=0 ∞ ∞ =1+ K(z) exp(–µk z) cos(λz) dz, Bsk = K(z) exp(–µk z) sin(λz) dz.

y(x) = sin(λx) Bck

0

52.

0



K(x – t)y(t) dt = 0.

y(x) + x

Eigenfunctions of this integral equation are determined by the roots of the following transcendental (algebraic) equation for the parameter λ:



K(–z)eλz dz = –1.

(1)

0

The left-hand side of this equation is the Laplace transform of the function K(–z) with parameter –λ. 1◦ . For a real simple root λk of equation (1) there is a corresponding eigenfunction yk (x) = exp(λk x). 2◦ . For a real root λk of multiplicity r there are corresponding r eigenfunctions yk1 (x) = exp(λk x),

yk2 (x) = x exp(λk x),

...,

ykr (x) = xr–1 exp(λk x).

3◦ . For a complex simple root λk = αk + iβk of equation (1) there is a corresponding eigenfunction pair yk(1) (x) = exp(αk x) cos(βk x),

yk(2) (x) = exp(αk x) sin(βk x).

4◦ . For a complex root λk = αk +iβk of multiplicity r there are corresponding r eigenfunction pairs (1) (2) yk1 (x) = exp(αk x) cos(βk x), (x) = exp(αk x) sin(βk x), yk1 (1) yk2 (x) = x exp(αk x) cos(βk x),

(2) yk2 (x) = x exp(αk x) sin(βk x),

⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅

⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅

(1) (x) = xr–1 exp(αk x) cos(βk x), ykr

(2) (x) = xr–1 exp(αk x) sin(βk x). ykr

The general solution is the combination (with arbitrary constants) of the eigenfunctions of the homogeneous integral equation.  For equations 2.9.53–2.9.62, only particular solutions are given. To obtain the general solution, one must add the general solution of the corresponding homogeneous equation 2.9.52 to the particular solution. 53.



y(x) +

K(x – t)y(t) dt = Axn ,

n = 0, 1, 2, . . .

x

This is a special case of equation 2.9.55 with λ = 0.

209

2.9. EQUATIONS WHOSE KERNELS CONTAIN ARBITRARY FUNCTIONS

1◦ . A solution with n = 0:



A y(x) = , B



AC A x– 2 , B B

K(–z) dz. 0

2◦ . A solution with n = 1: y(x) =



B =1+





B =1+

K(–z) dz,



C=

zK(–z) dz.

0

0

3◦ . A solution with n = 2: A 2 AC AC 2 AD x –2 2 x+2 3 – 2 , B B B B ∞ ∞ B =1+ K(–z) dz, C = zK(–z) dz, D = y2 (x) =

0

0





y(x) +

z 2 K(–z) dz.

0

4◦ . A solution with n = 3, 4, . . . is given by: n  λx 

e ∂ yn (x) = A , n ∂λ B(λ) λ=0 54.







B(λ) = 1 +

K(–z)eλz dz.

0

K(x – t)y(t) dt = Aeλx .

x

A solution:



A y(x) = eλx , B

55.

B =1+



K(–z)eλz dz = 1 + L{K(–z), –λ}.

0

The integral term in the expression for B is the Laplace transform of K(–z) with parameter –λ, which may be calculated using tables of Laplace transforms (e.g., see H. Bateman and A. Erd´elyi (vol. 1, 1954), V. A. Ditkin and A. P. Prudnikov (1965), and Supplement 5). ∞ y(x) + K(x – t)y(t) dt = Axn eλx , n = 1, 2, . . . x

1◦ . A solution with n = 1: A λx AC λx xe – 2 e , B B ∞ ∞ λz B =1+ K(–z)e dz, C = zK(–z)eλz dz. y1 (x) =

0

0

It is convenient to calculate B and C using tables of Laplace transforms (with parameter –λ). 2◦ . A solution with n = 2:

  A 2 λx AC λx AC 2 AD λx y2 (x) = x e – 2 2 xe + 2 3 – 2 e , B B B B ∞ ∞ ∞ λz λz B =1+ K(–z)e dz, C = zK(–z)e dz, D = z 2 K(–z)eλz dz. 0

0

0



3 . A solution with n = 3, 4, . . . is given by  λx  e ∂ ∂n yn–1 (x) = A n , yn (x) = ∂λ ∂λ B(λ)

B(λ) = 1 + 0



K(–z)eλz dz.

210

LINEAR EQUATIONS OF THE SECOND KIND WITH VARIABLE LIMIT OF INTEGRATION

56.



y(x) +

K(x – t)y(t) dt = A cosh(λx). x

A solution: y(x) =

1 A A –λx 1  A A A A λx cosh(λx) + sinh(λx), e + e = + – 2B+ 2B– 2 B+ B– 2 B+ B– ∞ ∞ B+ = 1 + K(–z)eλz dz, B– = 1 + K(–z)e–λz dz. 0

57.

0



K(x – t)y(t) dt = A sinh(λx).

y(x) + x

A solution: y(x) =

1 A A –λx 1  A A A A λx cosh(λx) + sinh(λx), e – e = – + 2B+ 2B– 2 B+ B– 2 B+ B– ∞ ∞ B+ = 1 + K(–z)eλz dz, B– = 1 + K(–z)e–λz dz. 0

58.

0



K(x – t)y(t) dt = A cos(λx).

y(x) + x

A solution:

 A B cos(λx) + B sin(λx) , y(x) = 2 c s Bc + Bs2 ∞ ∞ Bc = 1 + K(–z) cos(λz) dz, Bs = K(–z) sin(λz) dz. 0

59.

0



K(x – t)y(t) dt = A sin(λx).

y(x) + x

A solution:

 A Bc sin(λx) – Bs cos(λx) , y(x) = 2 2 Bc + Bs ∞ ∞ Bc = 1 + K(–z) cos(λz) dz, Bs = K(–z) sin(λz) dz. 0

60.



y(x) +

0

K(x – t)y(t) dt = Aeµx cos(λx).

x

A solution:

 A eµx Bc cos(λx) + Bs sin(λx) , 2 + Bs ∞ ∞ Bc = 1 + K(–z)eµz cos(λz) dz, Bs = K(–z)eµz sin(λz) dz. y(x) =

Bc2

0

61.



y(x) +

0

K(x – t)y(t) dt = Aeµx sin(λx).

x

A solution:

 A eµx Bc sin(λx) – Bs cos(λx) , 2 + Bs ∞ ∞ Bc = 1 + K(–z)eµz cos(λz) dz, Bs = K(–z)eµz sin(λz) dz. y(x) =

0

Bc2

0

2.9. EQUATIONS WHOSE KERNELS CONTAIN ARBITRARY FUNCTIONS

62.

211



K(x – t)y(t) dt = f (x).

y(x) + x

1◦ . For a polynomial right-hand side, f (x) =

n 

Ak xk , a solution has the form

k=0

y(x) =

n 

Bk xk ,

k=0

where the constants Bk are found by the method of undetermined coefficients. One can also make use of the formula given in item 4◦ of equation 2.9.53 to construct the solution. 2◦ . For f (x) = eλx

n 

Ak xk , a solution of the equation has the form

k=0

y(x) = eλx

n 

Bk xk ,

k=0

where the constants Bk are found by the method of undetermined coefficients. One can also make use of the formula given in item 3◦ of equation 2.9.55 to construct the solution. 3◦ . For f (x) =

n 

Ak exp(λk x), a solution of the equation has the form

k=0

y(x) =

n  Ak exp(λk x), Bk k=0

n 

4◦ . For f (x) = cos(λx)





Bk = 1 +

K(–z) exp(λk z) dz. 0

Ak xk a solution of the equation has the form

k=0

y(x) = cos(λx)

n 

Bk xk + sin(λx)

k=0

n 

Ck xk ,

k=0

where the constants Bk and Ck are found by the method of undetermined coefficients. 5◦ . For f (x) = sin(λx)

n 

Ak xk , a solution of the equation has the form

k=0

y(x) = cos(λx)

n 

Bk xk + sin(λx)

k=0

n 

Ck xk ,

k=0

where the Bk and Ck are found by the method of undetermined coefficients. 6◦ . For f (x) =

n 

Ak cos(λk x), a solution of the equation has the form

k=0

y(x) = Bck = 1 + 0

n 

 Ak Bck cos(λk x) + Bsk sin(λk x) , 2 + Bsk ∞ K(–z) cos(λk z) dz, Bsk = K(–z) sin(λk z) dz.

2 Bck k=0 ∞

0

212

LINEAR EQUATIONS OF THE SECOND KIND WITH VARIABLE LIMIT OF INTEGRATION

7◦ . For f (x) =

n 

Ak sin(λk x), a solution of the equation has the form

k=0 n 

 Ak Bck sin(λk x) – Bsk cos(λk x) , 2 + Bsk ∞ K(–z) cos(λk z) dz, Bsk = K(–z) sin(λk z) dz.

y(x) =

2 Bck k=0 ∞

Bck = 1 + 0

0

n 

8◦ . For f (x) = cos(λx)

Ak exp(µk x), a solution of the equation has the form

k=0 n n   Ak Bck Ak Bsk exp(µ x) + sin(λx) exp(µk x), k 2 2 2 + B2 Bck + Bsk Bck sk k=0 k=0 ∞ ∞ =1+ K(–z) exp(µk z) cos(λz) dz, Bsk = K(–z) exp(µk z) sin(λz) dz.

y(x) = cos(λx) Bck

0

0

9◦ . For f (x) = sin(λx)

n 

Ak exp(µk x), a solution of the equation has the form

k=0 n n   Ak Bck Ak Bsk exp(µ x) – cos(λx) exp(µk x), k 2 2 2 + B2 Bck + Bsk Bck sk k=0 k=0 ∞ ∞ =1+ K(–z) exp(µk z) cos(λz) dz, Bsk = K(–z) exp(µk z) sin(λz) dz.

y(x) = sin(λx)

Bck

0

0

10◦ . In the general case of arbitrary right-hand side f = f (x), the solution of the integral equation can be represented in the form y(x) =



˜ = f(p)

1 2πi



c+i∞

c–i∞

f (x)e–px dx,

f˜(p) epx dp, ˜ 1 + k(–p) ∞ ˜ k(–p) = K(–z)epz dz.

0

0

˜ To calculate f˜(p) and k(–p), it is convenient to use tables of Laplace transforms, and to determine y(x), tables of inverse Laplace transforms. 2.9-3. Other Equations. 63.

y(x) + 0

x

1 x

 f

t x

 y(t) dt = 0.

Eigenfunctions of this integral equation are determined by the roots of the following transcendental (algebraic) equation for the parameter λ:

1

f (z)z λ dz = –1. 0

1◦ . For a real simple root λk of equation (1) there is a corresponding eigenfunction yk (x) = xλk .

(1)

213

2.9. EQUATIONS WHOSE KERNELS CONTAIN ARBITRARY FUNCTIONS

2◦ . For a real root λk of multiplicity r there are corresponding r eigenfunctions yk1 (x) = xλk ,

yk2 (x) = xλk ln x,

...,

ykr (x) = xλk lnr–1 x.

3◦ . For a complex simple root λk = αk + iβk of equation (1) there is a corresponding eigenfunction pair yk(1) (x) = xαk cos(βk ln x),

yk(2) (x) = xαk sin(βk ln x).

4◦ . For a complex root λk = αk +iβk of multiplicity r there are corresponding r eigenfunction pairs (1) (2) yk1 (x) = xαk cos(βk ln x), (x) = xαk sin(βk ln x), yk1 (1) yk2 (x) = xαk ln x cos(βk ln x), ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅

(2) yk2 (x) = xαk ln x sin(βk ln x), ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅

(1) (x) = xαk lnr–1 x cos(βk ln x), ykr

(2) (x) = xαk lnr–1 x sin(βk ln x). ykr

The general solution is the combination (with arbitrary constants) of the eigenfunctions of the homogeneous integral equation.  For equations 2.9.64–2.9.71, only particular solutions are given. To obtain the general solution, one must add the general solution of the corresponding homogeneous equation 2.9.63 to the particular solution.   x t 1 64. y(x) + f y(t) dt = Ax + B. x 0 x A solution: 1 1 A B y(x) = x+ , I0 = f (t) dt, I1 = tf (t) dt. 1 + I1 1 + I0 0 0   x t 1 f y(t) dt = Axβ . 65. y(x) + x 0 x A solution: 1 A y(x) = xβ , B =1+ f (t)tβ dt. B 0   x t 1 f y(t) dt = A ln x + B. 66. y(x) + x 0 x A solution: y(x) = p ln x + q, where

67.

1 A B AIl p= , q= – , I0 = f (t) dt, 1 + I0 1 + I0 (1 + I0 )2 0   x t 1 f y(t) dt = Axβ ln x. y(x) + x 0 x A solution: y(x) = pxβ ln x + qxβ , where p=

A , 1 + I1

q=–

AI2 , (1 + I1 )2



f (t)tβ dt, 0

Il =

1

f (t) ln t dt. 0



1

I1 =



1

f (t)tβ ln t dt.

I2 = 0

214

LINEAR EQUATIONS OF THE SECOND KIND WITH VARIABLE LIMIT OF INTEGRATION

68.

x

y(x) + 0

1 f x



 t y(t) dt = A cos(ln x). x

A solution: AIc AIs cos(ln x) + 2 sin(ln x), Ic2 + Is2 Ic + Is2 1 1 f (t) cos(ln t) dt, Is = f (t) sin(ln t) dt. Ic = 1 + y(x) =

0

69.

x

y(x) +

1 x

0

 f

0

 t y(t) dt = A sin(ln x). x

A solution: AIs AIc cos(ln x) + 2 sin(ln x), 2 + Is Ic + Is2 1 1 f (t) cos(ln t) dt, Is = f (t) sin(ln t) dt. Ic = 1 + y(x) = –

Ic2

0

70.

x

y(x) +

1 x

0

 f

0

 t y(t) dt = Axβ cos(ln x) + Bxβ sin(ln x). x

A solution: y(x) = pxβ cos(ln x) + qxβ sin(ln x), where p=

AIc – BIs , Ic2 + Is2

1

f (t)tβ cos(ln t) dt,

Ic = 1 +

AIs + BIc , Ic2 + Is2 1 Is = f (t)tβ sin(ln t) dt.

q=

0

71.

0

 t 1 f y(t) dt = g(x). y(x) + x 0 x 1◦ . For a polynomial right-hand side,

x



g(x) =

N 

An xn

n=0

a solution bounded at zero is given by y(x) =

N  An n x , 1 + fn n=0



1

f (z)z n dz.

fn = 0

Here it is assumed that f0 < ∞ and fn ≠ –1 (n = 0, 1, 2, . . . ). If for some n the relation fn = –1 holds, then a solution differs from the above case in one term and has the form 1 n–1 N   Am m Am m An n y(x) = x + x + ¯ x ln x, f¯n = f (z)z n ln z dz. 1 + fm 1 + fm f n 0 m=0 m=n+1 For arbitrary g(x) expandable into power series, the formulas of item 1◦ can be used, in which one should set N = ∞. In this case, the convergence radius of the obtained solution y(x) is equal to that of the function g(x).

215

2.10. SOME FORMULAS AND TRANSFORMATIONS

2◦ . For g(x) = ln x

n 

Ak xk , a solution has the form

k=0

y(x) = ln x

n 

Bk xk +

k=0

n 

Ck xk ,

k=0

where the constants Bk and Ck are found by the method of undetermined coefficients. n   3◦ . For g(x) = Ak ln x)k , a solution of the equation has the form k=0

y(x) =

n 

 Bk ln x)k ,

k=0

where the Bk are found by the method of undetermined coefficients. n  4◦ . For g(x) = Ak cos(λk ln x), a solution of the equation has the form k=1

y(x) =

n 

Bk cos(λk ln x) +

k=1

n 

Ck sin(λk ln x),

k=1

where the Bk and Ck are found by the method of undetermined coefficients. n  5◦ . For g(x) = Ak sin(λk ln x) a solution of the equation has the form k=1

y(x) =

n 

Bk cos(λk ln x) +

k=1

n 

Ck sin(λk ln x),

k=1

where the Bk and Ck are found by the method of undetermined coefficients. 6◦ . For arbitrary right-hand side g(x), the transformation x = e–z ,

t = e–τ ,

y(x) = ez w(z),

f (ξ) = F (ln ξ),

g(x) = ez G(z)

leads to an equation with difference kernel of the form 2.9.62: ∞ w(z) + F (z – τ )w(τ ) dτ = G(z). z

7◦ . For arbitrary right-hand side g(x), the solution of the integral equation can be expressed via the inverse Mellin transform (see Example 2 in Subsection 11.6-4).

2.10. Some Formulas and Transformations Let the solution of the integral equation x y(x) + K(x, t)y(t) dt = f (x)

(1)

a



have the form

x

R(x, t)f (t) dt.

y(x) = f (x) + a

(2)

216

LINEAR EQUATIONS OF THE SECOND KIND WITH VARIABLE LIMIT OF INTEGRATION

Then the solution of the more complicated integral equation

x

K(x, t)

y(x) + a



has the form

g(x) y(t) dt = f (x) g(t)

(3)

g(x) f (t) dt. g(t)

(4)

x

R(x, t)

y(x) = f (x) + a

Below are formulas for the solutions of integral equations of the form (3) for some specific functions g(x). In all cases, it is assumed that the solution of equation (1) is known and is given by (2). 1◦ . The solution of the equation

x

K(x, t)(x/t)λy(t) dt = f (x)

y(x) + a



has the form

x

R(x, t)(x/t)λ f (t) dt.

y(x) = f (x) + a

2◦ . The solution of the equation

x

K(x, t)eλ(x–t) y(t) dt = f (x)

y(x) + a



has the form

x

R(x, t)eλ(x–t) f (t) dt.

y(x) = f (x) + a

Chapter 3

Linear Equations of the First Kind with Constant Limits of Integration  Notation: f = f (x), g = g(x), h = h(x), K = K(x), and M = M (x) are arbitrary functions (these may be composite functions of the argument depending on two variables x and t); A, B, C, a, b, c, k, α, β, γ, λ, and µ are free parameters; and n is a nonnegative integer.

3.1. Equations Whose Kernels Contain Power-Law Functions 3.1-1. Kernels Linear in the Arguments x and t.

1

|x – t| y(t) dt = f (x).

1. 0 ◦

1 . Let us remove the modulus in the integrand:



x

1

(x – t)y(t) dt +

(t – x)y(t) dt = f (x).

(1)

y(t) dt = fx (x).

(2)

x

0

Differentiating (1) with respect to x yields



x

1

y(t) dt – x

0

Differentiating (2) yields the solution  y(x) = 12 fxx (x).

(3)

2◦ . Let us demonstrate that the right-hand side f (x) of the integral equation must satisfy 1

certain relations. By setting x = 0 and x = 1 in (1), we obtain two corollaries 1

and

0

0

ty(t) dt = f (0)

(1 – t)y(t) dt = f (1), which can be rewritten in the form



1

ty(t) dt = f (0),

1

y(t) dt = f (0) + f (1).

0

(4)

0

In Section 3.1, we mean that kernels of the integral equations discussed may contain power-law functions or modulus of power-law functions.

217

218

LINEAR EQUATIONS OF THE FIRST KIND WITH CONSTANT LIMITS OF INTEGRATION

Substitute y(x) of (3) into (4). Integration by parts yields fx (1) = f (1)+f (0) and fx (1)–fx (0) = 2f (1) + 2f (0). Hence, we obtain the desired constraints for f (x): fx (1) = f (0) + f (1),

fx (0) + fx (1) = 0.

(5)

Conditions (5) make it possible to find the admissible general form of the right-hand side of the integral equation: A=

2.

– 12

Fx (1) +

f (x) = F (x) + Ax + B,

  Fx (0) , B = 12 Fx (1) – F (1) – F (0) ,

where F (x) is an arbitrary bounded twice differentiable function with bounded first derivative. b |x – t| y(t) dt = f (x), 0 ≤ a < b < ∞. a

This is a special case of equation 3.8.3 with g(x) = x. Solution:  y(x) = 12 fxx (x). The right-hand side f (x) of the integral equation must satisfy certain relations. The general form of f (x) is as follows: A=

3.

– 21

Fx (a) +

f (x) = F (x) + Ax + B,

  , B = 12 aFx (a) + bFx (b) – F (a) – F (b) ,

Fx (b)

where F (x) is an arbitrary bounded twice differentiable function (with bounded first derivative). a |λx – t| y(t) dt = f (x), λ > 0. 0

Here 0 ≤ x ≤ a and 0 ≤ t ≤ a. 1◦ . Let us remove the modulus in the integrand: λx a (λx – t)y(t) dt + (t – λx)y(t) dt = f (x).

(1)

λx

0

Differentiating (1) with respect to x, we find that a λx y(t) dt – λ y(t) dt = fx (x). λ 0

(2)

λx

 Differentiating (2) yields 2λ2 y(λx) = fxx (x). Hence, we obtain the solution 1   x  . (3) f y(x) = 2λ2 xx λ 2◦ . Let us demonstrate that the right-hand side f (x) of the integral equation must satisfy certain relations. By setting x = 0 in (1) and (2), we obtain two corollaries a a ty(t) dt = f (0), λ y(t) dt = –fx (0). (4) 0

0

Substitute y(x) from (3) into (4). Integrating by parts yields the desired constraints for f (x): (a/λ)fx (a/λ) = f (0) + f (a/λ),

fx (0) + fx (a/λ) = 0.

(5)

Conditions (5) make it possible to establish the admissible general form of the right-hand side of the integral equation: A=

– 12

f (x) = F (z) + Az + B, z = λx;

    1 + Fz (0) , B = 2 aFz (a) – F (a) – F (0) ,

Fz (a)

where F (x) is an arbitrary bounded twice differentiable function (with bounded first derivative).

3.1. EQUATIONS WHOSE KERNELS CONTAIN POWER-LAW FUNCTIONS



a

4.

|x – λt| y(t) dt = f (x),

219

λ > 0.

0

Here 0 ≤ x ≤ a and 0 ≤ t ≤ a. Solution:

 y(x) = 12 λfxx (λx).

The right-hand side f (x) of the integral equation must satisfy the relations aλfx (aλ) = f (0) + f (aλ),

fx (0) + fx (aλ) = 0.

Hence, it follows the general form of the right-hand side:

  f (x) = F (x) + Ax + B, A = – 12 Fx (λa) + Fx (0) , B = 12 aλFx (aλ) – F (λa) – F (0) , where F (x) is an arbitrary bounded twice differentiable function (with bounded first derivative). 3.1-2. Kernels Quadratic in the Arguments x and t.

a

5.

Ax + Bx2 – t y(t) dt = f (x),

A > 0,

B > 0.

0

This is a special case of equation 3.8.5 with g(x) = Ax + Bx2 .

a

6.

x – At – Bt2 y(t) dt = f (x),

A > 0,

B > 0.

0

This is a special case of equation 3.8.6 with g(x) = At + Bt2 .

b

7.

xt – t2 y(t) dt = f (x),

a

0 ≤ a < b < ∞.

The substitution w(t) = ty(t) leads to an equation of the form 3.1.2:

b

|x – t|w(t) dt = f (x). a



b

8.

2 2 x – t y(t) dt = f (x).

a

This is a special case of equation with g(x) = x2 .   3.8.3  d fx (x) . The right-hand side f (x) of the equation must satisfy Solution: y(x) = dx 4x certain constraints, given in 3.8.3.

a

9.

2 x – βt2 y(t) dt = f (x),

β > 0.

0

This is a special case of equation 3.8.4 with g(x) = x2 and β = λ2 . 10.

a

Ax + Bx2 – Aλt – Bλ2 t2 y(t) dt = f (x),

λ > 0.

0

This is a special case of equation 3.8.4 with g(x) = Ax + Bx2 .

220

LINEAR EQUATIONS OF THE FIRST KIND WITH CONSTANT LIMITS OF INTEGRATION

3.1-3. Kernels Containing Integer Powers of x and t or Rational Functions.

b

11.

x – t 3 y(t) dt = f (x).

a

Let us remove the modulus in the integrand:



x

b

(x – t)3 y(t) dt + a

(t – x)3 y(t) dt = f (x).

(1)

x

Differentiating (1) twice yields



x

b

(x – t)y(t) dt + 6

6 a

 (t – x)y(t) dt = fxx (x).

x

This equation can be rewritten in the form 3.1.2:

b

a

 |x – t| y(t) dt = 16 fxx (x).

(2)

Therefore the solution of the integral equation is given by y(x) =

1  12 yxxxx (x).

(3)

The right-hand side f (x) of the equation must satisfy certain conditions. To obtain these conditions, one must substitute solution (3) into (1) with x = a and x = b and into (2) with x = a and x = b, and then integrate the four resulting relations by parts.

b

12.

3 3 x – t y(t) dt = f (x).

a

This is a special case of equation 3.8.3 with g(x) = x3 .

b

13.

2 3 xt – t y(t) dt = f (x)

0 ≤ a < b < ∞.

a

The substitution w(t) = t2 y(t) leads to an equation of the form 3.1.2:

b

|x – t|w(t) dt = f (x). a



b

14.

2 x t – t3 y(t) dt = f (x).

a

The substitution w(t) = |t| y(t) leads to an equation of the form 3.1.8:

b

x2 – t2 w(t) dt = f (x).

a

15.

a

3 x – βt3 y(t) dt = f (x),

β > 0.

0

This is a special case of equation 3.8.4 with g(x) = x3 and β = λ3 .

221

3.1. EQUATIONS WHOSE KERNELS CONTAIN POWER-LAW FUNCTIONS



b

16.

x – t 2n+1 y(t) dt = f (x),

n = 0, 1, 2, . . .

a

Solution: y(x) =

1 f (2n+2) (x). 2(2n + 1)! x

(1)

The right-hand side f (x) of the equation must satisfy certain conditions. To obtain these conditions, one must substitute solution (1) into the relations



b

(t – a)2n+1 y(t) dt = f (a), a

b

(–1)k+1 (k+1) fx (a), Ak k = 0, 1, . . . , 2n,

(t – a)2n–k y(t) dt = a

Ak = (2n + 1)(2n) . . . (2n + 1 – k); and then integrate the resulting equations by parts.



17. 0

y(t) dt x+t

= f (x).

The left-hand side of this equation is the Stieltjes transform. 1◦ . By setting x = ez ,

t = eτ ,

y(t) = e–τ /2 w(τ ),

f (x) = e–z/2 g(z),

we obtain an integral equation with difference kernel of the form 3.8.15: ∞ w(τ ) dτ

1  = g(z), 2 cosh –∞ 2 (z – τ ) whose solution is given by ∞ 1 iux w(z) = √ cosh(πu) g(u)e ˜ du, 2π 3 –∞ 2◦ . Solution:

1 g(u) ˜ = √ 2π





g(z)e–iuz dz,

i2 = –1.

–∞

 1 lim f (–x – iε) – f (–x + iε) 2πi ε→+0  2k ∞

√  1  (–1)k π d x f (x) . = √ π x (2k)! x dx

y(x) =

k=0



3 . Under some assumptions, the solution of the original equation can be represented in the form

2n+1 (n) (n+1) (–1)n y(x) = lim x fx (x) x , (1) n→∞ (n + 1)!(n – 1) which is the real inversion of the Stieltjes transform. An alternative form of the solution is (–1)n  e 2n 2n (n) (n) y(x) = lim x fx (x) x . n→∞ 2π n

(2)

To obtain an approximate solution of the integral equation, one restricts oneself to a specific value of n in (1) or (2) instead of taking the limit. References: E. A. C. Paley and N. Wiener (1934), D. V. Widder (1939, 1971), I. I. Hirschman and D. V. Widder (1955), P. P. Zabreyko, A. I. Koshelev, et al. (1975), E. C. Titchmarsh (1986), Yu. A. Brychkov and A. P. Prudnikov (1989), A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 428).

222

LINEAR EQUATIONS OF THE FIRST KIND WITH CONSTANT LIMITS OF INTEGRATION

3.1-4. Kernels Containing Square Roots. a √ √ x – t y(t) dt = f (x), 18. 0 < a < ∞. 0

√ This is a special case of equation 3.8.3 with g(x) = x. Solution: d √   y(x) = x fx (x) . dx The right-hand side f (x) of the equation must satisfy certain conditions. The general form of the right-hand side is

 f (x) = F (x) + Ax + B, A = –Fx (a), B = 12 aFx (a) – F (a) – F (0) ,

19.

where F (x) is an arbitrary bounded twice differentiable function (with bounded first derivative). a √ √ x – β t y(t) dt = f (x), β > 0.

20.

This is a special case of equation 3.8.4 with g(x) = a √ x – t y(t) dt = f (x).

21.

This is a special case of equation 3.8.5 with g(x) = a √ x – t y(t) dt = f (x).

0

0

0

22.

23.

24.

This is a special case of equation 3.8.6 with g(t) = a y(t) dt = f (x), 0 < a ≤ ∞. √ |x – t| 0







x and β =

√ λ.

x (see item 3◦ of 3.8.5).

t (see item 3◦ of 3.8.6).

This is a special case of equation 3.1.30 with k = 12 . Solution:  a  t dt f (s) ds A d , y(x) = – 1/4 dx x (t – x)1/4 0 s 1/4 (t – s)1/4 x ∞ y(t) dt = f (x). √ |x – t| –∞

1 A= √ . 8π Γ2 (3/4)

This is a special case of equation 3.1.35 with λ = 12 . Solution: ∞ f (x) – f (t) 1 y(x) = dt. 4π –∞ |x – t|3/2 1 y(t) dt = f (x). √ 1 + x2 – 2xt –1 Solution: ∞ 1  2n + 1 (n) fx (0)Pn (x), y(x) = 2 n=0 n! where Pn (x) are the Legendre polynomials (see Supplement 11.11-1) 1 dn 2 (x – 1)n . Pn (x) = n! 2n dxn P. M. Morse and H. Feshbach (1953).

223

3.1. EQUATIONS WHOSE KERNELS CONTAIN POWER-LAW FUNCTIONS

3.1-5. Kernels Containing Arbitrary Powers.

a

25.

|xk – tk | y(t) dt = f (x),

0 < a < ∞.

0 < k < 1,

0 ◦

1 . Let us remove the modulus in the integrand:



x

a

(xk – tk )y(t) dt +

(tk – xk )y(t) dt = f (x).

(1)

x

0

Differentiating (1) with respect to x yields



x

k–1

a

k–1

y(t) dt – kx

kx

y(t) dt = fx (x).

(2)

x

0

Let us divide both sides of (2) by kxk–1 and differentiate the resulting equation. As a result, we obtain the solution 1 d 1–k   x fx (x) . y(x) = (3) 2k dx 2◦ . Let us demonstrate that the right-hand side f (x) of the integral equation must satisfy a

certain relations. By setting x = 0 and x = a, in (1), we obtain two corollaries a

and

0

0

tk y(t) dt = f (0)

(ak – tk )y(t) dt = f (a), which can be rewritten in the form



a

tk y(t) dt = f (0),

ak

0

a

y(t) dt = f (0) + f (a).

(4)

0

Substitute y(x) of (3) into (4). Integrating by parts yields the relations afx (a) = kf (a) + kf (0) and afx (a) = 2kf (a) + 2kf (0). Hence, the desired constraints for f (x) have the form f (0) + f (a) = 0,

fx (a) = 0.

(5)

Conditions (5) make it possible to find the admissible general form of the right-hand side of the integral equation: A = –Fx (a),

f (x) = F (x) + Ax + B,

26.

B=

1 2

  aFx (a) – F (a) – F (0) ,

where F (x) is an arbitrary bounded twice differentiable function with bounded first derivative.  The first derivative may be unbounded at x = 0, in which case the conditions x1–k Fx x=0 = 0 must hold. a |xk – βtk | y(t) dt = f (x), 0 < k < 1, β > 0. 0

This is a special case of equation 3.8.4 with g(x) = xk and β = λk . 27.

a

|xk tm – tk+m | y(t) dt = f (x),

0 < k < 1,

0 < a < ∞.

0

The substitution w(t) = tm y(t) leads to an equation of the form 3.1.25:

a

|xk – tk |w(t) dt = f (x). 0

224

LINEAR EQUATIONS OF THE FIRST KIND WITH CONSTANT LIMITS OF INTEGRATION



1

28.

|xk – tm | y(t) dt = f (x),

k > 0,

m > 0.

0

The transformation z = xk ,

τ = tm ,

w(τ ) = τ

leads to an equation of the form 3.1.1: 1 |z – τ |w(τ ) dτ = F (z),

1–m m

y(t)

F (z) = mf (z 1/k ).

0



b

29. a

|x – t|1+λ y(t) dt = f (x),

0 ≤ λ < 1.

For λ = 0, see equation 3.1.2. Assume that 0 < λ < 1. 1◦ . Let us remove the modulus in the integrand: b x 1+λ (x – t) y(t) dt + (t – x)1+λ y(t) dt = f (x). a

(1)

x

Let us differentiate (1) with respect to x twice and then divide both the sides by λ(λ + 1). As a result, we obtain x b 1 f  (x). (x – t)λ–1 y(t) dt + (t – x)λ–1 y(t) dt = (2) λ(λ + 1) xx a x Rewrite equation (2) in the form b y(t) dt 1 f  (x), = k |x – t| λ(λ + 1) xx a

k = 1 – λ.

(3)

See 3.1.30 and 3.1.31 for the solutions of equation (3) for various a and b. 2◦ . The right-hand side f (x) of the integral equation must satisfy certain relations. By setting x = a and x = b in (1), we obtain two corollaries b b (t – a)1+λ y(t) dt = f (a), (b – t)1+λ y(t) dt = f (b). (4) a

30.

a

On substituting the solution y(x) of (3) into (4) and then integrating by parts, we obtain the desired constraints for f (x). a y(t) dt = f (x), 0 < k < 1, 0 < a ≤ ∞. k 0 |x – t| 1◦ . Solution: " # 1–2k t a k–1 d t 2 dt f (s) ds y(x) = –Ax 2 , 1–k 1–k 1–k dx x 0 s 2 (t – s) 2 (t – x) 2   –2  πk  1 1+k cos Γ(k) Γ A= , 2π 2 2 where Γ(k) is the gamma function. 2◦ . The transformation x = z 2 , t = ξ 2 , w(ξ) = 2ξy(t) leads to an equation of the form 3.1.32: √a   w(ξ) dξ = f z 2 . 2 2 k |z – ξ | 0

225

3.1. EQUATIONS WHOSE KERNELS CONTAIN POWER-LAW FUNCTIONS

31.

b

y(t)

dt = f (x), 0 < k < 1. |x – t|k It is assumed that |a| + |b| < ∞. Solution: x x 1 f (t) dt Z(t)F (t) d 1 2 1 1 y(x) = cot( 2 πk) – cos ( 2 πk) dt, 2π dx a (x – t)1–k π 2 (x – t)1–k a a

where Z(t) = (t –

1+k a) 2

(b –

1–k t) 2

d F (t) = dt

,



t

a

dτ (t – τ )k



b τ

 f (s) ds . Z(s)(s – τ )1–k

Reference: F. D. Gakhov (1977).

32.

a

y(t)

– t2 |k Solution: |x2

0

dt = f (x),

0 < a ≤ ∞.

0 < k < 1,

  a 2–2k 2Γ(k) cos 12 πk k–1 d t F (t) dt y(x) = – ,

 1+k 2 x 1–k dx x (t2 – x2 ) 2 π Γ 2



t

F (t) = 0

s k f (s) ds (t2 – s 2 )

Reference: P. P. Zabreyko, A. I. Koshelev, et al. (1975).

33.

b

y(t)

dt = f (x), |xλ – tλ |k 1 . The transformation

0 < k < 1,

λ > 0.

a ◦

z = xλ ,

τ = tλ ,

w(τ ) = τ

1–λ λ

y(t)

leads to an equation of the form 3.1.31: B

w(τ ) dτ = F (z), |z – τ |k

A

where A = aλ , B = bλ , F (z) = λf (z 1/λ ). 2◦ . Solution with a = 0: y(x) =

λ(k–1) –Ax 2

d dx

"

b

t

λ(3–2k)–2 2

x

dt



t

s

λ(k+1)–2 2

f (s) ds

1–k

0 (tλ – xλ ) 2 (tλ – s λ )      –2 πk 1+k λ2 cos Γ(k) Γ A= , 2π 2 2

where Γ(k) is the gamma function. 34.

1

y(t)

dt = f (x), 0 < k < 1, λ > 0, m > 0. – tm |k The transformation 1–m z = xλ , τ = tm , w(τ ) = τ m y(t) 0

|xλ

leads to an equation of the form 3.1.31: 1 w(τ ) dτ = F (z), k 0 |z – τ |

F (z) = mf (z 1/λ ).

1–k 2

# ,

1–k 2

.

226

LINEAR EQUATIONS OF THE FIRST KIND WITH CONSTANT LIMITS OF INTEGRATION



35.



y(t) dt = f (x), 0 < Re λ < 1. |x – t|1–λ –∞ Solution:  πλ  ∞ f (x) – f (t) λ tan dt y(x) = 1+λ 2π 2 –∞ |x – t|  πλ  ∞ 2f (x) – f (x + t) – f (x – t) λ tan = dt. 2π 2 t1+λ 0 ∞

|f (x)|p dx < ∞ is satisfied for some p, 1 < p < 1/λ. It is assumed that the condition –∞ The integral equation and its solution form the Riesz transform pair (the Riesz potential). References: A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 428), S. G. Samko, A. A. Kilbas, and O. I. Marichev (1993).





36. –∞

37.

38.

y(t) |x3

– t|1–λ

dt = f (x),

0 < λ < 1.

The substitution z = x3 leads to an equation of the form 3.1.35: ∞   y(t) dt = f z 1/3 . 1–λ –∞ |z – t| ∞ y(t) dt = f (x), 0 < λ < 1. 3 3 1–λ –∞ |x – t | The transformation z = x3 , τ = t3 , w(τ ) = τ –2/3 y(t) leads to an equation of the form 3.1.35: ∞   w(τ ) dτ = F (z), F (z) = 3f z 1/3 . 1–λ –∞ |z – τ | ∞ sign(x – t) y(t) dt = f (x), 0 < Re λ < 1. 1–λ –∞ |x – t| Solution:  πλ  ∞ f (x) – f (t) λ cot sign(x – t) dt y(x) = 1+λ 2π 2 –∞ |x – t|  πλ  ∞ f (x + t) – f (x – t) λ cot = dt 2π 2 t1+λ 0  πλ  d ∞ f (t) λ cot = dt. 2π 2 dx –∞ |x – t|λ The integral equation and its solution form the Feller transform pair (the Feller potential). References: A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 428), S. G. Samko, A. A. Kilbas, and O. I. Marichev (1993).



39.



a + b sign(x – t) y(t) dt = f (x), 0 < Re λ < 1. |x – t|1–λ –∞ Solution: ∞  a + b sign(x – t) y(x) = Cλ f (x) – f (t) dt 1+λ |x – t| –∞ ∞

 = Cλ t–1–λ 2af (x) – (a + b)f (x – t) – (a – b)f (x + t) dt 0 ∞ b + a sign(x – t) d =C f (t) dt, dx –∞ |x – t|λ

3.1. EQUATIONS WHOSE KERNELS CONTAIN POWER-LAW FUNCTIONS

where C=

227

sin(πλ)  

  . 4π a2 cos2 12 πλ + b2 sin2 12 πλ

References: A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 431), S. G. Samko, A. A. Kilbas, and O. I. Marichev (1993).





40.

y(t) dt (ax + bt)k

0

= f (x),

a > 0,

b > 0,

k > 0.

1 2τ e , 2b

y(t) = be(k–2)τ w(τ ),

By setting x=

1 2z e , 2a

t=

f (x) = e–kz g(z),

we obtain an integral equation with the difference kernel of the form 3.8.15:



–∞





41.

w(τ ) dτ = g(z). coshk (z – τ )

tz–1 y(t) dt = f (z).

0

The left-hand side of this equation is the Mellin transform of y(t) (z is treated as a complex variable). Solution: c+i∞ 1 t–z f (z) dz, i2 = –1. y(t) = 2πi c–i∞ For specific f (z), one can use tables of Mellin and Laplace integral transforms to calculate the integral. References: H. Bateman and A. Erd´elyi (vol. 2, 1954), V. A. Ditkin and A. P. Prudnikov (1965).

3.1-6. Equations Containing the Unknown Function of a Complicated Argument.

1

y(xt) dt = f (x).

42. 0

Solution:

y(x) = xfx (x) + f (x).

 The function f (x) is assumed to satisfy the condition xf (x) x=0 = 0.

43.

1

tλ y(xt) dt = f (x).

0

x

The substitution ξ = xt leads to equation respect to x yields the solution

0

ξ λ y(ξ) dξ = xλ+1 f (x). Differentiating with

y(x) = xfx (x) + (λ + 1)f (x).

 The function f (x) is assumed to satisfy the condition xλ+1 f (x) x=0 = 0.

228

LINEAR EQUATIONS OF THE FIRST KIND WITH CONSTANT LIMITS OF INTEGRATION



1

44.

 Axk + Btm )y(xt) dt = f (x).

0

The substitution ξ = xt leads to an equation of the form 1.1.51: x  k+m  Ax + Bξ m y(ξ) dξ = xm+1 f (x). 0

45.



1



1

y(xt) dt = f (x). √ 1–t 0 The substitution ξ = xt leads to Abel’s equation 1.1.36: x y(ξ) dξ √ √ = x f (x). x–ξ 0

46. 0

y(xt) dt = f (x), (1 – t)λ

0 < λ < 1.

The substitution ξ = xt leads to the generalized Abel equation 1.1.47: x y(ξ) dξ = x1–λ f (x). λ 0 (x – ξ) 47.

1

tµ y(xt) (1 – t)λ

0

dt = f (x),

0 < λ < 1.

The transformation ξ = xt, w(ξ) = ξ µ y(ξ) leads to the generalized Abel equation 1.1.47: x w(ξ) dξ = x1+µ–λ f (x). (x – ξ)λ 0



48.

y(x + t) – y(x – t) t

0

dt = f (x).

Solution: y(x) = –

1 π2





0

f (x + t) – f (x – t) dt. t

References: V. A. Ditkin and A. P. Prudnikov (1965), A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 427).

3.1-7. Singular Equations. In this subsection, all singular integrals are understood in the sense of the Cauchy principal value. 49.



y(t) dt

t–x Solution:

= f (x).

–∞

y(x) = –

1 π2





–∞

f (t) dt . t–x

The integral equation and its solution form a Hilbert transform pair (in the asymmetric form). References: V. A. Ditkin and A. P. Prudnikov (1965), A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 427).

3.1. EQUATIONS WHOSE KERNELS CONTAIN POWER-LAW FUNCTIONS





50. 0

229

y(t) dt = f (x). t–x

Solution:

√ ∞ x f (t) √ y(x) = – 2 dt. π t (t – x) 0

The integral equation and its solution form a Hilbert transform pair on the semiaxis (in the asymmetric form). References: D. Hilbert (1953), A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 427), I. K. Lifanov, L. N. Poltavskii, and G. M. Vainikko (2004, p. 8).



b

51. a

y(t) dt t–x

= f (x).

This equation is encountered in hydrodynamics in solving the problem on the flow of an ideal inviscid fluid around a thin profile (a ≤ x ≤ b). It is assumed that |a| + |b| < ∞. 1◦ . The solution bounded at the endpoints is

1  y(x) = – 2 (x – a)(b – x) π provided that

a

b

a

b

f (t) dt √ , t (t – a)(b – t) – x

f (t) dt √ = 0. (t – a)(b – t)



2 . The solution bounded at the endpoint x = a and unbounded at the endpoint x = b is 1 y(x) = – 2 π





x–a b–x

b a



b – t f (t) dt. t–a t–x

3◦ . The solution unbounded at the endpoints is 1 y(x) = – 2 √ π (x – a)(b – x)



b

a

√  (t – a)(b – t) f (t) dt + C , t–x

b

where C is an arbitrary constant. The formula

a

y(t) dt = C/π holds.

Solutions that have a singularity point x = s inside the interval [a, b] can be found in Subsection 14.4-3. Reference: F. D. Gakhov (1977).



1

52. –1



 1 1 + y(t) dt = f (x), t–x x+t+2

–1 < x < 1.

Solution for f (x) = πq = const: 1+t y(t) = q √ . (1 – t)(3 + t) Reference: H. F. Bueckner (1966).

230

LINEAR EQUATIONS OF THE FIRST KIND WITH CONSTANT LIMITS OF INTEGRATION

1 53.

1 t–x

0

+

λ t+x

 y(t) dt = f (x),

0 < x < 1.

Solution for f (x) = πq = const: q y(x) = 2 sin( 21 πβ)



x √ 1 + 1 – x2

β 

–β     β β x √ √ √ +1 + –1 , 1 – x2 1 + 1 – x2 1 – x2

where β is given by 0 < β < 1.

cos(πβ) = –λ,

We assume that the following necessary condition holds

1

y(t) dt = 0. 0

References: H. F. Bueckner (1966), P. S. Theocaric and N. I. Ioakimidis (1977).

54.

1 πi



a

–a



1 t–x



λx xt – a

 y(t) dt = f (x), 2

(i2 = –1).

–a < x < a

1◦ . Solution: –β   x a–t 1 – f (t) dt –a – t t – x xt – a2 –a  –β β   a a–x 1 x a–t 1 + – f (t) dt, –a – x 2πi –a –a – t t – x xt – a2 

y(x) =

a–x –a – x

where λ = cos θ and β = 1 – 1 2πi



1 2πi

θ . We assume that the following necessary condition holds π

a

 e

a

–πiβ

–a

a–t –a – t



 –e

πiβ

a–t –a – t

–β 

f (t) dt = 0. t

2◦ . Solution for f (x) ≡ 0: y(x) = C1 Λ1 (x) + C2 Λ2 (x) + C3 Λ3 (x), where C1 , C2 , and C3 are arbitrary constants, and 1–β β  a–t a–t –iπβ + (1 – λ)e , Λ1 (x) = (1 + λ)e –a – t –a – t –1+β –β   a–t a–t Λ2 (x) = (1 + λ)e–iπβ + (1 – λ)eiπβ , –a – t –a – t 1–β –1+β   a–t a–t Λ3 (x) = eiπβ + e–iπβ . –a – t –a – t 

iπβ

Reference: D. I. Sherman (1969).

3.2. EQUATIONS WHOSE KERNELS CONTAIN EXPONENTIAL FUNCTIONS

55.

b

231

y(t)

dt = f (x), a ≤ x ≤ b. (x – t)2 The simple hypersingular equation of the first kind with Cauchy-type kernel. This equation governs circulation-free flow of an ideal incompressible fluid past the segment [a, b]. Let the conditions y(a) = y(b) = 0 be satisfied. Then the solution is √ b √ (b – t)(x – a) – (b – x)(t – a)  1 f (t) dt. √ y(x) = 2 ln √ π (b – t)(x – a) + (b – x)(t – a) t a

a

This equation is discussed in Subsection 14.6-3 in detail. Reference: I. K. Lifanov, L. N. Poltavskii, and G. M. Vainikko (2004, p. 7).

56.

1 1 1 u(x, y) dx dy = f (x0 , y0 ). π 2 –1 –1 (x0 – x)(y0 – y) A two-dimensional singular equation. A solution, which is bounded on the lines x = ±1 and y = ±1 but which is unbounded on the line x = q (–1 < q < 1), is given by the formula  (1 – x20 )(1 – y02 ) 1 1 f (x, y) dx dy  u(x0 , y0 ) = 2 2 π (1 – x )(1 – y 2 ) (x – x0 )(y – y0 ) –1 –1   (1 – x20 )(1 – y02 ) 1 dx  1 1 f (x, y) dy √  , – π 2 (q – x0 ) 1 – x2 π 2 –1 1 – y 2 (y – y0 ) –1 provided that



1

–1

f (x0 , y) dy  = 0, 1 – y2

–1 ≤ x0 ≤ 1.

Reference: I. K. Lifanov, L. N. Poltavskii, and G. M. Vainikko (2004, pp. 16–20).

3.2. Equations Whose Kernels Contain Exponential Functions 3.2-1. Kernels Containing Exponential Functions of the Form eλ|x–t| . ∞ 1. e–λ|x–t| y(t) dt = f (x), f (±∞) = 0. –∞

Solution: y(x) =

 1 2  λ f (x) – fxx (x) . 2λ

References: I. I. Hirschman and D. V. Widder (1955), F. D. Gakhov and Yu. I. Cherskii (1978), A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 433).

2.



e–λ|x–t| y(t) dt = f (x),

f (∞) = 0.

0 ◦

1 . Solution: y(x) = 2◦ . If fx (0) – λf (0) = 0 then

1 –λx d 2λx d –λx e e e f (x). 2λ dx dx

y(x) =

 1 2  λ f (x) – fxx (x) . 2λ

References: H. M. Srivastava and R. G. Buschman (1977), A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 433).

232

LINEAR EQUATIONS OF THE FIRST KIND WITH CONSTANT LIMITS OF INTEGRATION



b

3.

eλ|x–t| y(t) dt = f (x),

–∞ < a < b < ∞.

a ◦

1 . Let us remove the modulus in the integrand: b x eλ(x–t) y(t) dt + eλ(t–x) y(t) dt = f (x). a

(1)

x

Differentiating (1) with respect to x twice yields x 2 λ(x–t) 2 e y(t) dt + λ 2λy(x) + λ a

b

 eλ(t–x) y(t) dt = fxx (x).

(2)

x

By eliminating the integral terms from (1) and (2), we obtain the solution y(x) =

 1  f (x) – λ2 f (x) . 2λ xx

(3)

2◦ . The right-hand side f (x) of the integral equation must satisfy certain relations. By setting x = a and x = b in (1), we obtain two corollaries b b eλt y(t) dt = eλa f (a), e–λt y(t) dt = e–λb f (b). (4) a

a

On substituting the solution y(x) of (3) into (4) and then integrating by parts, we see that eλb fx (b) – eλa fx (a) = λeλa f (a) + λeλb f (b), e–λb fx (b) – e–λa fx (a) = λe–λa f (a) + λe–λb f (b). Hence, we obtain the desired constraints for f (x): fx (a) + λf (a) = 0,

fx (b) – λf (b) = 0.

(5)

The general form of the right-hand side satisfying conditions (5) is given by f (x) = F (x) + Ax + B,   1 1 A= Fx (a) + Fx (b) + λF (a) – λF (b) , B = – Fx (a) + λF (a) + Aaλ + A , bλ – aλ – 2 λ

where F (x) is an arbitrary bounded, twice differentiable function.

b

4.

 λ|x–t|  Ae + Beµ|x–t| y(t) dt = f (x),

–∞ < a < b < ∞.

a

Let us remove the modulus in the integrand and differentiate the resulting equation with respect to x twice to obtain b  2 λ|x–t|   2(Aλ + Bµ)y(x) + Aλ e + Bµ2 eµ|x–t| y(t) dt = fxx (x). (1) a

Eliminating the integral term with eµ|x–t| from (1) with the aid of the original integral equation, we find that b  2(Aλ + Bµ)y(x) + A(λ2 – µ2 ) eλ|x–t| y(t) dt = fxx (x) – µ2 f (x). (2) a

For Aλ + Bµ = 0, this is an equation of the form 3.2.3, and for Aλ + Bµ ≠ 0, this is an equation of the form 4.2.15. The right-hand side f (x) must satisfy certain relations, which can be obtained by setting x = a and x = b in the original equation (a similar procedure is used in 3.2.3).

233

3.2. EQUATIONS WHOSE KERNELS CONTAIN EXPONENTIAL FUNCTIONS



b

5. a

 n

   Ak exp λk |x – t| y(t) dt = f (x),

–∞ < a < b < ∞.

k=1

1◦ . Let us remove the modulus in the kth summand of the integrand: Ik (x) =

b

  exp λk |x – t| y(t) dt =

a





x

b

exp[λk (x – t)]y(t) dt + a

exp[λk (t – x)]y(t) dt. (1) x

Differentiating (1) with respect to x twice yields Ik = λk





x

b

exp[λk (x – t)]y(t) dt – λk a

Ik = 2λk y(x) + λ2k



exp[λk (t – x)]y(t) dt, x



x

a

(2)

b

exp[λk (x – t)]y(t) dt + λ2k

exp[λk (t – x)]y(t) dt, x

where the primes denote the derivatives with respect to x. By comparing formulas (1) and (2), we find the relation between Ik and Ik : Ik = 2λk y(x) + λ2k Ik ,

Ik = Ik (x).

(3)

2◦ . With the aid of (1), the integral equation can be rewritten in the form n 

Ak Ik = f (x).

(4)

k=1

Differentiating (4) with respect to x twice and taking into account (3), we obtain

σ1 y(x) +

n 

 Ak λ2k Ik = fxx (x),

k=1

σ1 = 2

n 

Ak λk .

(5)

k=1

Eliminating the integral In from (4) and (5) yields

σ1 y(x) +

n–1 

 Ak (λ2k – λ2n )Ik = fxx (x) – λ2n f (x).

(6)

k=1

Differentiating (6) with respect to x twice and eliminating In–1 from the resulting equation with the aid of (6), we obtain a similar equation whose right-hand side is a second-order n–2  linear differential operator (acting on y) with constant coefficients plus the sum Bk Ik . If k=1

we successively eliminate In–2 , In–3 , . . . , I1 with the aid of double differentiation, then we finally arrive at a linear nonhomogeneous ordinary differential equation of order 2(n – 1) with constant coefficients. 3◦ . The right-hand side f (x) must satisfy certain conditions. To find these conditions, one must set x = a in the integral equation and its derivatives. (Alternatively, these conditions can be found by setting x = a and x = b in the integral equation and all its derivatives obtained by means of double differentiation.)

234

LINEAR EQUATIONS OF THE FIRST KIND WITH CONSTANT LIMITS OF INTEGRATION

3.2-2. Kernels Containing Exponential Functions of the Forms eλx and eµt .

b

6. a

|eλx – eλt | y(t) dt = f (x),

λ > 0.

This is a special case of equation 3.8.3 with g(x) = eλx . Solution: 1 d –λx   e fx (x) . y(x) = 2λ dx The right-hand side f (x) of the integral equation must satisfy certain relations (see item 2◦ of equation 3.8.3).

a

7.

|eβx – eµt | y(t) dt = f (x),

β > 0,

µ > 0.

0

This is a special case of equation 3.8.4 with g(x) = eβx and λ = µ/β.

b

y(t) dt

8.

|eλx

a

– eλt |k

= f (x),

0 < k < 1.

The transformation z = eλx , τ = eλt , w(τ ) = e–λt y(t) leads to an equation of the form 3.1.31:

B

A

where A = eλa , B = eλb , F (z) = λf



9.

y(t) dt (eλx

0

+ eλt )k

= f (x),

w(τ ) dτ = F (z), |z – τ |k 1 λ

λ > 0,

 ln z . k > 0.

This equation can be rewritten as an equation with difference kernel in the form 3.8.16: 0



w(t) dt

coshk 12 λ(x

– t)

 = g(x),

    where w(t) = 2–k exp – 21 λkt y(t) and g(x) = exp 12 λkx f (x). 3.2-3. Kernels Containing Exponential Functions of the Form eλxt .



10.

e–xt y(t) dt = f (x).

–∞

Solution:

c+i∞ 1 y(t) = est f (s) ds 2πi c–i∞ ∞ ∞   2 2 1 = √ e–ξ /2 dξ e–x /2 cos ξ(x + t) f (x) dx. 2π 3 0 –∞

The integral equation and its solution form a two-side Laplace transform pair. References: B. Van der Pol and H. Bremmer (1955), A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 433).

3.2. EQUATIONS WHOSE KERNELS CONTAIN EXPONENTIAL FUNCTIONS





11.

eλxt y(t) dt = f (x),

235

λ ≠ 0.

–∞ ◦

1 . The transformation

1 x = – z, f (x) = F (z) λ leads to an equation of the form 3.2.10: ∞ e–zt y(t) dt = F (z). –∞ ◦

2 . The transformation y(t) = exp(–t2 )Y (t),

x=

2 ζ, λ

f (x) = exp(ζ 2 )Φ(ζ)

leads to an equation of the form 3.2.17: ∞ 2 e–(ζ–t) Y (t) dt = Φ(ζ). –∞





12.

e–ixt y(t) dt = f (x),

i2 = –1.

–∞

Solution:

1 y(t) = 2π





eixt f (x) dx.

–∞

Up to constant factors, the function f (x) and the solution y(t) are the Fourier transform pair. References: V. A. Ditkin and A. P. Prudnikov (1965), J. W. Miles (1971), B. Davis (1978), F. Oberhettinger (1980), Yu. A. Brychkov and A. P. Prudnikov (1989), W. H. Beyer (1991), I. Sneddon (1995), A. Pinkus and S. Zafrany (1997), R. Bracewell (1999), A. D. Poularikas (2000), R. J. Beerends, H. G. ter Morschem, and J. C. van den Berg (2003), L. Debnath and D. Bhatta (2007).

13.



e–zt y(t) dt = f (z).

0

The left-hand side of the equation is the Laplace transform of y(t) (z is treated as a complex variable). 1◦ . Solution: y(t) =

1 2πi



c+i∞

ezt f (z) dz,

i2 = –1.

c–i∞

For specific functions f (z), one may use tables of inverse Laplace transforms to calculate the integral (e.g., see Supplement 6). 2◦ . For real z = x, under some assumptions the solution of the original equation can be represented in the form (–1)n  n n+1 (n)  n  y(x) = lim , fx n→∞ n! x x which is the real inversion of the Laplace transform. To calculate the solution approximately, one should restrict oneself to a specific value of n in this formula instead of taking the limit. References: G. Doetsch (1950, 1956, 1958, 1974), H. Bateman and A. Erd´elyi (vol. 1, 1954), I. I. Hirschman and D. V. Widder (1955), V. A. Ditkin and A. P. Prudnikov (1965), J. W. Miles (1971), F. Oberhettinger (1973), B. Davis (1978), W. R. LePage (1980), R. Bellman and R. Roth (1984), Yu. A. Brychkov and A. P. Prudnikov (1989), W. H. Beyer (1991), A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, Vols 4 and 5), R. J. Beerends, H. G. ter Morschem, and J. C. van den Berg (2003).

236

LINEAR EQUATIONS OF THE FIRST KIND WITH CONSTANT LIMITS OF INTEGRATION

3.2-4. Kernels Containing Power-Law and Exponential Functions.

a

14.

λx ke – k – t y(t) dt = f (x).

0

This is a special case of equation 3.8.5 with g(x) = keλx – k. 15.

a

x – keλt – k y(t) dt = f (x).

0

16.

This is a special case of equation 3.8.6 with g(t) = keλt + k.   ∞ 2x – i –ix–1/2 π y(t) dt = f (x), i2 = –1. t exp 4 –∞ Solution: y(x) =

1 4π





–∞

  f (t) 2t + i π dt. xit–1/2 exp 4 cosh(πt)

Reference: A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 463).

2

3.2-5. Kernels Containing Exponential Functions of the Form eλ(x±t) .



17.

2

e–(x–t) y(t) dt = f (x).

–∞ ◦

1 . The transformation Y (t) = exp(–t2 )y(t),

z = –2x,

F (z) = exp(x2 )f (x)

leads to an equation of the form 3.2.10: ∞ e–zt Y (t) dt = F (z). –∞

2◦ . Solution: y(t) =

1 π 3/2





e 0

s 2 /4





ds

  cos s(t – x) f (x) dx

–∞

 k 2k    ∞ 1 d f (t) 1 1 d2 √ – = exp – √ f (t) ≡ . 4 π dt2 k! 4 π dt2k k=0

(See equation 3.2.18 for λ = 1.) 3◦ . Solution:

∞ 1  fx(n) (0) Hn (x), y(x) = √ π n=0 2n n!

where Hn (x) are the Hermite polynomials (see Supplement 11.17-3)   dm   Hm (x) = (–1)m exp x2 exp –x2 . m dx References: P. M. Morse and H. Feshbach (1953), I. I. Hirschman and D. V. Widder (1955), P. G. Rooney (1963), M. L. Krasnov (1975).

3.2. EQUATIONS WHOSE KERNELS CONTAIN EXPONENTIAL FUNCTIONS

18.

237

  ∞ 1 (x – t)2 y(t) dt = f (x). exp – √ λ πλ –∞ It is the Gauss transform (the Weierstrass transform for λ = 4). Solution: ∞   1 ∞ λs2 /4 e ds cos s(t – x) f (x) dx y(t) = π 0 –∞  k 2k    ∞ 2 λ d f (t) λ d 1 – = exp – f (t) ≡ . 4 dt2 k! 4 dt2k k=0

References: I. I. Hirschman and D. V. Widder (1955), P. G. Rooney (1963), A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 435).





19.

ei(x+t) y(t) dt = f (x), 2

i2 = –1.

–∞

Solution: y(x) =

1 π





2

e–i(x+t) f (t) dt.

–∞

References: E. A. C. Paley and N. Wiener (1934), A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 435).

3.2-6. Other Kernels.

b

20.

exp(λx2 ) – exp(λt2 ) y(t) dt = f (x),

λ > 0.

a

This is a special case of equation 3.8.3 with g(x) = exp(λx2 ). Solution:   1 d 1 2  exp(–λx )fx (x) . y(x) = 4λ dx x The right-hand side f (x) of the integral equation must satisfy certain relations (see item 2◦ of equation 3.8.3).

21.

1 √ πx

0



 2  t y(t) dt = f (x). exp – 4x

Applying the Laplace transformation to the equation, we obtain √ y( ˜ p) = f˜(p), √ p





f˜(p) =

e–pt f (t) dt.

0

˜ we find that y(p) ˜ = pf˜(p2 ). The inverse Substituting p by p2 and solving for the transform y, Laplace transform provides the solution of the original integral equation: y(t) = L–1 {pf˜(p2 )},

L–1 {g(p)} ≡

1 2πi



c+i∞

ept g(p) dp. c–i∞

238

LINEAR EQUATIONS OF THE FIRST KIND WITH CONSTANT LIMITS OF INTEGRATION

3.3. Equations Whose Kernels Contain Hyperbolic Functions 3.3-1. Kernels Containing Hyperbolic Cosine.

b

1.

cosh(λx) – cosh(λt) y(t) dt = f (x).

a

This is a special case of equation 3.8.3 with g(x) = cosh(λx). Solution:    fx (x) 1 d . y(x) = 2λ dx sinh(λx) The right-hand side f (x) of the integral equation must satisfy certain relations (see item 2◦ of equation 3.8.3).

a

2.

cosh(βx) – cosh(µt) y(t) dt = f (x),

β > 0,

µ > 0.

0

This is a special case of equation 3.8.4 with g(x) = cosh(βx) and λ = µ/β.

b

3.

coshk x – coshk t| y(t) dt = f (x),

0 < k < 1.

a

This is a special case of equation 3.8.3 with g(x) = coshk x. Solution:   1 d fx (x) y(x) = . 2k dx sinh x coshk–1 x The right-hand side f (x) of the integral equation must satisfy certain relations (see item 2◦ of equation 3.8.3).

b

4. a

y(t) |cosh(λx) – cosh(λt)|k

dt = f (x),

0 < k < 1.

This is a special case of equation 3.8.7 with g(x) = cosh(λx) + β, where β is an arbitrary number. 3.3-2. Kernels Containing Hyperbolic Sine.

b

5.

  sinh λ|x – t| y(t) dt = f (x),

a

–∞ < a < b < ∞.

1◦ . Let us remove the modulus in the integrand:



x

b

sinh[λ(x – t)]y(t) dt + a

sinh[λ(t – x)]y(t) dt = f (x).

(1)

x

Differentiating (1) with respect to x twice yields 2λy(x) + λ2



x

b

sinh[λ(x – t)]y(t) dt + λ2 a

x

 sinh[λ(t – x)]y(t) dt = fxx (x).

(2)

3.3. EQUATIONS WHOSE KERNELS CONTAIN HYPERBOLIC FUNCTIONS

239

Eliminating the integral terms from (1) and (2), we obtain the solution y(x) =

 1  f (x) – λ2 f (x) . 2λ xx

(3)

2◦ . The right-hand side f (x) of the integral equation must satisfy certain relations. By setting x = a and x = b in (1), we obtain two corollaries



b

sinh[λ(t – a)]y(t) dt = f (a), a

b

sinh[λ(b – t)]y(t) dt = f (b).

(4)

a

Substituting solution (3) into (4) and integrating by parts yields the desired conditions for f (x): sinh[λ(b – a)]fx (b) – λ cosh[λ(b – a)]f (b) = λf (a), sinh[λ(b – a)]fx (a) + λ cosh[λ(b – a)]f (a) = –λf (b).

(5)

The general form of the right-hand side is given by f (x) = F (x) + Ax + B,

(6)

where F (x) is an arbitrary bounded twice differentiable function, and the coefficients A and B are expressed in terms of F (a), F (b), Fx (a), and Fx (b) and can be determined by substituting formula (6) into conditions (5).

b

6.

     A sinh λ|x – t| + B sinh µ|x – t| y(t) dt = f (x),

a

–∞ < a < b < ∞.

Let us remove the modulus in the integrand and differentiate the equation with respect to x twice to obtain

b

2(Aλ + Bµ)y(x) +

     Aλ2 sinh λ|x – t| + Bµ2 sinh µ|x – t| y(t) dt = fxx (x).

(1)

a

  Eliminating the integral term with sinh µ|x – t| from (1) yields 2

b

2

2(Aλ + Bµ)y(x) + A(λ – µ )

   sinh λ|x – t| y(t) dt = fxx (x) – µ2 f (x).

(2)

a

For Aλ + Bµ = 0, this is an equation of the form 3.3.5, and for Aλ + Bµ ≠ 0, this is an equation of the form 4.3.26. The right-hand side f (x) must satisfy certain relations, which can be obtained by setting x = a and x = b in the original equation (a similar procedure is used in 3.3.5).

b

7.

sinh(λx) – sinh(λt) y(t) dt = f (x).

a

This is a special case of equation 3.8.3 with g(x) = sinh(λx). Solution:    fx (x) 1 d . y(x) = 2λ dx cosh(λx) The right-hand side f (x) of the integral equation must satisfy certain relations (see item 2◦ of equation 3.8.3).

240

LINEAR EQUATIONS OF THE FIRST KIND WITH CONSTANT LIMITS OF INTEGRATION



a

8.

sinh(βx) – sinh(µt) y(t) dt = f (x),

β > 0,

µ > 0.

0

9.

This is a special case of equation 3.8.4 with g(x) = sinh(βx) and λ = µ/β. b   sinh3 λ|x – t| y(t) dt = f (x). a

Using the formula sinh3 β = 14 sinh 3β – 34 sinh β, we arrive at an equation of the form 3.3.6: b

1   3   4 A sinh 3λ|x – t| – 4 A sinh λ|x – t| y(t) dt = f (x). a

10.

b  n a

   Ak sinh λk |x – t| y(t) dt = f (x),

–∞ < a < b < ∞.

k=1

1◦ . Let us remove the modulus in the kth summand of the integrand: b x b   sinh λk |x – t| y(t) dt = sinh[λk (x – t)]y(t) dt + sinh[λk (t – x)]y(t) dt. (1) Ik (x) = a

a

x

Differentiating (1) with respect to x twice yields x cosh[λk (x – t)]y(t) dt – λk Ik = λk a

Ik



= 2λk y(x) +

b

cosh[λk (t – x)]y(t) dt, x



x

λ2k

sinh[λk (x – t)]y(t) dt + a

(2)

b

λ2k

sinh[λk (t – x)]y(t) dt, x

where the primes denote the derivatives with respect to x. By comparing formulas (1) and (2), we find the relation between Ik and Ik : Ik = 2λk y(x) + λ2k Ik ,

Ik = Ik (x).

(3)



2 . With the aid of (1), the integral equation can be rewritten in the form n  Ak Ik = f (x).

(4)

k=1

Differentiating (4) with respect to x twice and taking into account (3), we find that n n    σ1 y(x) + Ak λ2k Ik = fxx (x), σ1 = 2 Ak λk . k=1

(5)

k=1

Eliminating the integral In from (4) and (5) yields σ1 y(x) +

n–1 

 Ak (λ2k – λ2n )Ik = fxx (x) – λ2n f (x).

(6)

k=1

Differentiating (6) with respect to x twice and eliminating In–1 from the resulting equation with the aid of (6), we obtain a similar equation whose right-hand side is a second-order n–2  linear differential operator (acting on y) with constant coefficients plus the sum Bk Ik . k=1

If we successively eliminate In–2 , In–3 , . . . , with the aid of double differentiation, then we finally arrive at a linear nonhomogeneous ordinary differential equation of order 2(n – 1) with constant coefficients. 3◦ . The right-hand side f (x) must satisfy certain conditions. To find these conditions, one should set x = a in the integral equation and its derivatives. (Alternatively, these conditions can be found by setting x = a and x = b in the integral equation and all its derivatives obtained by means of double differentiation.)

3.3. EQUATIONS WHOSE KERNELS CONTAIN HYPERBOLIC FUNCTIONS



b

11.

sinhk x – sinhk t y(t) dt = f (x),

241

0 < k < 1.

0

This is a special case of equation 3.8.3 with g(x) = sinhk x. Solution:   fx (x) 1 d y(x) = . 2k dx cosh x sinhk–1 x The right-hand side f (x) must satisfy certain conditions. As follows from item 3◦ of equation 3.8.3, the admissible general form of the right-hand side is given by f (x) = F (x) + Ax + B,

A = –Fx (b),

B=

1 2

 bFx (b) – F (0) – F (b) ,

where F (x) is an arbitrary bounded twice differentiable function (with bounded first derivative).

b

12. a

y(t) |sinh(λx) – sinh(λt)|k

dt = f (x),

0 < k < 1.

This is a special case of equation 3.8.7 with g(x) = sinh(λx) + β, where β is an arbitrary number.

a

13.

k sinh(λx) – t y(t) dt = f (x).

0

This is a special case of equation 3.8.5 with g(x) = k sinh(λx).

a

14.

x – k sinh(λt) y(t) dt = f (x).

0

This is a special case of equation 3.8.6 with g(x) = k sinh(λt).

3.3-3. Kernels Containing Hyperbolic Tangent.

b

15.

tanh(λx) – tanh(λt) y(t) dt = f (x).

a

This is a special case of equation 3.8.3 with g(x) = tanh(λx). Solution:  1 d cosh2 (λx)fx (x) . y(x) = 2λ dx The right-hand side f (x) of the integral equation must satisfy certain relations (see item 2◦ of equation 3.8.3). 16.

a

tanh(βx) – tanh(µt) y(t) dt = f (x),

β > 0,

µ > 0.

0

This is a special case of equation 3.8.4 with g(x) = tanh(βx) and λ = µ/β.

242

LINEAR EQUATIONS OF THE FIRST KIND WITH CONSTANT LIMITS OF INTEGRATION



b

17.

|tanhk x – tanhk t| y(t) dt = f (x),

0 < k < 1.

0

This is a special case of equation 3.8.3 with g(x) = tanhk x. Solution:  1 d cosh2 x cothk–1 x fx (x) . y(x) = 2k dx The right-hand side f (x) must satisfy certain conditions. As follows from item 3◦ of equation 3.8.3, the admissible general form of the right-hand side is given by

 f (x) = F (x) + Ax + B, A = –Fx (b), B = 12 bFx (b) – F (0) – F (b) ,

19.

where F (x) is an arbitrary bounded twice differentiable function (with bounded first derivative). b y(t) dt = f (x), 0 < k < 1. |tanh(λx) – tanh(λt)|k a This is a special case of equation 3.8.7 with g(x) = tanh(λx) + β, where β is an arbitrary number. a k tanh(λx) – t y(t) dt = f (x).

20.

This is a special case of equation 3.8.5 with g(x) = k tanh(λx). a x – k tanh(λt) y(t) dt = f (x).

18.

0

0

This is a special case of equation 3.8.6 with g(x) = k tanh(λt). 3.3-4. Kernels Containing Hyperbolic Cotangent.

b

21.

coth(λx) – coth(λt) y(t) dt = f (x).

a

22.

This is a special case of equation 3.8.3 with g(x) = coth(λx). b cothk x – cothk t y(t) dt = f (x), 0 < k < 1. 0

This is a special case of equation 3.8.3 with g(x) = cothk x.

3.4. Equations Whose Kernels Contain Logarithmic Functions 3.4-1. Kernels Containing Logarithmic Functions.

b

1.

ln(x/t) y(t) dt = f (x).

a

This is a special case of equation 3.8.3 with g(x) = ln x. Solution: 1 d   xfx (x) . y(x) = 2 dx The right-hand side f (x) of the integral equation must satisfy certain relations (see item 2◦ of equation 3.8.3).

243

3.4. EQUATIONS WHOSE KERNELS CONTAIN LOGARITHMIC FUNCTIONS



b

2. a

ln |x – t| y(t) dt = f (x).

Carleman’s equation. 1◦ . Solution with b – a ≠ 4: y(x) =

1 √ π 2 (x – a)(b – x)





b

a

(t – a)(b – t) ft (t) dt 1  + 1 t–x ln 4 (b – a)



b

a

 f (t) dt √ . (t – a)(b – t)

2◦ . If b – a = 4, then for the equation to be solvable, the condition

b

f (t)(t – a)–1/2 (b – t)–1/2 dt = 0 a

must be satisfied. In this case, the solution has the form 1 y(x) = 2 √ π (x – a)(b – x)

 a

b

√  (t – a)(b – t) ft (t) dt +C , t–x

where C is an arbitrary constant. Reference: F. D. Gakhov (1977).



b

3. a

  ln |x – t| + β y(t) dt = f (x).

By setting x = e–β z,

t = e–β τ ,

y(t) = Y (τ ),

f (x) = e–β g(z),

we arrive at an equation of the form 3.4.2:

B

ln |z – τ | Y (τ ) dτ = g(z),

A = aeβ , B = beβ .

A



a

 ln

4. –a

A  |x – t|

y(t) dt = f (x),

–a ≤ x ≤ a.

This is a special case of equation 3.4.3 with b = –a. Solution with 0 < a < 2A: y(x) =

  a d 1 w(t, a)f (t) dt w(x, a) 2M  (a) da –a   ξ 1 d 1 a d – w(x, ξ) w(t, ξ)f (t) dt dξ 2 |x| dξ M  (ξ) dξ –ξ  ξ  a w(x, ξ) 1 d – w(t, ξ) df (t) dξ, 2 dx |x| M  (ξ) –ξ

where M (ξ) =

–1  2A ln , ξ

and the prime stands for the derivative. Reference: I. C. Gohberg and M. G. Krein (1967).

w(x, ξ) =

π

M (ξ)  , ξ 2 – x2

244

LINEAR EQUATIONS OF THE FIRST KIND WITH CONSTANT LIMITS OF INTEGRATION

a x + t y(t) dt = f (x). ln x–t 0

5.

Solution: 2 d y(x) = – 2 π dx



a

x

F (t) dt √ , t2 – x2

d F (t) = dt



t

0

sf (s) ds √ . t2 – s 2

Reference: P. P. Zabreyko, A. I. Koshelev, et al. (1975).



b

6. a

1 + λx ln 1 + λt y(t) dt = f (x).

This is a special case of equation 3.8.3 with g(x) = ln(1 + λx). Solution:  1 d y(x) = (1 + λx)fx (x) . 2λ dx The right-hand side f (x) of the integral equation must satisfy certain relations (see item 2◦ of equation 3.8.3).

b

7.

β ln x – lnβ t y(t) dt = f (x),

0 < β < 1.

a

This is a special case of equation 3.8.3 with g(x) = lnβ x.

b

8. a

y(t) dt = f (x), |ln(x/t)|β

0 < β < 1.

This is a special case of equation 3.8.7 with g(x) = ln x + A, where A is an arbitrary number.

3.4-2. Kernels Containing Power-Law and Logarithmic Functions.

  ln |x – t| + βtk y(t) dt = f (x).

1

9. 0

See Example 3 in Subsection 12.6-2 with ψ(t) = βtk .

a

10.

k ln(1 + λx) – t y(t) dt = f (x).

0

This is a special case of equation 3.8.5 with g(x) = k ln(1 + λx).

a

11.

x – k ln(1 + λt) y(t) dt = f (x).

0

This is a special case of equation 3.8.6 with g(x) = k ln(1 + λt).



12. 0

x + t y(t) dt = f (x). ln t x–t

1

Solution: x d y(x) = 2 π dx

0



df (t) x2 ln 1 – 2 dt. dt t

Reference: P. P. Zabreyko, A. I. Koshelev, et al. (1975).

3.4. EQUATIONS WHOSE KERNELS CONTAIN LOGARITHMIC FUNCTIONS



13.

245



ln x – ln t y(t) dt = f (x). x–t 0 The left-hand side of this equation is the iterated Stieltjes transform. Under some assumptions, the solution of the integral equation can be represented in the form  e 4n 1 d y(x) = . lim Dn x2n D2n x2n Dn f (x), D = 4π 2 n→∞ n dx To calculate the solution approximately, one should restrict oneself to a specific value of n in this formula instead of taking the limit. Reference: I. I. Hirschman and D. V. Widder (1955).



b

14. a

ln |xβ – tβ | y(t) dt = f (x),

β > 0.

The transformation z = xβ ,

τ = tβ ,

w(τ ) = t1–β y(t)

leads to Carleman’s equation 3.4.2: B ln |z – τ |w(τ ) dτ = F (z),

A = aβ ,

B = bβ ,

A

  where F (z) = βf z 1/β .

1

15.

ln |xβ – tµ | y(t) dt = f (x),

β > 0, µ > 0.

0

The transformation z = xβ ,

τ = tµ ,

w(τ ) = t1–µ y(t)

leads to an equation of the form 3.4.2: 1 ln |z – τ |w(τ ) dτ = F (z), 16.

  F (z) = µf z 1/β .

0 ∞



1

xt ln(xt) Solution:

y(t) dt = f (x).

0



1 y(x) = – 2 π



0

1 √ f (t) dt. xt ln(xt)

References: E. C. Titchmarsh (1986), A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 450).

17.

d





dx –∞ Solution:

x ln 1 – y(t) dt = f (x). t 1 d y(x) = – 2 π dx





–∞

x ln 1 – f (t) dt. t

References: E. C. Titchmarsh (1986), A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 450).

18.



(xt)–[1+i ln(xt)]/2 y(t) dt = f (x),

i2 = –1.

0

Solution:

1 y(x) = 2π





(xt)–[1–i ln(xt)]/2 f (t) dt.

0

Reference: A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 452).

246

LINEAR EQUATIONS OF THE FIRST KIND WITH CONSTANT LIMITS OF INTEGRATION

3.4-3. Equation Containing the Unknown Function of a Complicated Argument.

1

19.

 A ln t + B)y(xt) dt = f (x).

0

The substitution ξ = xt leads to an equation of the form 1.9.3 with g(x) = –A ln x:

x

 A ln ξ – A ln x + B y(ξ) dξ = xf (x).

0

3.5. Equations Whose Kernels Contain Trigonometric Functions 3.5-1. Kernels Containing Cosine.



cos(xt)y(t) dt = f (x).

1. 0

2 ∞ cos(xt)f (t) dt. π 0 Up to constant factors, the function f (x) and the solution y(t) are the Fourier cosine transform pair.

Solution: y(x) =

References: E. A. C. Paley and N. Wiener (1934), S. Bochner and K. C. Chandrasekharan (1949), G. N. Watson (1952), H. Bateman and A. Erd´elyi (Vol. 1, 1954), S. Bochner (1959), V. A. Ditkin and A. P. Prudnikov (1965), B. Davis (1978), F. Oberhettinger (1980), E. C. Titchmarsh (1986), Ya. A. Brychkov and A. P. Prudnikov (1989), A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 440), I. Sneddon (1995), A. D. Poularikas (2000).



b

0 ≤ x < ∞.

cos(xt)y(t) dt = f (x),

2. a

Solution:

 y(t) =

2 π 0





cos(xt)f (x) dx if a < t < b, 0

if 0 < t < a or t > b,

where 0 ≤ a ≤ b ≤ ∞.

b

3.

cos(λx) – cos(λt) y(t) dt = f (x).

a

This is a special case of equation 3.8.3 with g(x) = cos(λx). Solution:    1 d fx (x) . y(x) = – 2λ dx sin(λx) The right-hand side f (x) of the integral equation must satisfy certain relations (see item 2◦ of equation 3.8.3). 4.

a

cos(βx) – cos(µt) y(t) dt = f (x),

β > 0,

µ > 0.

0

This is a special case of equation 3.8.4 with g(x) = cos(βx) and λ = µ/β.

3.5. EQUATIONS WHOSE KERNELS CONTAIN TRIGONOMETRIC FUNCTIONS



b

5.

k cos x – cosk t y(t) dt = f (x),

247

0 < k < 1.

a

This is a special case of equation 3.8.3 with g(x) = cosk x. Solution:   1 d fx (x) y(x) = – . 2k dx sin x cosk–1 x The right-hand side f (x) of the integral equation must satisfy certain relations (see item 2◦ of equation 3.8.3).

b

y(t)

6.

|cos(λx) – cos(λt)|k

a

7.

dt = f (x),

0 < k < 1.

This is a special case of equation 3.8.7 with g(x) = cos(λx) + β, where β is an arbitrary number.   ∞ 1 + 2ix π y(t) dt = f (x), i2 = –1. t–ix–1/2 cos 4 0 Solution:

1 y(t) = π





t

ix–1/2

–∞

  f (x) 1 – 2ix π dx. cos 4 cosh(πx)

Reference: A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 463).

3.5-2. Kernels Containing Sine.



sin(xt)y(t) dt = f (x).

8. 0

2 ∞ sin(xt)f (t) dt. π 0 Up to constant factors, the function f (x) and the solution y(t) are the Fourier sine transform pair.

Solution: y(x) =

References: E. A. C. Paley and N. Wiener (1934), S. Bochner and K. C. Chandrasekharan (1949), G. N. Watson (1952), H. Bateman and A. Erd´elyi (Vol. 1, 1954), S. Bochner (1959), V. A. Ditkin and A. P. Prudnikov (1965), B. Davis (1978), F. Oberhettinger (1980), E. C. Titchmarsh (1986), Ya. A. Brychkov and A. P. Prudnikov (1989), A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 440), I. Sneddon (1995), A. D. Poularikas (2000).



b

0 ≤ x < ∞.

sin(xt)y(t) dt = f (x),

9. a

Solution:

 y(t) =

10.

2 π 0





sin(xt)f (x) dx if a < t < b, 0

where 0 ≤ a ≤ b ≤ ∞. ∞   sin λ|x – t| y(t) dt = f (x),

if 0 < t < a or t > b,

f (±∞) = 0.

–∞

Solution: y(x) =

 1  f (x) + λ2 f (x) . 2λ xx

248

LINEAR EQUATIONS OF THE FIRST KIND WITH CONSTANT LIMITS OF INTEGRATION



b

11.

  sin λ|x – t| y(t) dt = f (x),

–∞ < a < b < ∞.

a ◦

1 . Let us remove the modulus in the integrand:



x

b

sin[λ(x – t)]y(t) dt + a

sin[λ(t – x)]y(t) dt = f (x).

(1)

x

Differentiating (1) with respect to x twice yields 2



x

b

2

sin[λ(x – t)]y(t) dt – λ

2λy(x) – λ

a

 sin[λ(t – x)]y(t) dt = fxx (x).

(2)

x

Eliminating the integral terms from (1) and (2), we obtain the solution y(x) =

 1  fxx (x) + λ2 f (x) . 2λ

(3)

2◦ . The right-hand side f (x) of the integral equation must satisfy certain relations. By setting x = a and x = b in (1), we obtain two corollaries b b sin[λ(t – a)]y(t) dt = f (a), sin[λ(b – t)]y(t) dt = f (b). (4) a

a

Substituting solution (3) into (4) followed by integrating by parts yields the desired conditions for f (x): sin[λ(b – a)]fx (b) – λ cos[λ(b – a)]f (b) = λf (a), (5) sin[λ(b – a)]fx (a) + λ cos[λ(b – a)]f (a) = –λf (b). The general form of the right-hand side of the integral equation is given by f (x) = F (x) + Ax + B,

(6)

where F (x) is an arbitrary bounded twice differentiable function, and the coefficients A and B are expressed in terms of F (a), F (b), Fx (a), and Fx (b) and can be determined by substituting formula (6) into conditions (5).

b

12.

     A sin λ|x – t| + B sin µ|x – t| y(t) dt = f (x),

a

–∞ < a < b < ∞.

Let us remove the modulus in the integrand and differentiate the equation with respect to x twice to obtain b  2      2(Aλ + Bµ)y(x) – Aλ sin λ|x – t| + Bµ2 sin µ|x – t| y(t) dt = fxx (x). (1) a

  Eliminating the integral term with sin µ|x – t| from (1) with the aid of the original equation, we find that b   2 2  2(Aλ + Bµ)y(x) + A(µ – λ ) sin λ|x – t| y(t) dt = fxx (x) + µ2 f (x). (2) a

For Aλ + Bµ = 0, this is an equation of the form 3.5.11 and for Aλ + Bµ ≠ 0, this is an equation of the form 4.5.29. The right-hand side f (x) must satisfy certain relations, which can be obtained by setting x = a and x = b in the original equation (a similar procedure is used in 3.5.11).

3.5. EQUATIONS WHOSE KERNELS CONTAIN TRIGONOMETRIC FUNCTIONS



b

13.

249

sin(λx) – sin(λt) y(t) dt = f (x).

a

This is a special case of equation 3.8.3 with g(x) = sin(λx). Solution:    fx (x) 1 d . y(x) = 2λ dx cos(λx) The right-hand side f (x) of the integral equation must satisfy certain relations (see item 2◦ of equation 3.8.3).

a

14.

sin(βx) – sin(µt) y(t) dt = f (x),

β > 0,

µ > 0.

0

This is a special case of equation 3.8.4 with g(x) = sin(βx) and λ = µ/β.

b

15.

  sin3 λ|x – t| y(t) dt = f (x).

a

Using the formula sin3 β = – 14 sin 3β +

16.

n b 

a

sin β, we arrive at an equation of the form 3.5.12:

b

    – 14 A sin 3λ|x – t| + 34 A sin λ|x – t| y(t) dt = f (x).

a



3 4

   Ak sin λk |x – t| y(t) dt = f (x),

–∞ < a < b < ∞.

k=1

1◦ . Let us remove the modulus in the kth summand of the integrand: Ik (x) =

b

  sin λk |x – t| y(t) dt =

a





x

b

sin[λk (x – t)]y(t) dt + a

sin[λk (t – x)]y(t) dt. (1) x

Differentiating (1) with respect to x yields Ik = λk





x

a

Ik

b

cos[λk (x – t)]y(t) dt – λk

= 2λk y(x) –



cos[λk (t – x)]y(t) dt, x



x

λ2k

sin[λk (x – t)]y(t) dt – a

(2)

b

λ2k

sin[λk (t – x)]y(t) dt, x

where the primes denote the derivatives with respect to x. By comparing formulas (1) and (2), we find the relation between Ik and Ik : Ik = 2λk y(x) – λ2k Ik ,

Ik = Ik (x).

(3)

2◦ . With the aid of (1), the integral equation can be rewritten in the form n 

Ak Ik = f (x).

(4)

k=1

Differentiating (4) with respect to x twice and taking into account (3), we find that σ1 y(x) –

n  k=1

 Ak λ2k Ik = fxx (x),

σ1 = 2

n  k=1

Ak λk .

(5)

250

LINEAR EQUATIONS OF THE FIRST KIND WITH CONSTANT LIMITS OF INTEGRATION

Eliminating the integral In from (4) and (5) yields σ1 y(x) +

n–1 

 Ak (λ2n – λ2k )Ik = fxx (x) + λ2n f (x).

(6)

k=1

Differentiating (6) with respect to x twice and eliminating In–1 from the resulting equation with the aid of (6), we obtain a similar equation whose left-hand side is a second-order n–2  linear differential operator (acting on y) with constant coefficients plus the sum Bk Ik . k=1

If we successively eliminate In–2 , In–3 , . . . , with the aid of double differentiation, then we finally arrive at a linear nonhomogeneous ordinary differential equation of order 2(n – 1) with constant coefficients.

17.

3◦ . The right-hand side f (x) must satisfy certain conditions. To find these conditions, one should set x = a in the integral equation and its derivatives. (Alternatively, these conditions can be found by setting x = a and x = b in the integral equation and all its derivatives obtained by means of double differentiation.) b k sin x – sink t y(t) dt = f (x), 0 < k < 1. 0

This is a special case of equation 3.8.3 with g(x) = sink x. Solution:   fx (x) 1 d . y(x) = 2k dx cos x sink–1 x The right-hand side f (x) must satisfy certain conditions. As follows from item 3◦ of equation 3.8.3, the admissible general form of the right-hand side is given by

 f (x) = F (x) + Ax + B, A = –Fx (b), B = 12 bFx (b) – F (0) – F (b) ,

19.

where F (x) is an arbitrary bounded twice differentiable function (with bounded first derivative). b y(t) dt = f (x), 0 < k < 1. k a |sin(λx) – sin(λt)| This is a special case of equation 3.8.7 with g(x) = sin(λx) +β, where β is an arbitrary number. a k sin(λx) – t y(t) dt = f (x).

20.

This is a special case of equation 3.8.5 with g(x) = k sin(λx). a x – k sin(λt) y(t) dt = f (x).

18.

0

0

21.

This is a special case of equation 3.8.6 with g(t) = k sin(λt). ∞ sin t [y(x + t) – y(x – t)] dt = f (x). t2 0 Solution:   1 ∞ cos t + Si(t) [f (x – t) – f (x + t)] dt, y(x) = π 0 t where Si(t) is sine integral (see Supplement 11.3-1). The integral equation and its solution form the Boas transform pair. Reference: A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 442).

3.5. EQUATIONS WHOSE KERNELS CONTAIN TRIGONOMETRIC FUNCTIONS





22.

 t–ix–1/2 sin

0

 1 + 2ix π y(t) dt = f (x), 4

Solution: y(t) =

1 π





 tix–1/2 sin

–∞

251

i2 = –1.

 f (x) 1 – 2ix π dx. 4 cosh(πx)

Reference: A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 463).

3.5-3. Kernels Containing Tangent.

b

23.

tan(λx) – tan(λt) y(t) dt = f (x).

a

This is a special case of equation 3.8.3 with g(x) = tan(λx). Solution:   1 d cos2 (λx)fx (x) . y(x) = 2λ dx The right-hand side f (x) of the integral equation must satisfy certain relations (see item 2◦ of equation 3.8.3).

a

24.

tan(βx) – tan(µt) y(t) dt = f (x),

β > 0,

µ > 0.

0

This is a special case of equation 3.8.4 with g(x) = tan(βx) and λ = µ/β.

b

25.

k tan x – tank t y(t) dt = f (x),

0 < k < 1.

0

This is a special case of equation 3.8.3 with g(x) = tank x. Solution:   1 d 2 k–1  cos x cot xfx (x) . y(x) = 2k dx The right-hand side f (x) must satisfy certain conditions. As follows from item 3◦ of equation 3.8.3, the admissible general form of the right-hand side is given by

 f (x) = F (x) + Ax + B, A = –Fx (b), B = 12 bFx (b) – F (0) – F (b) , where F (x) is an arbitrary bounded twice differentiable function (with bounded first derivative).

b

26. a

y(t) dt = f (x), |tan(λx) – tan(λt)|k

0 < k < 1.

This is a special case of equation 3.8.7 with g(x) = tan(λx)+β, where β is an arbitrary number.

a

27.

k tan(λx) – t y(t) dt = f (x).

0

This is a special case of equation 3.8.5 with g(x) = k tan(λx).

a

28.

x – k tan(λt) y(t) dt = f (x).

0

This is a special case of equation 3.8.6 with g(t) = k tan(λt).

252

LINEAR EQUATIONS OF THE FIRST KIND WITH CONSTANT LIMITS OF INTEGRATION

3.5-4. Kernels Containing Cotangent.

b

29.

cot(λx) – cot(λt) y(t) dt = f (x).

a

This is a special case of equation 3.8.3 with g(x) = cot(λx).

b

30.

k cot x – cotk t y(t) dt = f (x),

0 < k < 1.

a

This is a special case of equation 3.8.3 with g(x) = cotk x.

3.5-5. Kernels Containing a Combination of Trigonometric Functions.



31.

 cos(xt) + sin(xt) y(t) dt = f (x).

–∞

Solution: y(x) =

1 2π





 cos(xt) + sin(xt) f (t) dt.

–∞

Up to constant factors, the function f (x) and the solution y(t) are the Hartley transform pair. Reference: D. Zwillinger (1989).





32.

 sin(xt) – xt cos(xt) y(t) dt = f (x).

0

This equation can be reduced to a special case of equation 3.7.17 with ν = 32 . Solution: 2 ∞ sin(xt) – xt cos(xt) f (t) dt. y(x) = π 0 x2 t2



[sin(xt) + xt cos(xt)]y(t) dt = f (x).

33. 0

Solution: y(x) = –

2 π





si(xt)y(t) dt, 0

where si(z) is the sine integral (see Supplement 11.3-1). References: H. M. Srivastava and R. G. Buschman (1977), A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 457).





[1 – cos(xt) + xt sin(xt)]y(t) dt = f (x).

34. 0

Solution: y(x) =

2 π





ci(xt)f (t) dt, 0

where ci(z) is the cosine integral (see Supplement 11.3-2). References: H. M. Srivastava and R. G. Buschman (1977), A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 457).

3.5. EQUATIONS WHOSE KERNELS CONTAIN TRIGONOMETRIC FUNCTIONS





35.

 (xt)1/2

0

253

 sin(xt) + 2 cos(xt) y(t) dt = f (x). xt 

Solution: y(x) =

∞



2 π

0

 1 – S(xt) f (t) dt. 2

where S(z) is the Fresnel sine integral (see Supplement 11.3-3). References: H. M. Srivastava and R. G. Buschman (1977), A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 459).





36.

 (xt)1/2

0

cos(xt) – 1 xt

 – 2 sin(xt) y(t) dt = f (x). 

Solution: y(x) =

2 π

∞

0

 1 – C(xt) f (t) dt, 2

where C(z) is the Fresnel cosine integral (see Supplement 11.3-3). References: H. M. Srivastava and R. G. Buschman (1977), A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 460).





37.

(1 – ν) sin(xt) + xt cos(xt) (xt)ν

0

y(t) dt = f (x).

Solution: y(x) =

2 π





S(xt, ν)f (t) dt, 0

where S(z, ν) is the generalized Fresnel sine integral (see Supplement 11.3-3). References: H. M. Srivastava and R. G. Buschman (1977), A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 461).



38.



(1 – ν) cos(xt) – xt sin(xt) y(t) dt = f (x). (xt)ν 0 Solution: 2 ∞ y(x) = C(xt, ν)y(t) dt, π 0 where C(z, ν) is the generalized Fresnel cosine integral (see Supplement 11.3-3). References: H. M. Srivastava and R. G. Buschman (1977), A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 461).

39.

π



a sin(x + t)

a sin(x – t)



y(t) dt = f (x), 0 < a < 1. + 1 – 2a cos(x + t) + a2 1 – 2a cos(x – t) + a2 Solution: π ∞ 2  fn y(x) = C + 2 cos(nx), f = f (x) sin(nx) dx, n π n=1 an 0 0

where C is an arbitrary constant. Remark. The kernel of the integral equation can be represented as a series in powers of a: ∞

K(x, t) =

 a sin(x + t) a sin(x – t) + = 2 an sin(nx) cos(nt). 1 – 2a cos(x + t) + a2 1 – 2a cos(x – t) + a2 n=1

References: W. Schmeidler (1950, p. 169), S. Feny¨o and H. W. Stolle (1984, pp. 18–19).

254

LINEAR EQUATIONS OF THE FIRST KIND WITH CONSTANT LIMITS OF INTEGRATION

3.5-6. Equations Containing the Unknown Function of a Complicated Argument.

π/2

y(ξ) dt = f (x),

40.

ξ = x sin t.

0

Schl¨omilch equation. Solution: y(x) =

  π/2 2 f (0) + x fξ (ξ) dt , π 0

ξ = x sin t.

References: E. T. Whittaker and G. N. Watson (1958), F. D. Gakhov (1977).



π/2

ξ = x sink t.

y(ξ) dt = f (x),

41. 0

Generalized Schl¨omilch equation. This is a special case of equation 3.5.43 for λ = 0 and m = 0. Solution:   x 1 2k k–1 d x k xk y(x) = sin t f (ξ) dt , π dx 0

π/2

42.

sinλ t y(ξ) dt = f (x),

ξ = x sink t.

ξ = x sink t.

0

This is a special case of equation 3.5.43 for m = 0. Solution: y(x) = 43.

π/2

  x λ+1 2k k–λ–1 d x k x k sinλ+1 t f (ξ) dt , π dx 0

sinλ t cosm t y(ξ) dt = f (x),

ξ = x sink t.

ξ = x sink t.

0

1◦ . Let λ > –1, m > –1, and k > 0. The transformation 2

z = xk ,

ζ = z sin2 t,

w(ζ) = ζ

λ–1 2

 k y ζ2

leads to an equation of the form 1.1.44:

z

(z – ζ)

m–1 2

w(ζ) dζ = F (z),

F (z) = 2z

λ+m 2

 k f z2 .

0

2◦ . Solution with –1 < m < 1: y(x) =

  π(1 – m)  k–λ–1 d  λ+1 π/2 2k sin x k x k sinλ+1 t tanm t f (ξ) dt , π 2 dx 0

where ξ = x sink t.

3.6. EQUATIONS WHOSE KERNELS CONTAIN COMBINATIONS OF ELEMENTARY FUNCTIONS

255

3.5-7. Singular Equations.

44.

t – x y(t) dt = f (x), 0 ≤ x ≤ 2π. 2 0 Here the integral is understood in the sense of the Cauchy principal value and the right-hand 2π

cot



side is assumed to satisfy the condition f (t) dt = 0. 0 Solution: 2π  1 t –x y(x) = – 2 f (t) dt + C, cot 4π 0 2 where C is an arbitrary constant.



It follows from the solution that y(t) dt = 2πC. 0 The equation and its solution form a Hilbert transform pair (in the asymmetric form). Reference: F. D. Gakhov (1977).



π

 1 + cot

45. –π

 x – t  2

y(t) dt = f (x),

Hilbert–Plessner equation. Solution: y(x) =

1 4π 2

–π ≤ x ≤ π.

π  x – t  f (t) dt. 1 + cot 2 –π

Reference: S. Feny¨o and H. W. Stolle (1984, pp. 36–38).



46.

  ξ – x  –2 sin y(ξ) dξ = f (x), 0 ≤ x ≤ 2π. 2 0 The simple hypersingular equation of the first kind with Hilbert-type kernel. Let the periodic conditions y(0) = y(2π) be satisfied. Then the solution is 2π  ξ – x  1 y(x) = – 2 f (ξ) ln sin dξ + C, 4π 0 2 2π

where C is an arbitrary constant. This equation is discussed in Subsection 14.6-4 in detail. Reference: I. K. Lifanov, L. N. Poltavskii, and G. M. Vainikko (2004, p. 8).

3.6. Equations Whose Kernels Contain Combinations of Elementary Functions 3.6-1. Kernels Containing Hyperbolic and Logarithmic Functions.

b

1.

ln cosh(λx) – cosh(λt) y(t) dt = f (x).

a

This is a special case of equation 3.8.9 with g(x) = cosh(λx).

b

2.

ln sinh(λx) – sinh(λt) y(t) dt = f (x).

a

This is a special case of equation 3.8.9 with g(x) = sinh(λx).

256

LINEAR EQUATIONS OF THE FIRST KIND WITH CONSTANT LIMITS OF INTEGRATION



a

3. –a

   sinh 12 A   y(t) dt = f (x), ln 2 sinh 12 |x – t| 

–a ≤ x ≤ a.

Solution with 0 < a < A: y(x) =

  a d 1 w(t, a)f (t) dt w(x, a) 2M  (a) da –a   ξ 1 d 1 a d – w(x, ξ) w(t, ξ)f (t) dt dξ 2 |x| dξ M  (ξ) dξ –ξ  ξ  a w(x, ξ) 1 d – w(t, ξ) df (t) dξ, 2 dx |x| M  (ξ) –ξ

where the prime stands for the derivative with respect to the argument and   –1   sinh 12 A   M (ξ) = ln , sinh 12 ξ

  cosh 12 x M (ξ) . w(x, ξ) = √ π 2 cosh ξ – 2 cosh x

Reference: I. C. Gohberg and M. G. Krein (1967).



b

4.

ln tanh(λx) – tanh(λt) y(t) dt = f (x).

a

This is a special case of equation 3.8.9 with g(x) = tanh(λx).

a

5. –a

  ln coth 14 |x – t| y(t) dt = f (x),

Solution: y(x) =

–a ≤ x ≤ a.

  a 1 d w(t, a)f (t) dt w(x, a) 2M  (a) da –a   ξ 1 d d 1 a w(x, ξ) w(t, ξ)f (t) dt dξ – 2 |x| dξ M  (ξ) dξ –ξ  ξ  a w(x, ξ) 1 d – w(t, ξ) df (t) dξ, 2 dx |x| M  (ξ) –ξ

where the prime stands for the derivative with respect to the argument and M (ξ) =

P–1/2 (cosh ξ) , Q–1/2 (cosh ξ)

w(x, ξ) =

1 √ , πQ–1/2 (cosh ξ) 2 cosh ξ – 2 cosh x

and P–1/2 (cosh ξ) and Q–1/2 (cosh ξ) are the Legendre functions of the first and second kind, respectively. Reference: I. C. Gohberg and M. G. Krein (1967).

3.6-2. Kernels Containing Logarithmic and Trigonometric Functions.

b

6.

ln cos(λx) – cos(λt) y(t) dt = f (x).

a

This is a special case of equation 3.8.9 with g(x) = cos(λx).

3.6. EQUATIONS WHOSE KERNELS CONTAIN COMBINATIONS OF ELEMENTARY FUNCTIONS



b

7.

ln sin(λx) – sin(λt) y(t) dt = f (x).

a

8.

This is a special case of equation 3.8.9 with g(x) = sin(λx). π 1 – cos(x + t) y(t) dt = f (x), 0 ≤ x ≤ π. ln 1 – cos(x – t) 0 Solution: π ∞ 2  nfn sin(nx), fn = f (x) sin(nx) dx. y(x) = 2 π n=1 0 Reference: S. Feny¨o and H. W. Stolle (1984, p. 44).



a

9. –a

   sin 21 A   y(t) dt = f (x), ln 2 sin 12 |x – t| 

–a ≤ x ≤ a.

Solution with 0 < a < A: y(x) =

  a d 1 w(t, a)f (t) dt w(x, a) 2M  (a) da –a   ξ 1 d 1 a d – w(x, ξ) w(t, ξ)f (t) dt dξ 2 |x| dξ M  (ξ) dξ –ξ  ξ  a w(x, ξ) 1 d – w(t, ξ) df (t) dξ, 2 dx |x| M  (ξ) –ξ

where the prime stands for the derivative with respect to the argument and      1  –1 sin 2 A cos 12 ξ M (ξ) 1  . M (ξ) = ln , w(x, ξ) = √ π 2 cos x – 2 cos ξ sin 2 ξ Reference: I. C. Gohberg and M. G. Krein (1967).

10.

  x – t y(t) dt = f (x). ln 2 sin dx –π 2 Solution:  π  1 d x – t y(x) = – 2 f (t) dt, ln 2 sin π dx –π 2

d

π



π

y(t) dt = 0. –π

Reference: A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 452).

3.6-3. Kernels Containing Combinations of Exponential and Other Elementary Functions.

b

11. a

  ln |x – t| + Ae–αx–βt y(t) dt = f (x).

12.

This is a special case of equation 3.8.28 with ϕ(x) = Ae–αx and ψ(t) = e–βt . ∞ [sin(xt) + Ae–αx–βt ]y(t) dt = f (x).

13.

This is a special case of equation 3.8.29 with ϕ(x) = Ae–αx and ψ(t) = e–βt . ∞ [cos(xt) + Ae–αx–βt ]y(t) dt = f (x).

0

0

This is a special case of equation 3.8.30 with ϕ(x) = Ae–αx and ψ(t) = e–βt .

257

258

LINEAR EQUATIONS OF THE FIRST KIND WITH CONSTANT LIMITS OF INTEGRATION

3.7. Equations Whose Kernels Contain Special Functions∗ 3.7-1. Kernels Containing Error Function, Exponential Integral or Logarithmic Integral.



1.

 exp(i(x + t)2 ) erf(eπi/4 (x + t)) + exp(i(x – t)2 ) erf(eπi/4 (x – t)) y(t) dt = f (x).

0

Here erf z is the error function (see Supplement 11.2-1) and i2 = –1. Solution:  1 ∞ y(x) = – exp(–i(t + x)2 ) erf(e3πi/4 (t + x)) + exp(–i(t – x)2 ) erf(e3πi/4 (t – x)) f (t) dt. π 0 Reference: A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 459).





2.

e–ixt Ei(ixt)y(t) dt = f (x),

i2 = –1.

0

Here Ei(z) is the exponential integral (see Supplement 11.2-2). Solution:  ∞ √ 1 1+i ixt πi/4 e erf(e y(t) = xt) – √ f (x) dx. 2π 2 –∞ 2πxt Reference: A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 456).







3.

li 1

x t

 y(t) dt = f (x),

f (1) = f  (1) = 0.

Here li(z) is the logarithmic integral (see Supplement 11.2-3). Solution:  x  t  d 2 d –2 t f (t) dt, t ν ln –t y(t) = – x dt dt 1 ∞ ξ z dξ where ν(z) = . Γ(ξ + 1) 0 References: H. M. Srivastava and R. G. Buschman (1977), A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 457).

3.7-2. Kernels Containing Sine Integrals, Cosine Integrals, or Fresnel Integrals.



si(xt)y(t) dt = f (x).

4. 0

Here si(z) is the sine integral (see Supplement 11.3-1). Solution: 2 ∞ [sin(xt) + xt cos(xt)]f (t) dt. y(x) = – π 0 References: H. M. Srivastava and R. G. Buschman (1977), A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 457). * For notation and properties of special functions, see Supplement 11.

3.7. EQUATIONS WHOSE KERNELS CONTAIN SPECIAL FUNCTIONS



259



ci(xt)y(t) dt = f (x).

5. 0

Here ci(z) is the cosine integral (see Supplement 11.3-2). Solution: 2 ∞ y(x) = [1 – cos(xt) + xt sin(xt)]f (t) dt. π 0 References: H. M. Srivastava and R. G. Buschman (1977), A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 457).







6.

1 2

0

 – S(xt) y(t) dt = f (x).

Here S(z) is the Fresnel sine integral (see Supplement 11.3-3). Solution:  ∞   2 1/2 sin(xt) + 2 cos(xt) f (t) dt. y(x) = (xt) π 0 xt References: H. M. Srivastava and R. G. Buschman (1977), A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 459).





7. 0



 1 – C(xt) y(t) dt = f (x). 2

Here C(z) is the Fresnel cosine integral (see Supplement 11.3-3). Solution:  y(x) =

2 π







(xt) 0

1/2

 cos(xt) – 1 – 2 sin(xt) f (t) dt. xt

References: H. M. Srivastava and R. G. Buschman (1977), A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 460).





S(xt, ν)y(t) dt = f (x).

8. 0

Here S(z, ν) is the generalized Fresnel sine integral (see Supplement 11.3-3). Solution: 2 ∞ (1 – ν) sin(xt) + xt cos(xt) y(x) = f (t) dt. π 0 (xt)ν References: H. M. Srivastava and R. G. Buschman (1977), A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 461).





C(xt, ν)y(t) dt = f (x).

9. 0

Here C(z, ν) is the generalized Fresnel cosine integral (see Supplement 11.3-3). Solution: 2 ∞ (1 – ν) cos(xt) – xt sin(xt) y(x) = f (t) dt. π 0 (xt)ν References: H. M. Srivastava and R. G. Buschman (1977), A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 461).

260

LINEAR EQUATIONS OF THE FIRST KIND WITH CONSTANT LIMITS OF INTEGRATION

3.7-3. Kernels Containing Gamma Functions.



10.

(xt)–(π+1)/2 Γ(±i ln(xt))y(t) dt = f (x),

i2 = –1.

0

Here Γ(z) is the incomplete gamma function (see Supplement 11.4-1). Solution: ∞ 1 (xt)–(π+1)/2 Γ(∓i ln(xt))f (t) dt. y(x) = 4π 2 0 The integral equation and its solution form a Paley–Wiener transform pair (in the asymmetric form). References: E. C. Titchmarsh (1986), A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 453).





11.

e–π(x+t)/2 Γ(±i(x + t))y(t) dt = f (x).

–∞

Solution:

1 y(x) = 4π 2





e–π(x+t)/2 Γ(∓i(x + t))f (t) dt.

–∞

Reference: A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 453).





12.

Γ(α + i(x + t))Γ(α – i(x + t))y(t) dt = f (x).

–∞

Solution: α sin(2πα) y(x) = – 2π 3





Γ(–α + i(x + t))Γ(–α – i(x + t))f (t) dt,

–∞

where Re α < 0 (2α ≠ –1, –2, . . . ). References: J. Wimp (1971), A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 453).

3.7-4. Kernels Containing Incomplete Gamma Functions.



13.

(t – x)α–1 γ(1 – α, 2i(t – x))y(t) dt = f (x),

i2 = –1.

–∞

Here γ(ν, z) is the incomplete gamma function (see Supplement 11.5-1). Solution: ∞ 1 (t – x)–α–1 γ(1 + α, 2i(t – x))f (t) dt, y(x) = – 2 4π –∞ where –1/2 < Re α ≤ 0. Reference: A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 462).





14. –∞

     2x – i 1 –ix–1/2 ix–1/2 iat exp π t – ix, iat y(t) dt = f (x). + (b – a)a e Γ 4 2

Solution: 1 y(x) = 4π



∞

–∞

    f (t) 2t + i 1 it–1/2 –it–1/2 –ibx π x + it, –ibx dt. exp + (a – b)b e Γ 4 2 cosh(πt)

Reference: A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 463).

3.7. EQUATIONS WHOSE KERNELS CONTAIN SPECIAL FUNCTIONS 261   ∞ 1 + 2ix t–ix–1/2 sin π 4 0   

  i 1 1 + (b – a)aix–1/2 e–iat Γ – ix, –iat – eiat Γ – ix, iat y(t) dt = f (x). 2 2 2

15.

Solution: 1 y(t) = π

 1 – 2ix π t sin 4 –∞  

   f (x) 1 1 i –ix–1/2 –ibt ibt + ix, –ibt – e Γ + ix, ibt dx, e Γ + (a – b)b 2 2 2 cosh(πx)







ix–1/2

where a, b ∉ (–∞, 0) are complex numbers. Reference: A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 463).

16. 0

  1 + 2ix t–ix–1/2 cos π 4   

  1 1 1 + (b – a)aix–1/2 e–iat Γ – ix, –iat + eiat Γ – ix, iat y(t) dt = f (x). 2 2 2



Solution: 1 y(t) = π



  1 – 2ix t cos π 4 –∞  

   f (x) 1 1 1 + ix, –ibt + eibt Γ + ix, ibt dx, + (a – b)b–ix–1/2 e–ibt Γ 2 2 2 cosh(πx)



ix–1/2

where a, b ∉ (–∞, 0) are complex numbers. Reference: A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 463).

3.7-5. Kernels Containing Bessel Functions of the First Kind.



tJν (xt)y(t) dt = f (x).

17. 0

Here Jν (z) is the Bessel function of the first kind (see Supplement 11.6-1). Solution: ⎧ ∞ ⎪ tJν (xt)f (t) dt if Re ν ≥ –1 or ν = –2, –3, . . . , ⎪ ⎪ ⎪ ⎨ 0   ∞ n–1  y(x) = (–1)k (xt/2)2k+ν ⎪ f (t) dt if Re ν < –1 and ν ≠ –2, –3, . . . , t J (xt) – ν ⎪ ⎪ k!Γ(ν + k + 1) ⎪ ⎩ 0 k=0 where – n – 1 < Re ν < –n, n = 1, 2, . . . The functions f (x) and y(x) are the Hankel transform pair. References: E. C. Titchmarsh (1923), J. L. Griffith (1958), V. A. Ditkin and A. P. Prudnikov (1965), F. Oberhettinger (1972), I. Sneddon (1972), H. M. Srivastava and R. G. Buschman (1977), B. Davis (1978), A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 468), S. G. Samko, A. A. Kilbas, and O. I. Marichev (1993), I. Sneddon (1995).

262

LINEAR EQUATIONS OF THE FIRST KIND WITH CONSTANT LIMITS OF INTEGRATION



b

0 ≤ x < ∞.

tJν (xt)y(t) dt = f (x),

18. a



Solution: y(t) =



xJν (xt)f (x) dx if a < t < b, 0

if 0 < t < a or t > b,

0 where 0 ≤ a ≤ b ≤ ∞ and Re ν > –1.

References: A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 468), I. N. Sneddon (1995).





a ≤ x < ∞.

tJ0 (xt)y(t) dt = 0,

19. 0

Homogeneous integral equation of the first kind. Solution: a y(t) =

cos(xt)ϕ(x) dx, 0

where ϕ(x) is an arbitrary continuously differentiable function. Reference: Ya. S. Uflyand (1977).





a ≤ x < ∞.

tJν (xt)y(t) dt = 0,

20. 0

Homogeneous integral equation of the first kind, Re ν > –1/2. Solution:  a √ πt y(t) = x Jν–1/2 (xt)ϕ(x) dx, 2 0 where ϕ(x) is an arbitrary continuously differentiable function. Reference: Ya. S. Uflyand (1977).



b

21. a

22.

Jν (λx) – Jν (λt) y(t) dt = f (x).

This is a special case of equation 3.8.3 with g(x) = Jν (λx), where Jν (z) is the Bessel function of the first kind. ∞ Jν (λ(x – t))y(t) dt = f (x). 0 ◦

1 . If | Re ν| < 1 and f (0) = f  (0) = 0 then   2 x d 2 f (t) dt. J–ν (λ(x – t)) + λ y(x) = dt2 0 2◦ . If ν = n is a positive integer number and f (0) = f  (0) = · · · = f (n+1) (0) = 0 then y(x) =

1 λn





[(n–1)/2]

Cn2k+1

k=0

d dx

n–2k–1 

1 + n λ



d2 + λ2 dx2

f (x) 



[n/2]

x

J0 (λ(x – t)) 0

k+1

k=0

Cn2k

d dt

n–2k 

where [A] stands for the integer part of the number A and Cnk = coefficients (0! = 1).

d2 + λ2 dt2

k+1 f (t) dt,

n! are binomial k! (n – k)!

263

3.7. EQUATIONS WHOSE KERNELS CONTAIN SPECIAL FUNCTIONS

3◦ . If ν is not an integer, m – 1 < Re ν < m (m = 0, 1, 2, . . . ), and f (0) = f  (0) = · · · = f (m+1) (0) = 0 then y(x) =

m–ν λm



x

0

Jm–ν (λ(x – t)) x–t +

1 λm







[(m–1)/2] 2k+1 Cm

k=0

d dt



m–2k–1  

[m/2]

x

Jm–ν (λ(x – t)) 0

2k Cm

k=0

d dt

d2 + λ2 dt2

m–2k 

k+1 f (t) dt

d2 + λ2 dt2

k+1 f (t) dt.

References: H. M. Srivastava and R. G. Buschman (1977), A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 470), S. G. Samko, A. A. Kilbas, and O. I. Marichev (1993).





23.

|x – t|ν Jν (λ|x – t|)y(t) dt = f (x).

–∞

Solution: λ cos(νπ) y(x) = – 4 sin2 (νπ)





–∞

 sign(t – x) d  ν+1 |t – x| J (λ|t – x|)f (t) dt, –ν–1 |t – x|2ν+1 dt

where 0 < Re ν < 1/2. References: A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 469), S. G. Samko, A. A. Kilbas, and O. I. Marichev (1993).





G(x, t) = 2πx(t/x)n/2,

Jn/2–1 (2πxt)G(x, t)y(t) dt = f (x),

24.

n = 1, 2, . . .

0



Solution:



Jn/2–1 (2πxt)G(x, t)f (t) dt.

y(x) = 0

The functions f (x) and y(t) are the Bochner transform pair. Reference: Yu. A. Brychkov and A. P. Prudnikov (1979).



25.

 d  2 xJν (xt) ty(t) dt = f (x). dx 0 Solution: ∞ tJν (x, t)Yν (xt)f (t) dt y(x) = –2π ∞0   2 =π t sin(2νπ)[J–ν (xt) – Yν2 (xt)] – 2 cos(2νπ)J–ν (xt)Y–ν (xt) f (t) dt. ∞

0

References: I. I. Hirschman and D. V. Widder (1955), A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 474).

26.



 t J–µ (xt)J–ν (xt) ± Jµ (xt)Jν (xt) y(t) dt = f (x).

0

Solution: y(x) =

π

π   2 cos 2 (ν ± µ) sin π2 (ν ∓ µ)





t 0

 d t Jµ (xt)J–ν (xt)∓J–µ (xt)Jν (xt) f (t) dt, dt

where Re(µ + ν) < 3/2. References: I. I. Hirschman and D. V. Widder (1955), E. C. Titchmarsh (1986), A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 475).

264

LINEAR EQUATIONS OF THE FIRST KIND WITH CONSTANT LIMITS OF INTEGRATION





i2 = –1.

[Jix (t) + J–ix (t)]y(t) dt = f (x),

27. 0

Solution: 1 y(x) = 2x





t[Jit (x) + J–it (x)] f (t) dt. sinh(πt)

0

Reference: A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 469).





i2 = –1.

[Jit (x) + J–it (x)]y(t) dt = f (x),

28. 0

Solution:



x y(x) = 2 sinh(πx)



0

Jix (t) + J–ix (t) f (t) dt. t

Reference: A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 469).

3.7-6. Kernels Containing Bessel Functions of the Second Kind.



tYν (xt)y(t) dt = f (x).

29. 0

Here Yν (z) is the Bessel function of the second kind (see Supplement 11.6-1). 1◦ . If | Re ν| < 1 then





tHν (xt)f (t) dt,

y(x) = 0

where Hν (x) is the Struve function, which is defined as Hν (x) =

∞  j=0

(–1)j (x/2)ν+2j+1   .  Γ j + 32 Γ ν + j + 32

The function f (x) and the solution y(x) are the Yν -transform pair. 2◦ . If 1 < | Re ν| < 3 then y(x) =



 t Hν (xt) –

0

 (xt)ν–1 √ f (t) dt. 2ν–1 π Γ(ν + 1/2)

References: E. C. Titchmarsh (1948), G. N. Watson (1952), J. L. Griffith (1958), F. Oberhettinger (1972), A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 475).



b

30. a

Yν (λx) – Yν (λt) y(t) dt = f (x).

This is a special case of equation 3.8.3 with g(x) = Yν (λx), where Yν (z) is the Bessel function of the second kind.

3.7. EQUATIONS WHOSE KERNELS CONTAIN SPECIAL FUNCTIONS

265

3.7-7. Kernels Containing Combinations of the Bessel Functions.



[cos(pπ)Jν (xt) + sin(pπ)Yν (xt)]ty(t) dt = f (x).

31. 0

Solution:



y(x) =

Φ(xt)tf (t) dt,

Φ(z) =

0

∞  n=0

(–1)n(z/2)ν+2p+2n . Γ(p + n + 1)Γ(ν + p + n + 1)

The functions f (x) and y(x) are the Hardy transform pair. Reference: Yu. A. Brychkov and A. P. Prudnikov (1989).





tJν (xt)Yν (xt)y(t) dt = f (x).

32. 0



Solution:



t

y(x) = 2π 0

 d 2 tJν (xt) f (t) dt, dt

where Re ν > –1/4. References: E. C. Titchmarsh (1948), I. I. Hirschman and D. V. Widder (1955), A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 476).





t[Jν (ax)Yν (xt) – Yν (ax)Jν (xt)]y(t) dt = f (x).

33. a



Solution:



y(x) = 0

t[Jν (at)Yν (xt) – Yν (at)Jν (xt)] f (t) dt. Jν2 (at) + Yν2 (at)

The function f (x) and the solution y(x) are the Weber transform pair. References: G. N. Watson (1952), Yu. A. Brychkov and A. P. Prudnikov (1979, 1989), A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 477).





t[Jν (at)Yν (xt) – Yν (at)Jν (xt)]y(t) dt = f (x).

34. 0

Solution: x y(x) = 2 Jν (ax) + Yν2 (ax)





t[Jν (ax)Yν (xt) – Yν (ax)Jν (xt)]f (t) dt. 0

References: G. N. Watson (1952), A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 477).





35. –∞

(1) e±π(x–t)/2 Hi(t–x) (a)y(t) dt = f (x),

i2 = –1.

Here Hν(1) (z) = Jν (z) + iYν (z) is the Hankel function of the first kind (see Supplement 11.6-5). Solution: 1 ∞ ±π(t–x)/2 (1) y(x) = e Hi(t–x) (a)f (t) dt, 4 –∞ where a > 0. References: Vu Kim Tuan (1988), A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 479).

266

LINEAR EQUATIONS OF THE FIRST KIND WITH CONSTANT LIMITS OF INTEGRATION





36. –∞

(2) e±π(x–t)/2 Hi(t–x) (a)y(t) dt = f (x).

Here Hν(2) (z) = Jν (z) – iYν (z) is the Hankel function of the second kind (see Supplement 11.6-5). Solution: 1 ∞ ±π(t–x)/2 (2) y(x) = e Hi(t–x) (a)f (t) dt, 4 –∞ where a > 0. References: Vu Kim Tuan (1988), A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 479).

3.7-8. Kernels Containing Modified Bessel Functions of the First Kind.

b

37. a

Iν (λx) – Iν (λt) y(t) dt = f (x).

This is a special case of equation 3.8.3 with g(x) = Iν (λx), where Iν (z) is the modified Bessel function of the first kind (see Supplement 11.7-1). 38.



d

dx Solution: 0

2 Iit (x)y(t) dt = f (x),

y(x) =

i2 = –1. 2i x π





2 Kix (t)f (t) dt.

0

Reference: A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 485).





Ai(x + t)y(t) dt = f (x).

39. –∞

Here Ai(x) = Solution:

1√ 3 x

 I–1/3 (z) – I1/3 (z) is the Airy function (see Supplement 11.8-1). ∞ y(x) = Ai(x + t)f (t) dt. –∞

References: Vu Kim Tuan (1988), A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 485).

3.7-9. Kernels Containing Modified Bessel Functions of the Second Kind.



40.

  K0 |x – t| y(t) dt = f (x).

–∞

Here K0 (z) is the modified Bessel function of the second kind (the MacDonald function), see Supplement 11.7-1. Solution:  2  ∞   1 d y(x) = – 2 – 1 K0 |x – t| f (t) dt. 2 π dx –∞ Reference: D. Naylor (1986).



b

41. a

Kν (λx) – Kν (λt) y(t) dt = f (x).

This is a special case of equation 3.8.3 with g(x) = Kν (λx).

3.7. EQUATIONS WHOSE KERNELS CONTAIN SPECIAL FUNCTIONS





42.



267

zt Kν (zt)y(t) dt = f (z).

0

Here Kν (z) is the modified Bessel function of the second kind. Up to a constant factor, the left-hand side of this equation is the Meijer transform of y(t) (z is treated as a complex variable). Solution: c+i∞ √ 1 y(t) = zt Iν (zt)f (z) dz. πi c–i∞ For specific f (z), one may use tables of Meijer integral transforms to calculate the integral. Reference: V. A. Ditkin and A. P. Prudnikov (1965).





i2 = –1.

Kix (t)y(t) dt = f (x),

43. 0

Solution: y(x) =

2 π2 x





t sinh(πt)Kit (x)f (t) dt. 0

The function f (x) and the solution y(x) are the Kontorovich-Lebedev transform pair. References: V. A. Ditkin and A. P. Prudnikov (1965), F. Oberhettinger (1972), Yu. A. Brychkov and A. P. Prudnikov (1989), A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 487).





Kit (x)y(t) dt = f (x).

44. 0

Solution: y(x) =

2x sinh(πx) π2

0



Kix (t) f (t) dt. t

Reference: A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 487).





45. 0

2 Kit (x)y(t) dt = f (x).

Solution:



4x sinh(πx) y(x) = π2



0

 d [Iix (t) + I–ix (t)]Kix (t) f (t) dt. dt

Reference: A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 492).





Re Kix+1/2 (t)y(t) dt = f (x).

46. 0

Solution: y(x) =

4 π2





cosh(πt) Re Kit+1/2 (x)f (t) dt. 0

Reference: A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 488).





Im Kix+1/2 (t)y(t) dt = f (x).

47. 0

Solution: y(x) =

4 π2





cosh(πt) Im Kit+1/2 (x)f (t) dt. 0

Reference: A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 488).

268

LINEAR EQUATIONS OF THE FIRST KIND WITH CONSTANT LIMITS OF INTEGRATION





Re Kit+1/2 (x)y(t) dt = f (x).

48. 0

Solution: y(x) =

4 cosh(πx) π2





Re Kix+1/2 (t)f (t) dt. 0

Reference: A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 488).





Im Kit+1/2 (x)y(t) dt = f (x).

49. 0

Solution:

4 y(x) = 2 cosh(πx) π





Im Kix+1/2 (t)f (t) dt. 0

Reference: A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 488).





50.

eπ(x+t)/2 Ki(x+t) (a)y(t) dt = f (x).

–∞

Solution: y(x) =

1 π2





eπ(x+t)/2 Ki(x+t) (a)f (t) dt,

–∞

where a > 0. The function f (x) and the solution y(x) are a Crum transform pair (in the asymmetric form). Reference: A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 488).





Ki(x+t) (±ia)y(t) dt = f (x).

51. –∞

Solution:

1 y(x) = 2 π





Ki(x+t) (∓ia)f (t) dt. –∞

Reference: A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, pp. 488–489).





52. –∞

√   1 t– 4 (2ix+1) K 12 +ix 2iλ t y(t) dt = f (x).

Solution: y(x) =

λ π2





 √  1 x 4 (2it–1) K 1 –it 2iλ x f (t) dt,

–∞

2

√ √ where λ > 0 and x = –i |x| for x < 0. Reference: A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 489).

53. 0



 √ 1 (a + t)– 4 (2ix+1) K 21 +ix (2iλ a + t)

 √ 1 + (a – t)– 4 (2ix+1) K 21 +ix (2iλ a – t) y(t) dt = f (x).

Solution: y(t) =

λ π2



∞

–∞

 √ √ 1 1 (a + t) 4 (2ix–1) K 1 –ix (–2iλ a + t) + (a – t) 4 (2ix–1) K 1 –ix (–2iλ a – t) f (x) dx. 2

2

Reference: A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 489).

269

3.7. EQUATIONS WHOSE KERNELS CONTAIN SPECIAL FUNCTIONS





54. –∞

 √  1 x 4 (2it–1) K 21 –it 2iλ x y(t) dt = f (x),

Solution:

λ y(x) = 2 π



λ > 0.

 √  1 t– 4 (2ix+1) K 1 +ix 2iλ t f (t) dt.



2

–∞

Reference: A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 489).





55. –∞

 √ 1 (a + t) 4 (2it–1) K 12 –it (–2iλ a + x)

 √ 1 + (a – t) 4 (2it–1) K 12 –it (–2iλ a – x) y(t) dt = f (x).

Solution: ∞  √ √ 1 1 λ (a+x)– 4 (2it+1) K 1 +it (2iλ a + x)+(a–x)– 4 (2it+1) K 1 +it (2iλ a – x) f (x) dx. y(t) = 2 2 2 π 0 Reference: A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 490).





56.

exp –∞

 πx 2

 sign t Kix (|t|)y(t) dt = f (x).

Solution: y(x) =

1 π2 x





–∞

  πt sign x Kit (|x|)f (t) dt. t exp 2

Reference: A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 490).

3.7-10. Kernels Containing a Combination of Bessel and Modified Bessel Functions.



[Iix (t) + I–ix (t)]Kix (t)y(t) dt = f (x).

57. 0

4 d ∞ 2 t sinh(πt)Kit (x)f (t) dt. π 2 dx 0 The integral equation and its solution form the Lebedev transform pair.

Solution:

y(x) = –

Reference: A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 493).





[Kit (a)Iit (x) – Iit (a)Kit (x)]y(t) dt = f (x),

58.

0 < x < a.

0

Solution: 2t sinh(πt) y(t) = 2 π |Iia (a)|2



a

x–1 [Kit (a)Iit (x) – Iit (a)Kit (x)]f (x) dx,

t > 0.

0

Reference: A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 494).

  2 t Y0 (xt) – K0 (xt) y(t) dt = f (x). π 0 Solution: ∞ 

59.



 2 y(x) = t Y0 (xt) – K0 (xt) f (t) dt. π 0 The integral equation and its solution form the divisor transform pair.

References: F. Oberhettinger (1973), E. C. Titchmarsh (1986), A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 492).

270

LINEAR EQUATIONS OF THE FIRST KIND WITH CONSTANT LIMITS OF INTEGRATION

 ∞  2 t Y2n+1 (xt) ± K2n+1 (xt) y(t) dt = f (x), π 0

60.

Solution:





y(x) = 0

n = 1, 2, . . .



 2 t Y2n+1 (xt) ∓ K2n+1 (xt) f (t) dt. π

References: E. C. Titchmarsh (1986), A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 493).





61. 0

  2 t Y2n (xt) + K2n (xt) y(t) dt = f (x), π

Solution:





y(x) = 0

n = 1, 2, . . .



 2 t Y2n (xt) + K2n (xt) f (t) dt. π

References: E. C. Titchmarsh (1986), A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 493).

3.7-11. Kernels Containing Legendre Functions.



0 ≤ x < ∞.

P– 12 +ix (t)y(t) dt = f (x),

62. 1

Here Pν (x) is the Legendre function of the first kind (see Supplement 11.11-3) and i2 = –1. Solution: ∞ x tanh(πx)Pix–1/2 (t)f (x) dx.

y(t) = 0

The functions f (x) and y(t) are the Mehler–Fock transform pair. Remark. The Legendre function of the first kind can be represented in the form

P– 1 +ix (t) = 2

2 cosh(πx) π





cos(xs) ds √ , 2(t + cosh s)

0

1 ≤ t < ∞.

References: N. N. Lebedev (1965), V. A. Ditkin and A. P. Prudnikov (1965), Yu. A. Brychkov and A. P. Prudnikov (1989), A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 512).





P– 12 +it (x)y(t) dt = f (x),

63. 0

1 ≤ x < ∞.

Solution:



P– 1 +it (x)f (x) dx.

y(t) = t tanh(πt) 1

2

References: N. N. Lebedev (1965), V. A. Ditkin and A. P. Prudnikov (1965), Yu. A. Brychkov and A. P. Prudnikov (1989), A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 513).

64. 0



[P– 12 +ix (it) ± P– 12 +ix (–it)]y(t) dt = f (x).

Solution:

1 y(t) = 2

0



sinh(πx) [P– 21 +ix (–it) ± P– 12 +ix (it)]f (x) dx. cosh2 (πx)

Reference: A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 513).

3.7. EQUATIONS WHOSE KERNELS CONTAIN SPECIAL FUNCTIONS





65. 0

271

[P– 12 +it (ix) ± P– 12 +it (–ix)]y(t) dt = f (x).

Solution: y(t) =

t sinh(πt) 2 cosh2 (πt)

0



[P– 12 +it (–ix) ± P– 12 +it (ix)]f (x) dx.

Reference: A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 514).





66. –∞

[ie–iπx P– 12 +x (cos t) + P– 12 +x (– cos t)]y(t) dt = f (x).

Solution: 1 y(t) = sin t 2





–∞

x [ieiπx P– 1 +x (cos t) + P– 1 +x (– cos t)]f (x) dx. 2 2 sinh(2πx)

Reference: A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 513).





67. 0

[P– 12 +it (x)]2 y(t) dt = f (x),

1 ≤ x < ∞.

Solution:



y(t) = t tanh(πt) 1

  d 2 (x – 1)1/2 f (x) dx, P– 1 +it (x) Q– 1 +it (x) + Q– 1 –it (x) (x2 – 1)1/2 2 2 2 dx

where Qν (x) is the Legendre function of the second kind. Reference: A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 514).





68. 1

 P– 12 +ix (t) Q– 12 +ix (t) + Q– 12 –ix (t) y(t) dt = f (x),

0 ≤ x < ∞.

Here Qν (x) is the Legendre function of the second kind. Solution:   ∞ d y(t) = (t2 – 1)1/2 (t2 – 1)1/2 x tanh(πx)[P– 12 +ix (t)]2 f (x) dx . dt 0 Reference: A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 519).

3.7-12. Kernels Containing Associated Legendre Functions. 69. 1



P–µ1 +ix (t)y(t) dt = f (x), 2

0 ≤ x < ∞.

Here Pνµ (x) is the associated Legendre function of the first kind (see Supplement 11.11-3) and i2 = –1. Solution:     µ 1 ∞ y(t) = x sinh(πx)Γ 12 – µ + ix Γ 12 – µ – ix Pix–1/2 (t)f (x) dx. π 0 The functions f (x) and y(t) are the generalized Mehler–Fock transform pair. Reference: A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 518).

272

LINEAR EQUATIONS OF THE FIRST KIND WITH CONSTANT LIMITS OF INTEGRATION





70.

P–µ1 +it (x)y(t) dt = f (x),

1 ≤ x < ∞.

2

0

Solution:     1 y(t) = t sinh(πt)Γ 12 – µ + it Γ 12 – µ – it π

1



µ Pit–1/2 (x)f (x) dx.

Reference: A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 519).



1

71. –1

P–ix1 +ia (±t)y(t) dt = f (x),

–∞ < x < ∞.

2

Solution: 1 y(t) = 2πi(1 – t)







1 2

–∞

   + ia – ix Γ 12 – ia – ix P–ix1 +ia (∓t)f (x) dx. 2

Reference: A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 518).

72.



 x+t–1 y(t) dt = f (x), Re ν > –1. √ 2 2 xt 0 Here Qµν (x) is the associated Legendre function of the second kind (see Supplement 11.11-3). Solution:   ∞

–1/2 1 1 x+t–1 –1/2 2 √ y(t) = f (x) dx. (x + t – 1) (xt) – 4xt Q 1 ν– 2 4π 2 0 2 xt



–1/2 1 (x + t – 1) – 4xt Qν– 1 2

Reference: A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 520).

3.7-13. Kernels Containing Kummer Confluent Hypergeometric Functions.



F (a, b; ixt)y(t) dt = f (x).

73. 0

Here F (a, b; x) is the Kummer confluent hypergeometric function (see Supplement 11.9-1) and i2 = –1. Let Re(b – a) < n < Re b – 1/2. Then the solution is  n   ∞ Γ(a) b–1 d n–b+1 –ixt y(t) = t t e Ψ(n + a – b, n – b + 2; ixt)f (x) dx . 2πΓ(b) dt –∞ Reference: A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 530).





F

74.

1 2

0

 b ± ix, b; –it y(t) dt = f (x).

Solution: y(t) =

tb–1 2πΓ2 (b)





–∞

e∓πx Γ

1

2b

     + ix Γ 12 b – ix F 12 b ∓ ix, b; it f (x) dx,

where Re b > 0. Reference: A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 531).

3.7. EQUATIONS WHOSE KERNELS CONTAIN SPECIAL FUNCTIONS





75.

tix F

1

 + ix, b + ix; iαt y(t) dt = f (x).

2

0

273

Solution: y(t) =

   n ∞  Γ 1 + ix  d tb–1  Ψ n – b + 12 , n – b + 1 – ix; iαt f (x) dx, – tn–b–ix e–iαt  2 2π dt Γ b + ix –∞

where Im α = 0 and 0 < Re b – 1/2 < n < Re b. Reference: A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 531).





76.

  F a, b; iβ(x – t) y(t) dt = f (x).

–∞

Solution: β 2 (a – 1)(a – b + 1) sin(πb) y(t) = 4π(b – 1)(b – 2)(b – 3) sin(πa) sin[π(b – a)]





  F 2 – a, b – a – 1; iβ(x – t) f (x) dx,

–∞

where 1 < Re a < 3/2 and –1 < Re(b – a) < –1/2. Reference: A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 531).





F

77.

1 2

–∞

 ± ia, 12 ; ±i(x – t)2 y(t) dt = f (x),

Solution: y(t) =

eπa π cosh(πa)





F

 ∓ ia, 12 ; ∓i(x – t)2 f (x) dx.

1

–∞

a > 0.

2

References: Vu Kim Tuan (1988), A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 532).





F

78.

1 2

–∞

 b ± it, b; ix y(t) dt = f (x),

Re b > 0.

Solution:    e±πt  1 Γ 2 b + it Γ 12 b – it 2 2πΓ (b)

y(t) =





xb–1 F

1

0

2b

 ∓ it, b; –ix f (x) dx.

Reference: A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 532).





F

79.

1

–∞

2

 b ± it, b; –ix y(t) dt = f (x),

Re b > 0.

Solution: y(t) =

   e∓πt  1 Γ 2 b + it Γ 12 b – it 2 2πΓ (b)

0



xb–1 F

1

2b

 ∓ it, b; ix f (x) dx.

Reference: A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 532).





80. –∞

x–it F

1 2

 – it, b – it; iβx y(t) dt = f (x).

Solution:   ∞  Γ 12 (1 – it)   xn–b+it e–iβx Ψ n + y(t) = 1 2πΓ b – 2 it 0

1 2

   d n b–1  x f (x) dx. – b, n + 1 – b + it; iβx dx

Reference: A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 533).

274

LINEAR EQUATIONS OF THE FIRST KIND WITH CONSTANT LIMITS OF INTEGRATION

3.7-14. Kernels Containing Tricomi Confluent Hypergeometric Functions.



81.

tix Ψ(a + ix, 2ix + 1; t)y(t) dt = f (x).

0

Here Ψ(a, b; x) is the Tricomi confluent hypergeometric function (see Supplement 11.9-1) and i2 = –1. Solution: e–t ∞ y(t) = 2 x sinh(2πx)Γ(a – ix)Γ(a + ix)tix Ψ(a + ix, 2ix + 1; t)f (x) dx. π t 0 References: J. Wimp (1971), A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 534).





82.

xit Ψ(a + it, 2ix + 1; t)y(t) dt = f (x).

0

Solution: t y(t) = 2 sinh(2πt)Γ(a – it)Γ(a + it) π





x–1+it e–x Ψ(a + it, 2it + 1; x)f (x) dx.

0

References: J. Wimp (1971), A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 535).





83. –∞

 Ψ 12 + ix,

Solution:

3 2

 – iβ + ix; ±it y(t) dt = f (x),

1 y(t) = 4π





–∞

 1 Ψ 12 – ix, cosh(πx)

3 2

Im β = 0.  + iβ – ix; ∓it f (x) dx.

Reference: A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 536).

3.7-15. Kernels Containing Whittaker Confluent Hypergeometric Functions.

84.



1 Re ν > – . 2 0 Here Mµ,ν (z) is the Whittaker confluent hypergeometric function (see Supplement 11.9-3) and i2 = –1. Solution: ∞     1 y(t) = e∓πx Γ 12 + ν + ix Γ 12 + ν – ix M±ix,ν (–it)f (x) dx. 2 2πΓ (2ν + 1) t –∞ M±ix,ν (it)y(t) dt = f (x),

The integral equation and its solution form the Buchholz transform pair. References: H. Buchholz (1969), A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 523).





M±ix,ν (–it)y(t) dt = f (x),

85. 0

Solution: y(t) =

1 2πΓ2 (2ν + 1) t





–∞

1 Re ν > – . 2

e±πx Γ

1 2

   + ν + ix Γ 12 + ν – ix M∓ix,ν (it)f (x) dx.

Reference: A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, pp. 523–524).

3.7. EQUATIONS WHOSE KERNELS CONTAIN SPECIAL FUNCTIONS





M±it,ν (ix)y(t) dt = f (x),

86. –∞

1 Re ν > – . 2

Solution: y(t) =

275

    e∓πt Γ 12 + ν + it Γ 12 + ν – it 2 2πΓ (2ν + 1)





x–1 M∓it,ν (–ix)f (x) dx.

0

Reference: A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 524).





M±it,ν (–ix)y(t) dt = f (x),

87. –∞

1 Re ν > – . 2

Solution: y(t) =

    e±πt Γ 12 + ν + it Γ 12 + ν – it 2 2πΓ (2ν + 1)





x–1 M∓it,ν (ix)f (x) dx.

0

Reference: A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, pp. 524–525).





88. –∞

    Γ 12 + ν + ix – it Γ 12 + ν – ix + it Mit–ix,ν (a)y(t) dt = f (x).

Solution:    (2ν + 1) sin(2πν) ∞  1 y(t) = Γ – 2 – ν + ix – it Γ – 12 – ν – ix + it Mit–ix,–ν–1 (a)f (x) dx. 3 4π –∞ References: J. Wimp (1971), A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 526).





Wµ,ix (t)y(t) dt = f (x).

89. 0

Here Wµ,ν (z) is the Whittaker confluent hypergeometric function (see Supplement 11.9-3). Solution: ∞     1 y(t) = 2 2 x sinh(2πx)Γ 12 – µ – ix Γ 12 – µ + ix Wµ,ix (t)f (x) dx. π t 0 References: J. Wimp (1971), A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 527).





Wµ,it (x)y(t) dt = f (x).

90. 0

Solution: y(t) =

    t sinh(2πt)Γ 12 – µ – it Γ 12 – µ + it 2 π





x–2 Wµ,it (x)f (x) dx.

0

References: J. Wimp (1971), A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 527).





91.

e–ixt/2 Wµ,ν (ixt)y(t) dt = f (x).

–∞

Solution:

  Γ 32 – µ – ν (it)–n/2–1 y(t) = 2πΓ(1 + n – 2ν)  n ∞

ν–1/2  d (n–1)/2–ν ixt/2 x × x e Wµ+n/2–1,n/2–ν (ixt) f (x) dx, dx 0

where Re µ < Re ν + 1/2 < 3/4. Reference: A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 528).

276

LINEAR EQUATIONS OF THE FIRST KIND WITH CONSTANT LIMITS OF INTEGRATION

3.7-16. Kernels Containing Gauss Hypergeometric Functions. 92.



a

F

β

β+1

, µ;

4x2 t2



y(t) dt

= f (x). 2 2 + (x2 + t2 )β Here 0 < a ≤ ∞, 0 < β < µ < β + 1, and F (a, b, c; z) is the Gauss hypergeometric function (see Supplement 11.10-1). ,

0

(x2

t2 )2

1◦ . Solution:

a x2µ–2 tg(t) dt d , 2 Γ(1 + β – µ) dx x (t – x2 )µ–β t 2µ–1 2 Γ(β) sin[(β – µ)π] 1–2β d s f (s) ds t . g(t) = 2 πΓ(µ) dt 0 (t – s 2 )µ–β 2◦ . If a = ∞ and f (x) is a differentiable function, then the solution can be represented in the form   ∞ d (xt)2µ ft (t) 1–β 4x2 t2 β y(x) = A , µ + 1; 2 2 2 dt, F µ– , µ+ dt 0 (x2 + t2 )2µ–β 2 2 (x + t ) y(x) =

where A =

Γ(β) Γ(2µ – β) sin[(β – µ)π] . πΓ(µ) Γ(1 + µ)

Reference: P. P. Zabreyko, A. I. Koshelev, et al. (1975).





F (a + ix, a – ix, c; –t)y(t) dt = f (x),

93.

a, c > 0.

0

Solution: y(t) =

tc–1 (1 + t)2a–c π 2 Γ2 (c)





x sinh(2πx)|Γ(a + ix)Γ(c – a + ix)|2 F (a + ix, a – ix, c; –t)f (x) dx.

0

The integral equation and its solution form the Olevskii transform pair. Reference: A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 538).

3.7-17. Kernels Containing Parabolic Cylinder Functions.



94.

D–ix–1/2 (±e–πi/4 t)y(t) dt = f (x),

i2 = –1.

–∞

Here Dν (z) is the parabolic cylinder function (see Supplement 11.12-1). Solution: ∞ –πt/2 1 e y(x) = Dit–1/2 (±eπi/4 x)f (t) dt. 4π –∞ cosh(πt) Reference: A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 467).

   i(x – t)2 D±iα (e∓πi/4 (t – x)) – D±iα (e∓πi/4 (x – t)) y(t) dt = f (x). exp ± 4 –∞ Solution:   ∞ eπα/2 i(x – t)2 y(x) = D∓iα (e±πi/4 (t – x)) exp ∓ 4 8π cosh2 (πα/2) –∞  + D∓iα (e±πi/4 (x – t)) f (t) dt,

95.



where α > 0. Reference: A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 466).

3.7. EQUATIONS WHOSE KERNELS CONTAIN SPECIAL FUNCTIONS

96.

277

  ∞  i(x + t)2 exp D2iα (e3πi/4 (x + t)) – D2iα (e–πi/4 (x + t)) 4 0 

  i(x – t)2 3πi/4 –πi/4 D2iα (e (x – t)) – D2iα (e (x – t)) y(t) dt = f (x). + exp 4 Solution:  ∞   eπα i(x + t)2 y(x) = D–2iα (–eπi/4 (x + t)) – D–2iα (eπi/4 (x + t)) exp – 2 4 8π sinh (πα) 0 

  i(t – x)2 πi/4 πi/4 D–2iα (–e (t – x)) – D–2iα (e (t – x)) f (t) dt. + exp – 4 Reference: A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, pp. 465–466).

3.7-18. Kernels Containing Other Special Functions.

a

K

97. 0

 √  2 xt y(t) dt x+t

x+t

= f (x).

1

dt  is the complete elliptic integral of the first kind (see 0 (1 – t2 )(1 – z 2 t2 ) Supplement 11.13-1). Solution: a t 4 d tF (t) dt sf (s) ds d √ √ y(x) = – 2 , F (t) = . π dx x dt 0 t2 – x2 t2 – s 2

Here K(z) =

Reference: P. P. Zabreyko, A. I. Koshelev, et al. (1975).





98. 0

     1 1 1 ζ + ix, it – ζ + ix, + it y(t) dt = f (x). 2 2 2

Here ζ(z, v) =

∞  k=0

1 is the generalized Riemann zeta function (Re z > 1; v ≠ (v + k)z

0, –1, –2, . . . ). Solution: y(t) =

eπi/4 √ 4π t



∞ –∞

  ix–1/2  eπx/2 i 1+ 1+ tix f (x) dx. cosh(πx) 2t

Reference: A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 454).

99. 0



     1 1 – it 1 1 + it (1 + 2ix)π –ix–1/2 –ix–3/2 πx t ζ +2 + ix, –ζ + ix, sin e 4 2 2 2 2    

1 it 1 it –ζ + ix, – +ζ + ix, y(t) dt = f (x). 2 2 2 2

Here ζ(z, v) is the generalized Riemann zeta function (see Eq. 3.7.98). Solution:   

1 ∞ ix–1/2 (1 – 2ix)π 1 f (x) + sin – ix arctan t (t2 + 1)ix/2–1/4 dx. t y(t) = sin π –∞ 4 2 cosh(πx) Reference: A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 454).

278

LINEAR EQUATIONS OF THE FIRST KIND WITH CONSTANT LIMITS OF INTEGRATION

     ∞ 1 1 – it 1 1 + it (1 + 2ix)π –ix–1/2 –ix–3/2 πx t ζ – i2 + ix, +ζ + ix, cos e 4 2 2 2 2 0    

1 it 1 it –ζ + ix, – –ζ + ix, y(t) dt = f (x). 2 2 2 2

100.

Here ζ(z, v) is the generalized Riemann zeta function (see Eq. 3.7.98). Solution: y(t) =

1 π





tix–1/2 cos

–∞

  

1 f (x) (1 – 2ix)π +cos –ix arctan t (t2 +1)ix/2–1/4 dx. 4 2 cosh(πx)

Reference: A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992, p. 455).

3.8. Equations Whose Kernels Contain Arbitrary Functions 3.8-1. Equations with Degenerate Kernel.

b

1. a

 g1 (x)h1 (t) + g2 (x)h2 (t) y(t) dt = f (x).

This integral equation has solutions only if its right-hand side is representable in the form f (x) = A1 g1 (x) + A2 g2 (x),

A1 = const, A2 = const .

(1)

In this case, any function y = y(x) satisfying the normalization type conditions



b

h1 (t)y(t) dt = A1 ,

b

h2 (t)y(t) dt = A2

a

(2)

a

is a solution of the integral equation. Otherwise, the equation has no solutions.

b

2. a

 n

 gk (x)hk (t) y(t) dt = f (x).

k=0

This integral equation has solutions only if its right-hand side is representable in the form f (x) =

n 

Ak gk (x),

(1)

k=0

where the Ak are some constants. In this case, any function y = y(x) satisfying the normalization type conditions

b

hk (t)y(t) dt = Ak

(k = 1, . . . , n)

a

is a solution of the integral equation. Otherwise, the equation has no solutions.

(2)

3.8. EQUATIONS WHOSE KERNELS CONTAIN ARBITRARY FUNCTIONS

279

3.8-2. Equations Containing Modulus.

b

3. a

|g(x) – g(t)| y(t) dt = f (x).

Let a ≤ x ≤ b and a ≤ t ≤ b; it is assumed in items 1◦ and 2◦ that 0 < gx (x) < ∞. 1◦ . Let us remove the modulus in the integrand:

x

 g(x) – g(t) y(t) dt +



a

b

 g(t) – g(x) y(t) dt = f (x).

(1)

x

Differentiating (1) with respect to x yields gx (x)



x

y(t) dt – gx (x)

a



b

y(t) dt = fx (x).

(2)

x

Divide both sides of (2) by gx (x) and differentiate the resulting equation to obtain the solution   1 d fx (x) . (3) y(x) = 2 dx gx (x) 2◦ . Let us demonstrate that the right-hand side f (x) of the integral equation must satisfy certain relations. By setting x = a and x = b, in (1), we obtain two corollaries

b

 g(t) – g(a) y(t) dt = f (a),

a



b

 g(b) – g(t) y(t) dt = f (b).

(4)

a

Substitute y(x) of (3) into (4). Integrating by parts yields the desired constraints for f (x):

 fx (b) = f (a) + f (b), gx (b)

 f  (a) g(a) – g(b) x = f (a) + f (b). gx (a) g(b) – g(a)

(5)

Let us point out a useful property of these constraints: fx (b)gx (a) + fx (a)gx (b) = 0. Conditions (5) make it possible to find the admissible general form of the right-hand side of the integral equation: f (x) = F (x) + Ax + B, (6) where F (x) is an arbitrary bounded twice differentiable function (with bounded first derivative), and the coefficients A and B are given by gx (a)Fx (b) + gx (b)Fx (a) , gx (a) + gx (b)

 g(b) – g(a)  B = – 12 A(a + b) – 12 F (a) + F (b) – A + Fx (a) . 2gx (a) A=–

3◦ . If g(x) is representable in the form g(x) = O(x – a)k with 0 < k < 1 in the vicinity of the point x = a (in particular, the derivative gx is unbounded as x → a), then the solution of the integral equation is given by formula (3) as well. In this case, the right-hand side of the integral equation must satisfy the conditions f (a) + f (b) = 0,

fx (b) = 0.

(7)

280

LINEAR EQUATIONS OF THE FIRST KIND WITH CONSTANT LIMITS OF INTEGRATION

As before, the right-hand side of the integral equation is given by (6), with

 B = 12 (a + b)Fx (b) – F (a) – F (b) . A = –Fx (b), 4◦ . For gx (a) = 0, the right-hand side of the integral equation must satisfy the conditions



 g(b) – g(a) fx (b) = f (a) + f (b) gx (b). fx (a) = 0, As before, the right-hand side of the integral equation is given by (6), with A = –Fx (a), 4.

a

B=

1 2

 g(b) – g(a)   Fx (b) – Fx (a) . (a + b)Fx (a) – F (a) – F (b) +  2gx(b)

g(x) – g(λt) y(t) dt = f (x),

λ > 0.

0

Assume that 0 ≤ x ≤ a, 0 ≤ t ≤ a, and 0 < gx (x) < ∞. 1◦ . Let us remove the modulus in the integrand: x/λ a



 g(x) – g(λt) y(t) dt + g(λt) – g(x) y(t) dt = f (x).

(1)

x/λ

0

Differentiating (1) with respect to x yields x/λ gx (x) y(t) dt – gx (x) 0

a

y(t) dt = fx (x).

(2)

x/λ

Let us divide both sides of (2) by gx (x) and differentiate the resulting equation to obtain

 y(x/λ) = 12 λ fx (x)/gx (x) x . Substituting x by λx yields the solution   λ d fz (z) , z = λx. (3) y(x) = 2 dz gz (z) 2◦ . Let us demonstrate that the right-hand side f (x) of the integral equation must satisfy certain relations. By setting x = 0 in (1) and (2), we obtain two corollaries a a

 g(λt) – g(0) y(t) dt = f (0), gx (0) y(t) dt = –fx (0). (4) 0

0

Substitute y(x) of (3) into (4). Integrating by parts yields the desired constraints for f (x): fx (0)gx (λa) + fx (λa)gx (0) = 0,

 f  (λa) = f (0) + f (λa). g(λa) – g(0) x gx (λa)

(5)

Conditions (5) make it possible to find the admissible general form of the right-hand side of the integral equation: f (x) = F (x) + Ax + B, (6) where F (x) is an arbitrary bounded twice differentiable function (with bounded first derivative), and the coefficients A and B are given by gx (0)Fx (λa) + gx (λa)Fx (0) , gx (0) + gx (λa)

 g(λa) – g(0)  B = – 12 Aaλ – 12 F (0) + F (λa) – A + Fx (0) .  2gx (0) A=–

3.8. EQUATIONS WHOSE KERNELS CONTAIN ARBITRARY FUNCTIONS

281

3◦ . If g(x) is representable in the form g(x) = O(x)k with 0 < k < 1 in the vicinity of the point x = 0 (in particular, the derivative gx is unbounded as x → 0), then the solution of the integral equation is given by formula (3) as well. In this case, the right-hand side of the integral equation must satisfy the conditions fx (λa) = 0.

f (0) + f (λa) = 0,

(7)

As before, the right-hand side of the integral equation is given by (6), with A = –Fx (λa), 5.

a

B=

1 2

 aλFx (λa) – F (0) – F (λa) .

g(x) – t y(t) dt = f (x).

0

Assume that 0 ≤ x ≤ a, 0 ≤ t ≤ a; g(0) = 0, and 0 < gx (x) < ∞. 1◦ . Let us remove the modulus in the integrand:

g(x)

 g(x) – t y(t) dt +



a

 t – g(x) y(t) dt = f (x).

(1)

g(x)

0

Differentiating (1) with respect to x yields gx (x)



g(x)

y(t) dt – gx (x)



a

y(t) dt = fx (x).

(2)

g(x)

0

Let us divide both sides of (2) by gx (x) and differentiate the resulting equation to obtain    2gx (x)y g(x) = fx (x)/gx (x) x . Hence, we find the solution:   d fz (z) 1 , y(x) =  2gz (z) dz gz (z)

z = g –1 (x),

(3)

where g –1 is the inverse of g. 2◦ . Let us demonstrate that the right-hand side f (x) of the integral equation must satisfy certain relations. By setting x = 0 in (1) and (2), we obtain two corollaries

a

ty(t) dt = f (0),

gx (0)



0

a

y(t) dt = –fx (0).

(4)

0

Substitute y(x) of (3) into (4). Integrating by parts yields the desired constraints for f (x): fx (0)gx (xa ) + fx (xa )gx (0) = 0, g(xa )

xa = g –1 (a);

fx (xa ) = f (0) + f (xa ). gx (xa )

(5)

Conditions (5) make it possible to find the admissible general form of the right-hand side of the integral equation in question: f (x) = F (x) + Ax + B,

(6)

282

LINEAR EQUATIONS OF THE FIRST KIND WITH CONSTANT LIMITS OF INTEGRATION

where F (x) is an arbitrary bounded twice differentiable function (with bounded first derivative), and the coefficients A and B are given by gx (0)Fx (xa ) + gx (xa )Fx (0) , xa = g –1 (a), gx (0) + gx (xa )

 g(xa )  A + Fx (0) . B = – 12 Axa – 12 F (0) + F (xa ) –  2gx(0) A=–

3◦ . If g(x) is representable in the vicinity of the point x = 0 in the form g(x) = O(x)k with 0 < k < 1 (i.e., the derivative gx is unbounded as x → 0), then the solution of the integral equation is given by formula (3) as well. In this case, the right-hand side of the integral equation must satisfy the conditions fx (xa ) = 0.

f (0) + f (xa ) = 0,

(7)

As before, the right-hand side of the integral equation is given by (6), with

 B = 12 xa Fx (xa ) – F (0) – F (xa ) . A = –Fx (xa ), 6.

a

x – g(t) y(t) dt = f (x).

0

Assume that 0 ≤ x ≤ a, 0 ≤ t ≤ a; g(0) = 0, and 0 < gx (x) < ∞. 1◦ . Let us remove the modulus in the integrand:

g–1 (x)

 x – g(t) y(t) dt +



 g(t) – x y(t) dt = f (x),

a

(1)

g–1 (x)

0

where g –1 is the inverse of g. Differentiating (1) with respect to x yields



g–1 (x)

a

y(t) dt – g–1 (x)

0

y(t) dt = fx (x).

(2)

   (x). Hence, we obtain the Differentiating the resulting equation yields 2y g –1 (x) = gx (x)fxx solution  y(x) = 12 gz (z)fzz (z), z = g(x). (3) 2◦ . Let us demonstrate that the right-hand side f (x) of the integral equation must satisfy certain relations. By setting x = 0 in (1) and (2), we obtain two corollaries a a g(t)y(t) dt = f (0), y(t) dt = –fx (0). (4) 0

0

Substitute y(x) of (3) into (4). Integrating by parts yields the desired constraints for f (x): xa fx (xa ) = f (0) + f (xa ),

fx (0) + fx (xa ) = 0,

xa = g(a).

(5)

Conditions (5) make it possible to find the admissible general form of the right-hand side of the integral equation:

 A = – 12 Fx (0) + Fx (xa ) ,

f (x) = F (x) + Ax + B,

 B = 12 xa Fx (0) – F (xa ) – F (0) ,

xa = g(a),

where F (x) is an arbitrary bounded twice differentiable function (with bounded first derivative).

3.8. EQUATIONS WHOSE KERNELS CONTAIN ARBITRARY FUNCTIONS



b

7. a

y(t) |g(x) – g(t)|k

dt = f (x),

283

0 < k < 1.

Let gx ≠ 0. The transformation z = g(x),

τ = g(t),

leads to an equation of the form 3.1.31: B w(τ ) dτ = F (z), |z – τ |k A

w(τ ) =

1 y(t) gt (t)

A = g(a),

B = g(b),

where F = F (z) is the function which is obtained from z = g(x) and F = f (x) by eliminating x.

1

8. 0

y(t) dt = f (x), |g(x) – h(t)|k

0 < k < 1.

Let g(0) = 0, g(1) = 1, gx > 0; h(0) = 0, h(1) = 1, and ht > 0. The transformation z = g(x),

τ = h(t),

w(τ ) =

1

y(t) ht (t)

leads to an equation of the form 3.1.30: 1 0

w(τ ) dτ = F (z), |z – τ |k

where F = F (z) is the function which is obtained from z = g(x) and F = f (x) by eliminating x.

b

9. a

y(t) ln |g(x) – g(t)| dt = f (x).

Let gx ≠ 0. The transformation z = g(x),

τ = g(t),

leads to Carleman’s equation 3.4.2: B ln |z – τ |w(τ ) dτ = F (z),

w(τ ) =

1 y(t) gt (t)

A = g(a),

B = g(b),

A

where F = F (z) is the function which is obtained from z = g(x) and F = f (x) by eliminating x.

1

y(t) ln |g(x) – h(t)| dt = f (x).

10. 0

Let g(0) = 0, g(1) = 1, gx > 0; h(0) = 0, h(1) = 1, and ht > 0. The transformation z = g(x),

τ = h(t),

w(τ ) =

1 y(t) ht (t)

leads to an equation of the form 3.4.2: 1 ln |z – τ |w(τ ) dτ = F (z), 0

where F = F (z) is the function which is obtained from z = g(x) and F = f (x) by eliminating x.

284

LINEAR EQUATIONS OF THE FIRST KIND WITH CONSTANT LIMITS OF INTEGRATION

3.8-3. Equations with Difference Kernel: K(x, t) = K(x – t).



11.

K(x – t)y(t) dt = Axn ,

n = 0, 1, 2, . . .

–∞

1◦ . Solution with n = 0:

A y(x) = , B



B=

K(x) dx. –∞

2◦ . Solution with n = 1: AC A y(x) = x + 2 , B B







B=

K(x) dx,



C=

–∞

xK(x). –∞

3◦ . Solution with n ≥ 2: y(x) =



12.



 dn Aeλx , dλn B(λ) λ=0





B(λ) =

K(x)e–λx dx.

–∞

K(x – t)y(t) dt = Aeλx .

–∞

Solution:



A y(x) = eλx , B



13.

K(x – t)y(t) dt = Axn eλx ,



B=

K(x)e–λx dx.

–∞

n = 1, 2, . . .

–∞

1◦ . Solution with n = 1: A λx AC λx xe + 2 e , B B ∞ ∞ –λx B= K(x)e dx, C = xK(x)e–λx dx. y(x) =

–∞

–∞

2◦ . Solution with n ≥ 2: y(x) =

  dn Aeλx , dλn B(λ)





B(λ) =

K(x)e–λx dx.

–∞



K(x – t)y(t) dt = A cos(λx) + B sin(λx).

14. –∞

Solution: BIc – AIs AIc + BIs cos(λx) + sin(λx), y(x) = Ic2 + Is2 Ic2 + Is2 ∞ ∞ Ic = K(z) cos(λz) dz, Is = K(z) sin(λz) dz. –∞

–∞

3.8. EQUATIONS WHOSE KERNELS CONTAIN ARBITRARY FUNCTIONS



285



K(x – t)y(t) dt = f (x).

15. –∞

The Fourier transform is used to solve this equation. 1◦ . Solution:



f˜(u) iux e du, ˜ –∞ K(u) ∞ ∞ 1 1 –iux ˜ ˜ √ √ f(u) = f (x)e dx, K(u) = K(x)e–iux dx. 2π –∞ 2π –∞ 1 y(x) = 2π

The following statement is valid. Let f (x) ∈ L2 (–∞, ∞) and K(x) ∈ L1 (–∞, ∞). Then for a solution y(x) ∈ L2 (–∞, ∞) of the integral equation to exist, it is necessary and sufficient ˜ that f˜(u)/K(u) ∈ L2 (–∞, ∞). 2◦ . Let the function P (s) defined by the formula 1 = P (s)





e–st K(t) dt

–∞

be a polynomial of degree n with real roots of the form  s   s  s  1– ... 1 – . P (s) = 1 – a1 a2 an Then the solution of the integral equation is given by y(x) = P (D)f (x),

D=

d . dx

References: I. I. Hirschman and D. V. Widder (1955), V. A. Ditkin and A. P. Prudnikov (1965).





K(x – t)y(t) dt = f (x).

16. 0

The Wiener–Hopf equation of the first kind. This equation is discussed in Subsection 12.8-1 in detail. 3.8-4. Other Equations of the Form



17.

b a

K(x, t)y(t) dt = F (x).

K(ax – t)y(t) dt = Aeλx .

–∞

Solution: y(x) =

λ  A exp x , B a





B= –∞

 λ  K(z) exp – z dz. a



K(ax – t)y(t) dt = f (x).

18. –∞

The substitution z = ax leads to an equation of the form 3.8.15:



K(z – t)y(t) dt = f (z/a). –∞

286

LINEAR EQUATIONS OF THE FIRST KIND WITH CONSTANT LIMITS OF INTEGRATION





19.

K(ax + t)y(t) dt = Aeλx .

–∞

Solution:



 λ  A exp – x , y(x) = B a



B= –∞

 λ  K(z) exp – z dz. a



K(ax + t)y(t) dt = f (x).

20. –∞

The transformation τ = –t, z = ax, y(t) = Y (τ ) leads to an equation of the form 3.8.15: ∞ K(z – τ )Y (τ ) dt = f (z/a). –∞





21.

[eβt K(ax + t) + eµt M (ax – t)]y(t) dt = Aeλx .

–∞

Solution: y(x) = A

Ik (q)epx – Im (p)eqx , Ik (p)Ik (q) – Im (p)Im (q)

where





K(z)e

Ik (q) =

p=–

(β+q)z

dz,



Im (q) =

–∞



λ – β, a

q=

λ – µ, a

M (z)e–(µ+q)z dz.

–∞



g(xt)y(t) dt = f (x).

22. 0

By setting x = ez ,

y(t) = eτ w(τ ),

t = e–τ ,

g(ξ) = G(ln ξ),

f (ξ) = F (ln ξ),

we arrive at an integral equation with difference kernel of the form 3.8.15: ∞ G(z – τ )w(τ ) dτ = F (z). –∞

23.



g

x

t By setting

y(t) dt = f (x).

0

x = ez ,

t = eτ ,

y(t) = e–τ w(τ ),

g(ξ) = G(ln ξ),

f (ξ) = F (ln ξ),

we arrive at an integral equation with difference kernel of the form 3.8.15: ∞ G(z – τ )w(τ ) dτ = F (z). –∞

24.



  g xβ tλ y(t) dt = f (x),

β > 0,

λ > 0.

0

By setting x = ez/β ,

t = e–τ /λ ,

y(t) = eτ /λ w(τ ),

g(ξ) = G(ln ξ),

f (ξ) =

we arrive at an integral equation with difference kernel of the form 3.8.15: ∞ G(z – τ )w(τ ) dτ = F (z). –∞

1 λ F (β ln ξ),

287

3.8. EQUATIONS WHOSE KERNELS CONTAIN ARBITRARY FUNCTIONS

 ∞ xβ y(t) dt = f (x), g tλ 0 

25.

β > 0,

λ > 0.

By setting x = ez/β ,

t = eτ /λ ,

y(t) = e–τ /λ w(τ ),

g(ξ) = G(ln ξ),

f (ξ) =

1 λ F (β ln ξ),

we arrive at an integral equation with difference kernel of the form 3.8.15:



G(z – τ )w(τ ) dτ = F (z). –∞



a



26. 0

 1 + ϕ(x)ψ(t) y(t) dt = f (x), |x – t|k

0 < k < 1.

The solution can be obtained by the methods described in Subsection 12.6-2; it must be taken into account that the truncated equation, with ϕ(x) = 0, coincides with equation 3.1.30.



27.

exp[–g(x)t2 ]y(t) dt = f (x).

0

Assume that g(0) = ∞, g(∞) = 0, and gx < 0. 1 The substitution z = leads to equation 3.2.21: 4g(x) 1 √ πz



∞ 0

 2 t y(t) dt = F (z), exp – 4z

 2 1 where the function F (z) is determined by the relations F = √ f (x) g(x) and z = π 4g(x) by means of eliminating x.

b

28. a

  ln |x – t| + ϕ(x)ψ(t) y(t) dt = f (x).

The solution can be obtained by the methods described in Subsection 12.6-2; it must be taken into account that the truncated equation, with ϕ(x) = 0, coincides with equation 3.4.2. See also Example 3 in Subsection 12.6-2.



[sin(xt) + ϕ(x)ψ(t)]y(t) dt = f (x).

29. 0

The solution can be obtained by the methods described in Subsection 12.6-2; it must be taken into account that the truncated equation, with ϕ(x) = 0, coincides with equation 3.5.8. Solution: y(t) = yf (t) + Ayϕ (t), where

yf (t) =

2 π





sin(xt)f (x) dx, yϕ (t) = 0

2 π







sin(xt)ϕ(x) dx, A = – 0

0

1+

ψ(t)yf (t) dt

∞ 0

. ψ(t)yϕ (t) dt

288

LINEAR EQUATIONS OF THE FIRST KIND WITH CONSTANT LIMITS OF INTEGRATION





[cos(xt) + ϕ(x)ψ(t)]y(t) dt = f (x).

30. 0

The solution can be obtained by the methods described in Subsection 12.6-2; it must be taken into account that the truncated equation, with ϕ(x) = 0, coincides with equation 3.5.1. Solution: y(t) = yf (t) + Ayϕ (t), where

yf (t) =



31.



2 π



cos(xt)f (x) dx, yϕ (t) = 0

2 π







0

cos(xt)ϕ(x) dx, A = –

1+

0



ta–1 cos ϕ(x)ta y(t) dt = f (x),

ψ(t)yf (t) dt

∞ 0

. ψ(t)yϕ (t) dt

a > 0.

0

Transformation z = ϕ(x),

τ = ta ,

Y (τ ) = y(t),

F (z) = af (x)

leads to an equation of the form 3.5.1:



cos(zτ )Y (τ ) dτ = F (z). 0





32.



ta–1 sin ϕ(x)ta y(t) dt = f (x),

a > 0.

0

Transformation z = ϕ(x),

τ = ta ,

Y (τ ) = y(t),

F (z) = af (x)

leads to an equation of the form 3.5.8:



sin(zτ )Y (τ ) dτ = F (z). 0





[tJν (xt) + ϕ(x)ψ(t)]y(t) dt = f (x),

33.

ν > –1.

0

Here Jν (z) is the Bessel function of the first kind. The solution can be obtained by the methods described in Subsection 12.6-2; it must be taken into account that the truncated equation, with ϕ(x) = 0, coincides with equation 3.7.17. Solution: y(t) = yf (t) + Ayϕ (t), where yf (t) =





xJν (xt)f (x) dx, 0

yϕ (t) =





xJν (xt)ϕ(x) dx, 0

A=–

0

1+

ψ(t)yf (t) dt

∞ 0

. ψ(t)yϕ (t) dt

289

3.8. EQUATIONS WHOSE KERNELS CONTAIN ARBITRARY FUNCTIONS

3.8-5. Equations of the Form

b a

K(x, t)y(· · ·) dt = F (x).

b

f (t)y(xt) dt = Ax + B.

34. a

Solution: y(x) =

b

35.



A B x+ , I1 I0



b

I0 =

f (t) dt,

b

I1 =

tf (t) dt.

a

a

f (t)y(xt) dt = Axβ .

a

Solution: y(x) =



A β x , B

b

f (t)tβ dt.

B= a

b

f (t)y(xt) dt = A ln x + B.

36. a

Solution: y(x) = p ln x + q, where A p= , I0

b

37.

B AIl q= – 2 , I0 I0

I0 =



b

f (t) dt,

b

Il =

f (t) ln t dt.

a

a

f (t)y(xt) dt = Axβ ln x.

a

Solution: y(x) = pxβ ln x + qxβ , where p=

A , I1

q=–

AI2 , I12





b

f (t)tβ dt,

I1 = a

b

f (t)tβ ln t dt.

I2 = a

b

f (t)y(xt) dt = A cos(ln x).

38. a

Solution: AIc AIs cos(ln x) + 2 sin(ln x), Ic2 + Is2 Ic + Is2 b b Ic = f (t) cos(ln t) dt, Is = f (t) sin(ln t) dt. y(x) =

a



a

b

f (t)y(xt) dt = A sin(ln x).

39. a

Solution: AIs AIc cos(ln x) + 2 sin(ln x), 2 + Is Ic + Is2 b b f (t) cos(ln t) dt, Is = f (t) sin(ln t) dt. Ic = y(x) = –

a

Ic2

a

290

LINEAR EQUATIONS OF THE FIRST KIND WITH CONSTANT LIMITS OF INTEGRATION



b

40.

f (t)y(xt) dt = Axβ cos(ln x) + Bxβ sin(ln x).

a

Solution: y(x) = pxβ cos(ln x) + qxβ sin(ln x), where AIc – BIs , Ic2 + Is2

p=

b

f (t)tβ cos(ln t) dt,

Ic =

AIs + BIc , Ic2 + Is2 b Is = f (t)tβ sin(ln t) dt.

q=

a



a

b

f (t)y(x – t) dt = Ax + B.

41. a

Solution: y(x) = px + q, where p=

b

42.

A , I0

q=

AI1 B + , I0 I02





b

I0 =

f (t) dt,

b

I1 =

a

tf (t) dt. a

f (t)y(x – t) dt = Aeλx .

a

Solution: y(x) =

A λx e , B

B=

b

f (t) exp(–λt) dt. a

b

f (t)y(x – t) dt = A cos(λx).

43. a

Solution: AIc AIs sin(λx) + 2 cos(λx), Ic2 + Is2 Ic + Is2 b b Ic = f (t) cos(λt) dt, Is = f (t) sin(λt) dt. y(x) = –

a



a

b

f (t)y(x – t) dt = A sin(λx).

44. a

Solution: AIs AIc sin(λx) + 2 cos(λx), 2 + Is Ic + Is2 b b f (t) cos(λt) dt, Is = f (t) sin(λt) dt. Ic = y(x) =

a

Ic2

a

3.8. EQUATIONS WHOSE KERNELS CONTAIN ARBITRARY FUNCTIONS



b

45.

f (t)y(x – t) dt = eµx (A sin λx + B cos λx).

a

Solution: y(x) = eµx (p sin λx + q cos λx), where p=

AIc – BIs , Ic2 + Is2

AIs + BIc , Ic2 + Is2 b Is = f (t)e–µt sin(λt) dt. q=

b

f (t)e–µt cos(λt) dt,

Ic = a



a

b

f (t)y(x – t) dt = g(x).

46. a

1◦ . For g(x) =

n 

Ak exp(λk x), the solution of the equation has the form

k=1

y(x) =



n  Ak exp(λk x), Bk

b

Bk =

f (t) exp(–λk t) dt. a

k=1

2◦ . For a polynomial right-hand side, g(x) =

n 

Ak xk , the solution has the form

k=0

y(x) =

n 

Bk xk ,

k=0

where the constants Bk are found by the method of undetermined coefficients. n  3◦ . For g(x) = eλx Ak xk , the solution has the form k=0

y(x) = eλx

n 

Bk xk ,

k=0

where the constants Bk are found by the method of undetermined coefficients. n  4◦ . For g(x) = Ak cos(λk x), the solution has the form k=1

y(x) =

n 

Bk cos(λk x) +

k=1

n 

Ck sin(λk x),

k=1

where the constants Bk and Ck are found by the method of undetermined coefficients. n  5◦ . For g(x) = Ak sin(λk x), the solution has the form k=1

y(x) =

n  k=1

Bk cos(λk x) +

n 

Ck sin(λk x),

k=1

where the constants Bk and Ck are found by the method of undetermined coefficients.

291

292

LINEAR EQUATIONS OF THE FIRST KIND WITH CONSTANT LIMITS OF INTEGRATION

6◦ . For g(x) = cos(λx)

n 

Ak xk , the solution has the form

k=0

y(x) = cos(λx)

n 

Bk xk + sin(λx)

k=0

n 

Ck xk ,

k=0

where the constants Bk and Ck are found by the method of undetermined coefficients. n  7◦ . For g(x) = sin(λx) Ak xk , the solution has the form k=0

y(x) = cos(λx)

n 

Bk xk + sin(λx)

k=0

n 

Ck xk ,

k=0

where the constants Bk and Ck are found by the method of undetermined coefficients. n  8◦ . For g(x) = eµx Ak cos(λk x), the solution has the form k=1

y(x) = eµx

n 

Bk cos(λk x) + eµx

k=1

n 

Ck sin(λk x),

k=1

where the constants Bk and Ck are found by the method of undetermined coefficients. n  9◦ . For g(x) = eµx Ak sin(λk x), the solution has the form k=1

y(x) = eµx

n 

Bk cos(λk x) + eµx

k=1

n 

Ck sin(λk x),

k=1

where the constants Bk and Ck are found by the method of undetermined coefficients. n  10◦ . For g(x) = cos(λx) Ak exp(µk x), the solution has the form k=1

y(x) = cos(λx)

n 

Bk exp(µk x) + sin(λx)

k=1

n 

Bk exp(µk x),

k=1

where the constants Bk and Ck are found by the method of undetermined coefficients. n  11◦ . For g(x) = sin(λx) Ak exp(µk x), the solution has the form k=1

y(x) = cos(λx)

n 

Bk exp(µk x) + sin(λx)

k=1

n 

Bk exp(µk x),

k=1

where the constants Bk and Ck are found by the method of undetermined coefficients.

b

f (t)y(x + βt) dt = Ax + B.

47. a

Solution: y(x) = px + q, where p=

A , I0

q=

B AI1 β – , I0 I02





b

I0 =

f (t) dt, a

I1 =

b

tf (t) dt. a

293

3.8. EQUATIONS WHOSE KERNELS CONTAIN ARBITRARY FUNCTIONS



b

48.

f (t)y(x + βt) dt = Aeλx .

a

Solution: A y(x) = eλx , B



b

B=

f (t) exp(λβt) dt. a

b

f (t)y(x + βt) dt = A sin λx + B cos λx.

49. a

Solution: y(x) = p sin λx + q cos λx, where p=

AIc + BIs , Ic2 + Is2

b

f (t) cos(λβt) dt,

Ic =

BIc – AIs , Ic2 + Is2 b Is = f (t) sin(λβt) dt. q=

a



a

1

y(ξ) dt = f (x),

50.

ξ = g(x)t.

0

Assume that g(0) = 0, g(1) = 1, and gx ≥ 0. 1

1◦ . The substitution z = g(x) leads to an equation of the form 3.1.42: y(zt) dt = F (z), 0 where the function F (z) is obtained from z = g(x) and F = f (x) by eliminating x. 2◦ . Solution y = y(z) in the parametric form: y(z) =

1

51.

tλ y(ξ) dt = f (x),

g(x)  f (x) + f (x), gx (x) x

z = g(x).

ξ = g(x)t.

0

Assume that g(0) = 0, g(1) = 1, and gx ≥ 0. 1

1◦ . The substitution z = g(x) leads to an equation of the form 3.1.43: tλ y(zt) dt = F (z), 0 where the function F (z) is obtained from z = g(x) and F = f (x) by eliminating x. 2◦ . Solution y = y(z) in the parametric form: y(z) =

b

52.

f (t)y(ξ) dt = Axβ ,

g(x)  f (x) + (λ + 1)f (x), gx (x) x

z = g(x).

ξ = xϕ(t).

a

Solution: y(x) =

A β x , B



b

B= a

β f (t) ϕ(t) dt.

(1)

294

LINEAR EQUATIONS OF THE FIRST KIND WITH CONSTANT LIMITS OF INTEGRATION



b

f (t)y(ξ) dt = g(x),

53.

ξ = xϕ(t).

a

1◦ . For g(x) =

n 

Ak xk , the solution of the equation has the form

k=0



n  Ak k x , y(x) = Bk n 

k f (t) ϕ(t) dt.

b

λ f (t) ϕ(t) k dt.

a

k=0

2◦ . For g(x) =

b

Bk =

Ak xλk , the solution has the form

k=0

y(x) =



n  Ak λk x , Bk

Bk = a

k=0

3◦ . For g(x) = ln x

n 

Ak xk , the solution has the form

k=0

y(x) = ln x

n 

Bk xk +

k=0

n 

Ck xk ,

k=0

where the constants Bk and Ck are found by the method of undetermined coefficients. 4◦ . For g(x) =

n 

 Ak ln x)k , the solution has the form

k=0

y(x) =

n 

 Bk ln x)k ,

k=0

where the constants Bk are found by the method of undetermined coefficients. 5◦ . For g(x) =

n 

Ak cos(λk ln x), the solution has the form

k=1

y(x) =

n 

Bk cos(λk ln x) +

k=1

n 

Ck sin(λk ln x),

k=1

where the constants Bk and Ck are found by the method of undetermined coefficients. 6◦ . For g(x) =

n 

Ak sin(λk ln x), the solution has the form

k=1

y(x) =

n  k=1

Bk cos(λk ln x) +

n 

Ck sin(λk ln x),

k=1

where the constants Bk and Ck are found by the method of undetermined coefficients.

3.8. EQUATIONS WHOSE KERNELS CONTAIN ARBITRARY FUNCTIONS



b

f (t)y(ξ) dt = g(x),

54.

ξ = x + ϕ(t).

a

1◦ . For g(x) =

n 

Ak exp(λk x), the solution of the equation has the form

k=1

y(x) =



n  Ak exp(λk x), Bk

b

Bk =

 f (t) exp λk ϕ(t) dt.

a

k=1

2◦ . For a polynomial right-hand side, g(x) =

n 

Ak xk , the solution has the form

k=0

y(x) =

n 

Bk xk ,

k=0

where the constants Bk are found by the method of undetermined coefficients. n  3◦ . For g(x) = eλx Ak xk , the solution has the form k=0

y(x) = eλx

n 

Bk xk ,

k=0

where the constants Bk are found by the method of undetermined coefficients. n  4◦ . For g(x) = Ak cos(λk x) the solution has the form k=1

y(x) =

n 

Bk cos(λk x) +

k=1

n 

Ck sin(λk x),

k=1

where the constants Bk and Ck are found by the method of undetermined coefficients. n  5◦ . For g(x) = Ak sin(λk x), the solution has the form k=1

y(x) =

n 

Bk cos(λk x) +

k=1

n 

Ck sin(λk x),

k=1

where the constants Bk and Ck are found by the method of undetermined coefficients. n  6◦ . For g(x) = cos(λx) Ak xk , the solution has the form k=0

y(x) = cos(λx)

n 

Bk xk + sin(λx)

k=0

n 

Ck xk ,

k=0

where the constants Bk and Ck are found by the method of undetermined coefficients. n  7◦ . For g(x) = sin(λx) Ak xk , the solution has the form k=0

y(x) = cos(λx)

n  k=0

Bk xk + sin(λx)

n 

Ck xk ,

k=0

where the constants Bk and Ck are found by the method of undetermined coefficients.

295

296

LINEAR EQUATIONS OF THE FIRST KIND WITH CONSTANT LIMITS OF INTEGRATION

8◦ . For g(x) = eµx

n 

Ak cos(λk x), the solution has the form

k=1

y(x) = eµx

n 

Bk cos(λk x) + eµx

k=1

n 

Ck sin(λk x),

k=1

where the constants Bk and Ck are found by the method of undetermined coefficients. n  9◦ . For g(x) = eµx Ak sin(λk x), the solution has the form k=1

y(x) = e

µx

n 

Bk cos(λk x) + e

µx

k=1

n 

Ck sin(λk x),

k=1

where the constants Bk and Ck are found by the method of undetermined coefficients. n  10◦ . For g(x) = cos(λx) Ak exp(µk x), the solution has the form k=1

y(x) = cos(λx)

n 

Bk exp(µk x) + sin(λx)

k=1

n 

Bk exp(µk x),

k=1

where the constants Bk and Ck are found by the method of undetermined coefficients. n  11◦ . For g(x) = sin(λx) Ak exp(µk x), the solution has the form k=1

y(x) = cos(λx)

n 

Bk exp(µk x) + sin(λx)

k=1

n 

Bk exp(µk x),

k=1

where the constants Bk and Ck are found by the method of undetermined coefficients.

3.9. Dual Integral Equations of the First Kind 3.9-1. Kernels Containing Trigonometric Functions.



cos(xt)y(t) dt = f (x) for 0 < x < 1,

1. 0 ∞

for 1 < x < ∞.

sin(xt)y(t) dt = 0 0

Solution: y(x) =

2x π





1

tJ0 (xt) 0

0

t

 f (s) ds √ dt. t2 – s 2

References: C. Nasim and B. D. Aggarwala (1984), B. N. Mandal and N. Mandal (1999, pp. 134–136).





cos(xt)y(t) dt = 0

2. 0





for 0 < x < 1,

sin(xt)y(t) dt = f (x) for 1 < x < ∞.

0

Solution: y(x) =

2x π







tJ0 (xt) 1

t



 f (s) ds √ dt. s 2 – t2

References: C. Nasim and B. D. Aggarwala (1984), B. N. Mandal and N. Mandal (1999, pp. 136–137).

297

3.9. DUAL INTEGRAL EQUATIONS OF THE FIRST KIND





cos(xt)y(t) dt = f (x) for 0 < x < 1,

3. 0 ∞

for 1 < x < ∞.

t cos(xt)y(t) dt = 0 0

Solution: y(x) =

2x π





1

t

tJ0 (xt) 0

0

 1 f (s) ds f (s) ds 2 √ √ dt – J1 (x) . π t2 – s 2 1 – s2 0

References: I. W. Busbridge (1938), B. N. Mandal and N. Mandal (1999, pp. 138–139).





t cos(xt)y(t) dt = f (x) for 0 < x < 1,

4. 0 ∞

for 1 < x < ∞.

cos(xt)y(t) dt = 0 0

Solution: 2 y(x) = π





1

t

tJ0 (xt) 0

0

 f (s) ds √ dt. t2 – s 2

References: I. W. (1937, p. 339), B. N. Mandal and N. Mandal (1999, pp. 139–140).





sin(xt)y(t) dt = f (x) for 0 < x < 1,

5. 0 ∞

for 1 < x < ∞.

t sin(xt)y(t) dt = 0 0

It is assumed that f (0) = 0. Solution: 2 y(x) = π





1

t

tJ0 (xt) 0

0

 fs (s) ds √ dt. t2 – s 2

References: I. W. Busbridge (1938), B. N. Mandal and N. Mandal (1999, pp. 140–141).





t sin(xt)y(t) dt = f (x) for 0 < x < 1,

6.

0



sin(xt)y(t) dt = 0

for 1 < x < ∞.

0

Solution: 2 y(x) = π





1

t

J1 (xt) 0

0

 sf (s) ds √ dt. t2 – s 2

References: B. Noble (1963), B. N. Mandal and N. Mandal (1999, pp. 141–142).





[a sin(xt) + t cos(xt)]y(t) dt = f (x) for 0 < x < 1,

7.

0



t[a sin(xt) + t cos(xt)]y(t) dt = 0

for 1 < x < ∞.

0

Solution:

1

tJ0 (xt)F (t) dt +

y(x) = 0

F (1) K0 (a)







tJ0 (xt) 1

t



 e–as ds √ dt, s 2 – t2

298

LINEAR EQUATIONS OF THE FIRST KIND WITH CONSTANT LIMITS OF INTEGRATION

where

z zϕ(z) dz 2 t ϕz (z) dz √ √ = , ϕ(z) = e–az eas f (s) ds π 0 t2 – z 2 t2 – z 2 0 0 and K0 (x) is the modified Bessel functions of the second kind. F (t) =

2 d π dt



t

(0 < z < 1),

References: B. D. Aggarwala and C. Nasim (1996), B. N. Mandal and N. Mandal (1999, pp. 143–145).





t[a sin(xt) + t cos(xt)]y(t) dt = f (x) for 0 < x < 1,

8. 0 ∞

for 1 < x < ∞.

[a sin(xt) + t cos(xt)]y(t) dt = g(x) 0

Solution for a ≠ 0: 1 y(x) = J1 (xt)F (t) dt– 0

where



1

2Da tJ1 (xt)G(t) dt+ π

2 t zϕ(z) dz √ F (t) = , π 0 t2 – z 2 ∞ ψ(z) dz 2 d √ G(t) = , π dt t z 2 – t2





J1 (xt)K1 (at) dt, t

ϕ(z) = e

D=

F (1) + G(1) , aK1 (a)

z

–az

eas f (s) ds

(0 < z < 1),

eas g(s) ds

(1 < z < ∞),

0



z

ψ(z) = e–az 1

and K0 (x) is the modified Bessel functions of the second kind. References: B. D. Aggarwala and C. Nasim (1996), B. N. Mandal and N. Mandal (1999, pp. 143, 147–148).

3.9-2. Kernels Containing Bessel Functions of the First Kind.



J0 (xt)y(t) dt = f (x) for 0 < x < a,

9. 0 ∞

for a < x < ∞.

tJ0 (xt)y(t) dt = 0 0

Solution: y(x) =

2 π





a

cos(xt) 0

d dt



t 0

 sf (s) ds √ dt. t2 – s 2



tJ0 (xt)y(t) dt = f (x) for 0 < x < a,

10. 0 ∞

for a < x < ∞.

J0 (xt)y(t) dt = 0 0

Solution: 2 y(x) = π

0

a





d sin(xt) dt

0

t

 sf (s) ds √ dt. t2 – s 2



11. 0



tJµ (xt)y(t) dt = f (x)

for 0 < x < a,

Jµ (xt)y(t) dt = 0

for a < x < ∞.



0



Solution: y(x) =

2x π





a

t3/2 Jµ+ 1 (xt) 0

2

0

π/2

 sinµ+1 θf (t sin θ) dθ dt.

3.9. DUAL INTEGRAL EQUATIONS OF THE FIRST KIND



299



Jµ (xt)y(t) dt = f (x) for 0 < x < 1,

12. 0 ∞

t2 Jµ (xt)y(t) dt = 0

for 1 < x < ∞.

0

Solution:



1

y(x) = f (1)Jµ–1 (x) + x

tJµ (xt)f (t) dt. 0

Reference: B. N. Mandal and N. Mandal (1999, p. 31).





13.

0

t2β Jµ (xt)y(t) dt = f (x)

for 0 < x < 1,

Jµ (xt)y(t) dt = 0

for 1 < x < ∞.



0 ◦

1 . Solution for β > 0: (2x)1–β 1 1+β t Jµ+β (xt)F (t) dt, y(x) = Γ(β) 0



1

f (tζ)ζ µ+1 (1 – ζ 2 )β–1 dζ.

F (t) =

(1)

0

2◦ . Solution for β > –1:   1 1 (2x)–β x1+β Jµ+β (x) tµ+1 (1 – t2 )β f (t) dt + tµ+1 (1 – t2 )β Φ(x, t) dt , (2) y(x) = Γ(1 + β) 0 0 1 (xξ)2+β Jµ+β+1 (xξ)f (ξt) dξ. Φ(x, t) = 0

14.

Formula (2) holds for β > –1 and for –µ – 12 < 2β < µ + 32 . It can be shown that for β > 0 the solution of Eq. (2) can be reduced to the form (1). ∞ t–2α Jµ (xt)y(t) dt = f (x) for 0 < x < 1, 0 ∞ t–2β Jµ (xt)y(t) dt = g(x) for 1 < x < ∞. 0

Solution for 0 < β – α < 1:  t  21+α–β x1+α+β 1 1+α–β–µ y(x) = t Jµ+β–α (xt) s 1+µ (t2 – s 2 )β–α–1 f (s) ds dt Γ(β – α) 0 0  ∞  ∞ β–α d 2 x1+α+β tµ+β–α Jµ+β–α (xt) s 1–µ (s 2 – t2 )α–β g(s) ds dt. – Γ(1 + α – β) dt t 1 References: C. Nasim and B. D. Aggarwala (1984), B. N. Mandal and N. Mandal (1999, pp. 40–44).





J0 (xt)y(t) dt = f (x)

15.

0



for 0 < x < a,

cos(xt)y(t) dt = g(x) for a < x < ∞.

0

Solution: 2 y(x) = π

0

a



d cos(xt) dt

0

t

 sf (s) ds 2 ∞ √ dt + cos(xt)g(t) dt. π a t2 – s 2

Reference: B. N. Mandal and N. Mandal (1999, pp. 194–195).

300

LINEAR EQUATIONS OF THE FIRST KIND WITH CONSTANT LIMITS OF INTEGRATION





tJ0 (xt)y(t) dt = f (x) for 0 < x < a,

16. 0 ∞

sin(xt)y(t) dt = g(x)

for a < x < ∞.

0

Solution: 2 y(x) = π





a

t

sin(xt) 0

0

 sf (s) ds 2 ∞ √ dt + sin(xt)g(t) dt. π a t2 – s 2

Reference: B. N. Mandal and N. Mandal (1999, pp. 195–196).

3.9-3. Kernels Containing Bessel Functions of the Second Kind.



17.

0



t–2α Yµ (xt)y(t) dt = f (x) for 0 < x < 1, t–2β Yν (xt)y(t) dt = g(x)

for 1 < x < ∞.

0

Let 2(α – β) = ν – µ > 0, |µ| < 12 , |ν| < 12 . 1◦ . Solution for 0 < ν – µ < 1:

 1  1 d 2ν–µ x2β+1 tν Hν (xt) s 1–µ (s 2 – t2 )µ–ν f (s) ds dt Γ(1 + µ – ν) dt t 0  ∞  ∞ µ–ν 2 + t1+µ Hµ (xt) s 1–ν (s 2 – t2 )ν–µ–1 g(s) ds dt, Γ(µ – ν) 1 t where Hµ (x) is the Struve function, which is defined as y(x) = –

Hµ (x) =

∞  j=0

(–1)j (x/2)µ+2j+1   .  Γ j + 32 Γ µ + j + 32



2 . Solution for –1 < ν – µ < 0:  1  21–ν–µ 2β+1 1 ν+1 1–µ 2 2 µ–ν–1 x t Hν (xt) s (s – t ) f (s) ds dt y(x) = Γ(µ – ν) 0 t   ∞ ∞ 2µ–ν + x2α+1 tµ Hµ (xt) s 1–ν (s 2 – t2 )ν–µ g(s) ds dt. Γ(1 – µ + ν) 1 t References: C. Nasim and B. D. Aggarwala (1984), B. N. Mandal and N. Mandal (1999, pp. 58–59).

3.9-4. Kernels Containing Legendre Spherical Functions of the First Kind, i2 = –1.



18. 0 ∞ 0

tP– 12 +it (cosh x)y(t) dt = f (x)

for 0 < x < a,

tanh(πt)P– 12 +it (cosh x)y(t) dt = 0

for a < x < ∞.

√ a  t  2 f (s) sinh s √ y(x) = sin(xt) ds dt. π 0 cosh t – cosh s 0 Note that √ x 2 cos(ts) √ P– 1 +it (cosh x) = ds, x > 0, 2 π 0 cosh x – cosh s where the integral on the right-hand side is called the Meler integral.

Solution:

Chapter 4

Linear Equations of the Second Kind with Constant Limits of Integration  Notation: f = f (x), g = g(x), h = h(x), v = v(x), w = w(x), K = K(x) are arbitrary functions; A, B, C, D, E, a, b, c, l, α, β, γ, δ, µ, and ν are arbitrary parameters; n is a nonnegative integer; and i is the imaginary unit.  Preliminary remarks. A number λ is called a characteristic value of the integral equation

b

y(x) – λ

K(x, t)y(t) dt = f (x) a

if there exist nontrivial solutions of the corresponding homogeneous equation (with f (x) ≡ 0). The nontrivial solutions themselves are called the eigenfunctions of the integral equation corresponding to the characteristic value λ. If λ is a characteristic value, the number 1/λ is called an eigenvalue of the integral equation. A value of the parameter λ is said to be regular if for this value the homogeneous equation has only the trivial solution. Sometimes the characteristic values and the eigenfunctions of a Fredholm integral equation are called the characteristic values and the eigenfunctions of the kernel K(x, t). In the above equation, it is usually assumed that a ≤ x ≤ b.

4.1. Equations Whose Kernels Contain Power-Law Functions 4.1-1. Kernels Linear in the Arguments x and t. 1.

b

y(x) – λ

(x – t)y(t) dt = f (x). a

Solution: y(x) = f (x) + λ(A1 x + A2 ), where A1 =

12f1 + 6λ (f1 ∆2 – 2f2 ∆1 ) –12f2 + 2λ (3f2 ∆2 – 2f1 ∆3 ) , A2 = , λ2 ∆41 + 12 λ2 ∆41 + 12 b b f1 = f (x) dx, f2 = xf (x) dx, ∆n = bn – an . a

a

301

302

LINEAR EQUATIONS OF THE SECOND KIND WITH CONSTANT LIMITS OF INTEGRATION

2.

b

y(x) – λ

(x + t)y(t) dt = f (x). a

The characteristic values of the equation:  6(b + a) + 4 3(a2 + ab + b2 ) λ1 = , (a – b)3

 6(b + a) – 4 3(a2 + ab + b2 ) λ2 = . (a – b)3

1◦ . Solution with λ ≠ λ1,2 : y(x) = f (x) + λ(A1 x + A2 ), where A1 =

12f1 – 6λ(f1 ∆2 – 2f2 ∆1 ) 12f2 – 2λ(3f2 ∆2 – 2f1 ∆3 ) , A2 = , 12 – 12λ∆2 – λ2 ∆41 12 – 12λ∆2 – λ2 ∆41 b b f1 = f (x) dx, f2 = xf (x) dx, ∆n = bn – an . a

a

2◦ . Solution with λ = λ1 ≠ λ2 and f1 = f2 = 0: y(x) = f (x) + Cy1 (x), where C is an arbitrary constant and y1 (x) is an eigenfunction of the equation corresponding to the characteristic value λ1 : 1 b+a y1 (x) = x + – . λ1 (b – a) 2 3◦ . Solution with λ = λ2 ≠ λ1 and f1 = f2 = 0 is given by the formulas of item 2◦ in which one must replace λ1 and y1 (x) by λ2 and y2 (x), respectively.

3.

4◦ . The equation has no multiple characteristic values. b y(x) – λ (Ax + Bt)y(t) dt = f (x). a

The characteristic values of the equation:  3(A + B)(b + a) ± 9(A – B)2 (b + a)2 + 48AB(a2 + ab + b2 ) λ1,2 = . AB(a – b)3 1◦ . Solution with λ ≠ λ1,2 : y(x) = f (x) + λ(A1 x + A2 ), where the constants A1 and A2 are given by A1 =

12Af1 – 6ABλ(f1 ∆2 – 2f2 ∆1 ) 12Bf2 – 2ABλ(3f2 ∆2 – 2f1 ∆3 ) , A2 = , 4 2 12 – 6(A + B)λ∆2 – ABλ ∆1 12 – 6(A + B)λ∆2 – ABλ2 ∆41 b b f1 = f (x) dx, f2 = xf (x) dx, ∆n = bn – an . a

a

2◦ . Solution with λ = λ1 ≠ λ2 and f1 = f2 = 0: y(x) = f (x) + Cy1 (x), where C is an arbitrary constant and y1 (x) is an eigenfunction of the equation corresponding to the characteristic value λ1 : b+a 1 – . y1 (x) = x + λ1 A(b – a) 2 3◦ . Solution with λ = λ2 ≠ λ1 and f1 = f2 = 0 is given by the formulas of item 2◦ in which one must replace λ1 and y1 (x) by λ2 and y2 (x), respectively.

303

4.1. EQUATIONS WHOSE KERNELS CONTAIN POWER-LAW FUNCTIONS

4◦ . Solution with λ = λ1,2 = λ∗ and f1 = f2 = 0 provided that A ≠ ±B, where the characteristic 4 is double: value λ∗ = (A + B)(b2 – a2 ) y(x) = f (x) + Cy∗ (x), where C is an arbitrary constant and y∗ (x) is an eigenfunction of the equation corresponding to λ∗ : (A – B)(b + a) . y∗ (x) = x – 4A The equation has no multiple characteristic values if A = ±B. 4.

b

y(x) – λ

[A + B(x – t)]y(t) dt = f (x). a

This is a special case of equation 4.9.8 with h(t) = 1. Solution: y(x) = f (x) + λ(A1 + A2 x), where A1 and A2 are the constants determined by the formulas presented in 4.9.8. 5.

b

(Ax + Bt + C)y(t) dt = f (x).

y(x) – λ a

This is a special case of equation 4.9.7 with g(x) = x and h(t) = 1. Solution: y(x) = f (x) + λ(A1 x + A2 ), where A1 and A2 are the constants determined by the formulas presented in 4.9.7. 6.

y(x) + A

b

a

|x – t| y(t) dt = f (x).

This is a special case of equation 4.9.36 with g(t) = A. 1◦ . The function y = y(x) obeys the following second-order linear nonhomogeneous ordinary differential equation with constant coefficients:   yxx + 2Ay = fxx (x).

(1)

The boundary conditions for (1) have the form (see 4.9.36) yx (a) + yx (b) = fx (a) + fx (b),

(2)

y(a) + y(b) + (b – a)yx (a) = f (a) + f (b) + (b – a)fx (a).

Equation (1) under the boundary conditions (2) determines the solution of the original integral equation. 2◦ . For A < 0, the general solution of equation (1) is given by x y(x) = C1 cosh(kx) + C2 sinh(kx) + f (x) + k sinh[k(x – t)]f (t) dt,

k=

√ –2A,

(3)

k=

√ 2A.

(4)

a

where C1 and C2 are arbitrary constants. For A > 0, the general solution of equation (1) is given by x y(x) = C1 cos(kx) + C2 sin(kx) + f (x) – k sin[k(x – t)]f (t) dt, a

The constants C1 and C2 in solutions (3) and (4) are determined by conditions (2).

304

LINEAR EQUATIONS OF THE SECOND KIND WITH CONSTANT LIMITS OF INTEGRATION

3◦ . In the special case a = 0 and A > 0, the solution of the integral equation is given by formula (4) with Is (1 + cos λ) – Ic (λ + sin λ) Is sin λ + Ic (1 + cos λ) , C2 = k , 2 + 2 cos λ + λ sin λ 2 + 2 cos λ + λ sin λ b b √ k = 2A, λ = bk, Is = sin[k(b – t)]f (t) dt, Ic = cos[k(b – t)]f (t) dt. C1 = k

0

0

4.1-2. Kernels Quadratic in the Arguments x and t. 7.

b

y(x) – λ

(x2 + t2 )y(t) dt = f (x).

a

The characteristic values of the equation: λ1 =

1 3 3 (b

 – a3 ) +

1 1 5 5 (b

, – a5 )(b – a)

λ2 =

1 3 3 (b

 – a3 ) –

1 1 5 5 (b

. – a5 )(b – a)

1◦ . Solution with λ ≠ λ1,2 : y(x) = f (x) + λ(A1 x2 + A2 ), where the constants A1 and A2 are given by     f1 – λ 13 f1 ∆3 – f2 ∆1 f2 – λ 13 f2 ∆3 – 15 f1 ∆5     , A2 = 2 1 2 1 , A1 = 2 1 2 1 λ 9 ∆3 – 5 ∆1 ∆5 – 23 λ∆3 + 1 λ 9 ∆3 – 5 ∆1 ∆5 – 23 λ∆3 + 1 b b f (x) dx, f2 = x2 f (x) dx, ∆n = bn – an . f1 = a

a

2◦ . Solution with λ = λ1 ≠ λ2 and f1 = f2 = 0: $ y(x) = f (x) + Cy1 (x),

b 5 – a5 , 5(b – a)

y1 (x) = x2 +

where C is an arbitrary constant and y1 (x) is an eigenfunction of the equation corresponding to the characteristic value λ1 . 3◦ . Solution with λ = λ2 ≠ λ1 and f1 = f2 = 0: $ y(x) = f (x) + Cy2 (x),

y2 (x) = x2 –

b 5 – a5 , 5(b – a)

where C is an arbitrary constant and y2 (x) is an eigenfunction of the equation corresponding to the characteristic value λ2 . 4◦ . The equation has no multiple characteristic values.

305

4.1. EQUATIONS WHOSE KERNELS CONTAIN POWER-LAW FUNCTIONS

8.

b

y(x) – λ

(x2 – t2 )y(t) dt = f (x).

a

The characteristic values of the equation: λ1,2 = ± 

1 1 3 9 (b

.

– a3 )2 – 15 (b5 – a5 )(b – a)

1◦ . Solution with λ ≠ λ1,2 : y(x) = f (x) + λ(A1 x2 + A2 ), where the constants A1 and A2 are given by     f1 + λ 13 f1 ∆3 – f2 ∆1 –f2 + λ 13 f2 ∆3 – 15 f1 ∆5    A1 = 2  1 , A2 = , λ 5 ∆1 ∆5 – 19 ∆22 + 1 λ2 15 ∆1 ∆5 – 19 ∆22 + 1 b b f (x) dx, f2 = x2 f (x) dx, ∆n = bn – an . f1 = a

a



2 . Solution with λ = λ1 ≠ λ2 and f1 = f2 = 0: y(x) = f (x) + Cy1 (x),

y1 (x) = x2 +

3 – λ1 (b3 – a3 ) , 3λ1 (b – a)

where C is an arbitrary constant and y1 (x) is an eigenfunction of the equation corresponding to the characteristic value λ1 . 3◦ . The solution with λ = λ2 ≠ λ1 and f1 = f2 = 0 is given by the formulas of item 2◦ in which one must replace λ1 and y1 (x) by λ2 and y2 (x), respectively.

9.

4◦ . The equation has no multiple characteristic values. b y(x) – λ (Ax2 + Bt2 )y(t) dt = f (x). a

The characteristic values of the equation:  1 1 4 2 2 (A + B)∆ ± 3 3 9 (A – B) ∆3 + 5 AB∆1 ∆5  1 2 1 , λ1,2 = 2AB 9 ∆3 – 5 ∆1 ∆5

∆n = bn – an .

1◦ . Solution with λ ≠ λ1,2 : y(x) = f (x) + λ(A1 x2 + A2 ), where the constants A1 and A2 are given by

  Af1 – ABλ 13 f1 ∆3 – f2 ∆1   A1 = , ABλ2 19 ∆23 – 15 ∆1 ∆5 – 13 (A + B)λ∆3 + 1   Bf2 – ABλ 13 f2 ∆3 – 15 f1 ∆5   , A2 = ABλ2 19 ∆23 – 15 ∆1 ∆5 – 13 (A + B)λ∆3 + 1 b b f (x) dx, f2 = x2 f (x) dx. f1 = a

a

2◦ . Solution with λ = λ1 ≠ λ2 and f1 = f2 = 0: y(x) = f (x) + Cy1 (x),

y1 (x) = x2 +

3 – λ1 A(b3 – a3 ) , 3λ1 A(b – a)

where C is an arbitrary constant and y1 (x) is an eigenfunction of the equation corresponding to the characteristic value λ1 .

306

LINEAR EQUATIONS OF THE SECOND KIND WITH CONSTANT LIMITS OF INTEGRATION

3◦ . The solution with λ = λ2 ≠ λ1 and f1 = f2 = 0 is given by the formulas of item 2◦ in which one must replace λ1 and y1 (x) by λ2 and y2 (x), respectively. 4◦ . Solution with λ = λ1,2 = λ∗ and f1 = f2 = 0 provided that A ≠ ±B, where λ∗ = 6 is the double characteristic value: (A + B)(b3 – a3 ) y(x) = f (x) + C1 y∗ (x),

10.

where C1 is an arbitrary constant and y∗ (x) is an eigenfunction of the equation corresponding to λ∗ : (A – B)(b3 – a3 ) . y∗ (x) = x2 – 6A(b – a) The equation has no multiple characteristic values if A = ±B. b y(x) – λ (xt – t2 )y(t) dt = f (x). a

This is a special case of equation 4.9.8 with A = 0, B = 1, and h(t) = t. Solution: y(x) = f (x) + λ(A1 + A2 x),

11.

where A1 and A2 are the constants determined by the formulas presented in 4.9.8. b y(x) – λ (x2 – xt)y(t) dt = f (x). a

This is a special case of equation 4.9.10 with A = 0, B = 1, and h(x) = x. Solution: y(x) = f (x) + λ(E1 x2 + E2 x),

12.

where E1 and E2 are the constants determined by the formulas presented in 4.9.10. b y(x) – λ (Bxt + Ct2 )y(t) dt = f (x). a

This is a special case of equation 4.9.9 with A = 0 and h(t) = t. Solution: y(x) = f (x) + λ(A1 + A2 x),

13.

where A1 and A2 are the constants determined by the formulas presented in 4.9.9. b y(x) – λ (Bx2 + Cxt)y(t) dt = f (x). a

This is a special case of equation 4.9.11 with A = 0 and h(x) = x. Solution: y(x) = f (x) + λ(A1 x2 + A2 x),

14.

where A1 and A2 are the constants determined by the formulas presented in 4.9.11. b y(x) – λ (Axt + Bx2 + Cx + D)y(t) dt = f (x). a

This is a special case of equation 4.9.18 with g1 (x) = Bx2 + Cx + D, h1 (t) = 1, g2 (x) = x, and h2 (t) = At. Solution: y(x) = f (x) + λ[A1 (Bx2 + Cx + D) + A2 x], where A1 and A2 are the constants determined by the formulas presented in 4.9.18.

307

4.1. EQUATIONS WHOSE KERNELS CONTAIN POWER-LAW FUNCTIONS

15.

b

y(x) – λ

(Ax2 + Bt2 + Cx + Dt + E)y(t) dt = f (x).

a

This is a special case of equation 4.9.18 with g1 (x) = Ax2 + Cx, h1 (t) = 1, g2 (x) = 1, and h2 (t) = Bt2 + Dt + E. Solution: y(x) = f (x) + λ[A1 (Ax2 + Cx) + A2 ], where A1 and A2 are the constants determined by the formulas presented in 4.9.18. 16.

b

[Ax + B + (Cx + D)(x – t)]y(t) dt = f (x).

y(x) – λ a

This is a special case of equation 4.9.18 with g1 (x) = Cx2 + (A + D)x + B, h1 (t) = 1, g2 (x) = Cx + D, and h2 (t) = –t. Solution: y(x) = f (x) + λ[A1 (Cx2 + Ax + Dx + B) + A2 (Cx + D)], where A1 and A2 are the constants determined by the formulas presented in 4.9.18. 17.

b

[At + B + (Ct + D)(t – x)]y(t) dt = f (x).

y(x) – λ a

This is a special case of equation 4.9.18 with g1 (x) = 1, h1 (t) = Ct2 + (A + D)t + B, g2 (x) = x, and h2 (t) = –(Ct + D). Solution: y(x) = f (x) + λ(A1 + A2 x), where A1 and A2 are the constants determined by the formulas presented in 4.9.18. 18.

b

y(x) – λ

(x – t)2 y(t) dt = f (x).

a

This is a special case of equation 4.9.19 with g(x) = x, h(t) = –t, and m = 2. 19.

b

y(x) – λ

(Ax + Bt)2 y(t) dt = f (x).

a

This is a special case of equation 4.9.19 with g(x) = Ax, h(t) = Bt, and m = 2. 4.1-3. Kernels Cubic in the Arguments x and t. 20.

b

y(x) – λ

(x3 + t3 )y(t) dt = f (x).

a

The characteristic values of the equation: λ1 =

1 4 4 (b

 – a4 ) +

1 1 7 7 (b

, – a7 )(b – a)

λ2 =

1 4 4 (b

 – a4 ) –

1◦ . Solution with λ ≠ λ1,2 : y(x) = f (x) + λ(A1 x3 + A2 ),

1 1 7 7 (b

. – a7 )(b – a)

308

LINEAR EQUATIONS OF THE SECOND KIND WITH CONSTANT LIMITS OF INTEGRATION

where the constants A1 and A2 are given by     f1 – λ 14 f1 ∆4 – f2 ∆1 f2 – λ 14 f2 ∆4 – 17 f1 ∆7   A1 = 2  1 2 1 , A2 = 2  1 2 1 , λ 16 ∆4 – 7 ∆1 ∆7 – 12 λ∆4 + 1 λ 16 ∆4 – 7 ∆1 ∆7 – 12 λ∆4 + 1 b b f (x) dx, f2 = x3 f (x) dx, ∆n = bn – an . f1 = a

a



2 . Solution with λ = λ1 ≠ λ2 and f1 = f2 = 0:

$ b 7 – a7 , 7(b – a)

y1 (x) = x3 +

y(x) = f (x) + Cy1 (x),

where C is an arbitrary constant and y1 (x) is an eigenfunction of the equation corresponding to the characteristic value λ1 . 3◦ . Solution with λ = λ2 ≠ λ1 and f1 = f2 = 0:

$ b 7 – a7 , 7(b – a)

y2 (x) = x3 –

y(x) = f (x) + Cy2 (x),

where C is an arbitrary constant and y2 (x) is an eigenfunction of the equation corresponding to the characteristic value λ2 . 4◦ . The equation has no multiple characteristic values. 21.

b

y(x) – λ

(x3 – t3 )y(t) dt = f (x).

a

The characteristic values of the equation: λ1,2 = ± 

1 1 4 4 (a

.

– b4 )2 – 17 (a7 – b7 )(b – a)

1◦ . Solution with λ ≠ λ1,2 : y(x) = f (x) + λ(A1 x3 + A2 ), where the constants A1 and A2 are given by   f1 + λ 14 f1 ∆4 – f2 ∆1  A1 = 2  1 , 1 λ 7 ∆1 ∆7 – 16 ∆24 + 1 b f (x) dx, f2 = f1 = a

  –f2 + λ 14 f2 ∆4 – 17 f1 ∆7  A2 = 2  1 , 1 λ 7 ∆1 ∆7 – 16 ∆24 + 1 b

x3 f (x) dx,

∆n = bn – an .

a

2◦ . Solution with λ = λ1 ≠ λ2 and f1 = f2 = 0: y(x) = f (x) + Cy1 (x),

y1 (x) = x3 +

4 – λ1 (b4 – a4 ) , 4λ1 (b – a)

where C is an arbitrary constant and y1 (x) is an eigenfunction of the equation corresponding to the characteristic value λ1 . 3◦ . The solution with λ = λ2 ≠ λ1 and f1 = f2 = 0 is given by the formulas of item 2◦ in which one must replace λ1 and y1 (x) by λ2 and y2 (x), respectively. 4◦ . The equation has no multiple characteristic values.

309

4.1. EQUATIONS WHOSE KERNELS CONTAIN POWER-LAW FUNCTIONS

22.

b

y(x) – λ

(Ax3 + Bt3 )y(t) dt = f (x).

a

The characteristic values of the equation:

λ1,2 =

1 4 (A

+ B)∆4 ±

 

2AB

1 4 2 2 16 (A – B) ∆4 + 7 AB∆1 ∆7  1 1 2 16 ∆4 – 7 ∆1 ∆7

,

∆n = bn – an .

1◦ . Solution with λ ≠ λ1,2 : y(x) = f (x) + λ(A1 x3 + A2 ), where the constants A1 and A2 are given by   Af1 – ABλ 14 f1 ∆4 – f2 ∆1  , 1 ABλ2 16 ∆24 – 17 ∆1 ∆7 – 14 λ(A + B)∆4 + 1   Bf2 – ABλ 14 f2 ∆4 – 17 f1 ∆7 1 2 1  , A2 = ABλ2 16 ∆4 – 7 ∆1 ∆7 – 14 λ(A + B)∆4 + 1 b b f (x) dx, f2 = x3 f (x) dx. f1 = A1 =



a

a

2◦ . Solution with λ = λ1 ≠ λ2 and f1 = f2 = 0: y(x) = f (x) + Cy1 (x),

y1 (x) = x3 +

4 – λ1 A(b4 – a4 ) , 4λ1 A(b – a)

where C is an arbitrary constant and y1 (x) is an eigenfunction of the equation corresponding to the characteristic value λ1 . 3◦ . The solution with λ = λ2 ≠ λ1 and f1 = f2 = 0 is given by the formulas of item 2◦ in which one must replace λ1 and y1 (x) by λ2 and y2 (x), respectively. 4◦ . Solution with λ = λ1,2 = λ∗ and f1 = f2 = 0 provided that A ≠ ±B, where λ∗ = 8 is the double characteristic value: (A + B)(b4 – a4 ) y(x) = f (x) + Cy∗ (x),

y∗ (x) = x3 –

(A – B)(b4 – a4 ) , 8A(b – a)

where C is an arbitrary constant and y∗ (x) is an eigenfunction of the equation corresponding to λ∗ . The equation has no multiple characteristic values if A = ±B. 23.

b

y(x) – λ

(xt2 – t3 )y(t) dt = f (x).

a

This is a special case of equation 4.9.8 with A = 0, B = 1, and h(t) = t2 . Solution: y(x) = f (x) + λ(A1 + A2 x), where A1 and A2 are the constants determined by the formulas presented in 4.9.8.

310

LINEAR EQUATIONS OF THE SECOND KIND WITH CONSTANT LIMITS OF INTEGRATION

24.

b

y(x) – λ

(Bxt2 + Ct3 )y(t) dt = f (x).

a

This is a special case of equation 4.9.9 with A = 0 and h(t) = t2 . Solution: y(x) = f (x) + λ(A1 + A2 x), where A1 and A2 are the constants determined by the formulas presented in 4.9.9. 25.

b

y(x) – λ

(Ax2 t + Bxt2 )y(t) dt = f (x).

a

This is a special case of equation 4.9.17 with g(x) = x2 and h(x) = x. Solution: y(x) = f (x) + λ(A1 x2 + A2 x), where A1 and A2 are the constants determined by the formulas presented in 4.9.17. 26.

b

y(x) – λ

(Ax3 + Bxt2 )y(t) dt = f (x).

a

This is a special case of equation 4.9.18 with g1 (x) = x3 , h1 (t) = A, g2 (x) = x, and h2 (t) = Bt2 . Solution: y(x) = f (x) + λ(A1 x3 + A2 x), where A1 and A2 are the constants determined by the formulas presented in 4.9.18. 27.

b

y(x) – λ

(Ax3 + Bx2 t + Cx2 + D)y(t) dt = f (x).

a

This is a special case of equation 4.9.18 with g1 (x) = Ax3 + Cx2 + D, h1 (t) = 1, g2 (x) = x2 , and h2 (t) = Bt. Solution: y(x) = f (x) + λ[A1 (Ax3 + Cx2 + D) + A2 x2 ], where A1 and A2 are the constants determined by the formulas presented in 4.9.18. 28.

b

y(x) – λ

(Axt2 + Bt3 + Ct2 + D)y(t) dt = f (x).

a

This is a special case of equation 4.9.18 with g1 (x) = x, h1 (t) = At2 , g2 (x) = 1, and h2 (t) = Bt3 + Ct2 + D. Solution: y(x) = f (x) + λ(A1 x + A2 ), where A1 and A2 are the constants determined by the formulas presented in 4.9.18. 29.

b

y(x) – λ

(x – t)3 y(t) dt = f (x).

a

This is a special case of equation 4.9.19 with g(x) = x, h(t) = –t, and m = 3. 30.

b

y(x) – λ

(Ax + Bt)3 y(t) dt = f (x).

a

This is a special case of equation 4.9.19 with g(x) = Ax, h(t) = Bt, and m = 3.

4.1. EQUATIONS WHOSE KERNELS CONTAIN POWER-LAW FUNCTIONS

311

4.1-4. Kernels Containing Higher-Order Polynomials in x and t. 31.

b

y(x) – λ

(xn + tn )y(t) dt = f (x),

n = 1, 2, . . .

a

The characteristic values of the equation: 1 √ , λ1,2 = ∆n ± ∆0 ∆2n 1◦ . Solution with λ ≠ λ1,2 :

where ∆n =

1 (bn+1 – an+1 ). n+1

y(x) = f (x) + λ(A1 xn + A2 ), where the constants A1 and A2 are given by f1 – λ(f1 ∆n – f2 ∆0 ) f2 – λ(f2 ∆n – f1 ∆2n ) , A2 = 2 2 , A1 = 2 2 λ (∆n – ∆0 ∆2n ) – 2λ∆n + 1 λ (∆n – ∆0 ∆2n ) – 2λ∆n + 1 b b 1 (bn+1 – an+1 ). f1 = f (x) dx, f2 = xn f (x) dx, ∆n = n + 1 a a 2◦ . Solution with λ = λ1 ≠ λ2 and f1 = f2 = 0:  y1 (x) = xn + ∆2n /∆0 , y(x) = f (x) + Cy1 (x), where C is an arbitrary constant and y1 (x) is an eigenfunction of the equation corresponding to the characteristic value λ1 . 3◦ . Solution with λ = λ2 ≠ λ1 and f1 = f2 = 0: y(x) = f (x) + Cy2 (x),

y2 (x) = xn –

 ∆2n /∆0 ,

where C is an arbitrary constant and y2 (x) is an eigenfunction of the equation corresponding to the characteristic value λ2 .

32.

4◦ . The equation has no multiple characteristic values. b y(x) – λ (xn – tn )y(t) dt = f (x), n = 1, 2, . . . a

The characteristic values of the equation:  –1/2 1 1 n+1 n+1 2 2n+1 2n+1 (b (b – a ) – – a )(b – a) . λ1,2 = ± (n + 1)2 2n + 1 1◦ . Solution with λ ≠ λ1,2 : y(x) = f (x) + λ(A1 xn + A2 ), where the constants A1 and A2 are given by f1 + λ(f1 ∆n – f2 ∆0 ) –f2 + λ(f2 ∆n – f1 ∆2n ) , A2 = , A1 = 2 2 λ (∆0 ∆2n – ∆n ) + 1 λ2 (∆0 ∆2n – ∆2n ) + 1 b b 1 f1 = f (x) dx, f2 = xn f (x) dx, ∆n = (bn+1 – an+1 ). n + 1 a a 2◦ . Solution with λ = λ1 ≠ λ2 and f1 = f2 = 0: 1 – λ1 ∆n y1 (x) = xn + , y(x) = f (x) + Cy1 (x), λ1 ∆0 where C is an arbitrary constant and y1 (x) is an eigenfunction of the equation corresponding to the characteristic value λ1 . 3◦ . The solution with λ = λ2 ≠ λ1 and f1 = f2 = 0 is given by the formulas of item 2◦ in which one must replace λ1 and y1 (x) by λ2 and y2 (x), respectively. 4◦ . The equation has no multiple characteristic values.

312

LINEAR EQUATIONS OF THE SECOND KIND WITH CONSTANT LIMITS OF INTEGRATION

33.

b

y(x) – λ

(Axn + Btn )y(t) dt = f (x),

n = 1, 2, . . .

a

The characteristic values of the equation:  (A + B)∆n ± (A – B)2 ∆2n + 4AB∆0 ∆2n , λ1,2 = 2AB(∆2n – ∆0 ∆2n )

∆n =

1 (bn+1 – an+1 ). n+1

1◦ . Solution with λ ≠ λ1,2 : y(x) = f (x) + λ(A1 xn + A2 ), where the constants A1 and A2 are given by Af1 – ABλ(f1 ∆n – f2 ∆0 ) , ABλ2 (∆2n – ∆0 ∆2n ) – (A + B)λ∆n + 1 Bf2 – ABλ(f2 ∆n – f1 ∆2n ) A2 = , 2 ABλ (∆2n – ∆0 ∆2n ) – (A + B)λ∆n + 1 b b f1 = f (x) dx, f2 = xn f (x) dx.

A1 =

a

a



2 . Solution with λ = λ1 ≠ λ2 and f1 = f2 = 0: y(x) = f (x) + Cy1 (x),

y1 (x) = xn +

1 – Aλ1 ∆n , Aλ1 ∆0

where C is an arbitrary constant and y1 (x) is an eigenfunction of the equation corresponding to the characteristic value λ1 . 3◦ . The solution with λ = λ2 ≠ λ1 and f1 = f2 = 0 is given by the formulas of item 2◦ in which one must replace λ1 and y1 (x) by λ2 and y2 (x), respectively. 4◦ . Solution with λ = λ1,2 = λ∗ and f1 = f2 = 0 provided that A ≠ ±B, where the characteristic value λ∗ = 2/[(A + B)∆n ] is double: y(x) = f (x) + Cy∗ (x),

y∗ (x) = xn –

(A – B)∆n . 2A∆0

Here C is an arbitrary constant and y∗ (x) is an eigenfunction of the equation corresponding to λ∗ . The equation has no multiple characteristic values if A = ±B. 34.

b

y(x) – λ

(x – t)tm y(t) dt = f (x),

m = 1, 2, . . .

a

This is a special case of equation 4.9.8 with A = 0, B = 1, and h(t) = tm . Solution: y(x) = f (x) + λ(A1 + A2 x), where A1 and A2 are the constants determined by the formulas presented in 4.9.8. 35.

b

y(x) – λ

(x – t)xm y(t) dt = f (x),

m = 1, 2, . . .

a

This is a special case of equation 4.9.10 with A = 0, B = 1, and h(x) = xm . Solution: y(x) = f (x) + λ(A1 xm+1 + A2 xm ), where A1 and A2 are the constants determined by the formulas presented in 4.9.10.

313

4.1. EQUATIONS WHOSE KERNELS CONTAIN POWER-LAW FUNCTIONS

36.

b

y(x) – λ

(Axm+1 + Bxm t + Cxm + D)y(t) dt = f (x),

m = 1, 2, . . .

a

This is a special case of equation 4.9.18 with g1 (x) = Axm+1 +Cxm +D, h1 (t) = 1, g2 (x) = xm , and h2 (t) = Bt. Solution: y(x) = f (x) + λ[A1 (Axm+1 + Cxm + D) + A2 xm ], where A1 and A2 are the constants determined by the formulas presented in 4.9.18. 37.

b

y(x) – λ

(Axtm + Btm+1 + Ctm + D)y(t) dt = f (x),

m = 1, 2, . . .

a

This is a special case of equation 4.9.18 with g1 (x) = x, h1 (t) = Atm , g2 (x) = 1, and h2 (t) = Btm+1 + Ctm + D. Solution: y(x) = f (x) + λ(A1 x + A2 ), where A1 and A2 are the constants determined by the formulas presented in 4.9.18. 38.

b

y(x) – λ

(Axn tn + Bxm tm )y(t) dt = f (x),

n, m = 1, 2, . . . ,

a

n ≠ m.

This is a special case of equation 4.9.14 with g(x) = xn and h(t) = tm . Solution: y(x) = f (x) + λ(A1 xn + A2 xm ), where A1 and A2 are the constants determined by the formulas presented in 4.9.14. 39.

b

y(x) – λ

(Axn tm + Bxm tn )y(t) dt = f (x),

n, m = 1, 2, . . . ,

a

n ≠ m.

This is a special case of equation 4.9.17 with g(x) = xn and h(t) = tm . Solution: y(x) = f (x) + λ(A1 xn + A2 xm ), where A1 and A2 are the constants determined by the formulas presented in 4.9.17. 40.

b

y(x) – λ

(x – t)m y(t) dt = f (x),

m = 1, 2, . . .

a

This is a special case of equation 4.9.19 with g(x) = x and h(t) = –t. 41.

b

y(x) – λ

(Ax + Bt)m y(t) dt = f (x),

m = 1, 2, . . .

a

This is a special case of equation 4.9.19 with g(x) = Ax and h(t) = Bt. 42.

y(x) + A

b a

|x – t|tk y(t) dt = f (x).

This is a special case of equation 4.9.36 with g(t) = Atk . Solving the integral equation   is reduced to solving the ordinary differential equation yxx + 2Axk y = fxx (x), the general solution of which can be expressed via Bessel functions or modified Bessel functions (the boundary conditions are given in 4.9.36).

314 43.

LINEAR EQUATIONS OF THE SECOND KIND WITH CONSTANT LIMITS OF INTEGRATION

y(x) + A

b

a

|x – t|2n+1 y(t) dt = f (x),

n = 0, 1, 2, . . .

Let us remove the modulus in the integrand:



x

b

(x – t)2n+1 y(t) dt + A

y(x) + A a

(t – x)2n+1 y(t) dt = f (x).

(1)

x

The k-fold differentiation of (1) with respect to x yields



x

b

(x – t)2n+1–k y(t) dt + (–1)k ABk

yx(k) (x) + ABk a

(t – x)2n+1–k y(t) dt = fx(k) (x), x

Bk = (2n + 1)(2n) . . . (2n + 2 – k),

(2)

k = 1, 2, . . . , 2n + 1.

Differentiating (2) with k = 2n + 1, we arrive at the following linear nonhomogeneous differential equation with constant coefficients for y = y(x): yx(2n+2) + 2(2n + 1)! Ay = fx(2n+2) (x).

(3)

Equation (3) must satisfy the initial conditions which can be obtained by setting x = a in (1) and (2):

b

(t – a)2n+1 y(t) dt = f (a),

y(a) + A a

yx(k) (a)

k

+ (–1) ABk

(4)

b

(t – a)

2n+1–k

y(t) dt =

fx(k) (a),

k = 1, 2, . . . , 2n + 1.

a

These conditions can be reduced to a more habitual form containing no integrals. To this end, y must be expressed from equation (3) in terms of yx(2n+2) and fx(2n+2) and substituted into (4), and then one must integrate the resulting expressions by parts (sufficiently many times).

4.1-5. Kernels Containing Rational Functions. 44.

b



y(x) – λ

1 x

a

+

1 t

 y(t) dt = f (x).

This is a special case of equation 4.9.2 with g(x) = 1/x. Solution:   A1 y(x) = f (x) + λ + A2 , x where A1 and A2 are the constants determined by the formulas presented in 4.9.2. 45.

b

y(x) – λ a



 1 1 – y(t) dt = f (x). x t

This is a special case of equation 4.9.3 with g(x) = 1/x. Solution:   A1 y(x) = f (x) + λ + A2 , x where A1 and A2 are the constants determined by the formulas presented in 4.9.3.

4.1. EQUATIONS WHOSE KERNELS CONTAIN POWER-LAW FUNCTIONS

b 46.

y(x) – λ a

47.

48.

49.

50.

51.

 A B + y(t) dt = f (x). x t

This is a special case of equation 4.9.4 with g(x) = 1/x. Solution:   A1 + A2 , y(x) = f (x) + λ x where A1 and A2 are the constants determined by the formulas presented in 4.9.4.  b A B + y(t) dt = f (x). y(x) – λ x+α t+β a B A and h(t) = . This is a special case of equation 4.9.5 with g(x) = x+α t+β Solution:   A y(x) = f (x) + λ A1 + A2 , x+α where A1 and A2 are the constants determined by the formulas presented in 4.9.5.  b x t – y(t) dt = f (x). y(x) – λ t x a This is a special case of equation 4.9.16 with g(x) = x and h(t) = 1/t. Solution:   A2 , y(x) = f (x) + λ A1 x + x where A1 and A2 are the constants determined by the formulas presented in 4.9.16.  b Ax Bt + y(t) dt = f (x). y(x) – λ t x a This is a special case of equation 4.9.17 with g(x) = x and h(t) = 1/t. Solution:   A2 , y(x) = f (x) + λ A1 x + x where A1 and A2 are the constants determined by the formulas presented in 4.9.17.  b x+α t+α A +B y(t) dt = f (x). y(x) – λ t+β x+β a 1 . This is a special case of equation 4.9.17 with g(x) = x + α and h(t) = t+β Solution:   A2 y(x) = f (x) + λ A1 (x + α) + , x+β where A1 and A2 are the constants determined by the formulas presented in 4.9.17.  b (x + α)n (t + α)n A y(t) dt = f (x), n, m = 0, 1, 2, . . . +B y(x) – λ (t + β)m (x + β)m a This is a special case of equation 4.9.17 with g(x) = (x + α)n and h(t) = (t + β)–m . Solution:   A2 y(x) = f (x) + λ A1 (x + α)n + , (x + β)m where A1 and A2 are the constants determined by the formulas presented in 4.9.17.

315

316

LINEAR EQUATIONS OF THE SECOND KIND WITH CONSTANT LIMITS OF INTEGRATION

52.

y(x) – λ 1



y(t) dt = f (x), x+t

1 ≤ x < ∞,



Solution:



y(x) = 0 ∞

–∞ < πλ < 1.

τ sinh(πτ ) F (τ ) P 1 (x) dτ , cosh(πτ ) – πλ – 2 +iτ

F (τ ) =

f (x)P– 1 +iτ (x) dx, 2   1 where Pν (x) = F –ν, ν + 1, 1; 2 (1 – x) is the Legendre spherical function of the first kind, for which the integral representation cos(τ s) ds 2 α √ P– 1 +iτ (cosh α) = (α ≥ 0) 2 π 0 2(cosh α – cosh s) can be used. 1

Reference: V. A. Ditkin and A. P. Prudnikov (1965).

53.

2

2

(x + b )y(x) =

λ





a3 y(t)

dt. π –∞ a2 + (x – t)2 This equation is encountered in atomic and nuclear physics. We seek the solution in the form ∞  Am x y(x) = . x2 + (am + b)2

(1)

m=0

The coefficients Am obey the equations   m + 2b + λAm–1 = 0, mAm a

∞ 

Am = 0.

(2)

m=0

Using the first equation of (2) to express all Am via A0 (A0 can be chosen arbitrarily), substituting the result into the second equation of (2), and dividing by A0 , we obtain ∞  (–λ)m 1 = 0. (3) 1+ m! (1 + 2b/a)(2 + 2b/a) . . . (m + 2b/a) m=1

It follows from the definitions of the Bessel functions of the first kind that equation (3) can be rewritten in the form  √  λ–b/a J2b/a 2 λ = 0. (4) In this sort of problem, a and λ are usually assumed to be given and b, which is proportional to the system energy, to be unknown. The quantity b can be determined by tables of zeros of Bessel functions. In some cases, b and a are given and λ is unknown. Reference: I. Sneddon (1995).

54.

1

y(x) – y(t)

dt = λy(x). |x – t| The characteristic values of the equation:  1 1 , λn = 2 1 + + · · · + 2 n The eigenfunctions of the equation: –1

where n = 1, 2, . . .

yn (x) = Pn (x), where n = 1, 2, . . . 1 dn 2 (x – 1)n are the Legendre polynomials. Here Pn (x) = n! 2n dxn Reference: A. G. Petrov (1986).

4.1. EQUATIONS WHOSE KERNELS CONTAIN POWER-LAW FUNCTIONS

317

4.1-6. Kernels Containing Arbitrary Powers. 55.

b

y(x) – λ

(x – t)tµ y(t) dt = f (x).

a

This is a special case of equation 4.9.8 with A = 0, B = 1, and h(t) = tµ . Solution: y(x) = f (x) + λ(A1 + A2 x), where A1 and A2 are the constants determined by the formulas presented in 4.9.8. 56.

b

y(x) – λ

(x – t)xν y(t) dt = f (x).

a

This is a special case of equation 4.9.10 with A = 0, B = 1, and h(x) = xν . Solution: y(x) = f (x) + λ(E1 xν+1 + E2 xν ), where E1 and E2 are the constants determined by the formulas presented in 4.9.10. 57.

b

y(x) – λ

(xµ – tµ )y(t) dt = f (x).

a

This is a special case of equation 4.9.3 with g(x) = xµ . Solution: y(x) = f (x) + λ(A1 xµ + A2 ), where A1 and A2 are the constants determined by the formulas presented in 4.9.3. 58.

b

y(x) – λ

(Axν + Btν )tµ y(t) dt = f (x).

a

This is a special case of equation 4.9.6 with g(x) = xν and h(t) = tµ . Solution: y(x) = f (x) + λ(A1 xν + A2 ), where A1 and A2 are the constants determined by the formulas presented in 4.9.6. 59.

b

y(x) – λ

(Dxν + Etµ )xγ y(t) dt = f (x).

a

This is a special case of equation 4.9.18 with g1 (x) = xν+γ , h1 (t) = D, g2 (x) = xγ , and h2 (t) = Etµ . Solution: y(x) = f (x) + λ(A1 xν+γ + A2 xγ ), where A1 and A2 are the constants determined by the formulas presented in 4.9.18. 60.

b

y(x) – λ

(Axν tµ + Bxγ tδ )y(t) dt = f (x).

a

This is a special case of equation 4.9.18 with g1 (x) = xν , h1 (t) = Atµ , g2 (x) = xγ , and h2 (t) = Btδ . Solution: y(x) = f (x) + λ(A1 xν + A2 xγ ), where A1 and A2 are the constants determined by the formulas presented in 4.9.18.

318

LINEAR EQUATIONS OF THE SECOND KIND WITH CONSTANT LIMITS OF INTEGRATION

61.

b

y(x) – λ

(A + Bxtµ + Ctµ+1 )y(t) dt = f (x).

a

This is a special case of equation 4.9.9 with h(t) = tµ . Solution: y(x) = f (x) + λ(A1 + A2 x), where A1 and A2 are the constants determined by the formulas presented in 4.9.9. 62.

b

y(x) – λ

(Atα + Bxβ tµ + Ctµ+γ )y(t) dt = f (x).

a

This is a special case of equation 4.9.18 with g1 (x) = 1, h1 (t) = Atα + Ctµ+γ , g2 (x) = xβ , and h2 (t) = Btµ . Solution: y(x) = f (x) + λ(A1 + A2 xβ ), where A1 and A2 are the constants determined by the formulas presented in 4.9.18. 63.

b

y(x) – λ

(Axα tγ + Bxβ tγ + Cxµ tν )y(t) dt = f (x).

a

This is a special case of equation 4.9.18 with g1 (x) = Axα + Bxβ , h1 (t) = tγ , g2 (x) = xµ , and h2 (t) = Ctν . Solution: y(x) = f (x) + λ[A1 (Axα + Bxβ ) + A2 xµ ], where A1 and A2 are the constants determined by the formulas presented in 4.9.18. 64.

b

 A

y(x) – λ a

 (x + p1 )β (x + p2 )µ y(t) dt = f (x). + B (t + q1 )γ (t + q2 )δ

This is a special case of equation 4.9.18 with g1 (x) = (x + p1 )β , h1 (t) = A(t + q1 )–γ , g2 (x) = (x + p2 )µ , and h2 (t) = B(t + q2 )–δ . Solution: 

y(x) = f (x) + λ A1 (x + p1 )β + A2 (x + p2 )µ , where A1 and A2 are the constants determined by the formulas presented in 4.9.18. 65.

b

y(x) – λ a

  xµ + a xγ + c A ν +B δ y(t) dt = f (x). t +b t +d

A This is a special case of equation 4.9.18 with g1 (x) = xµ + a, h1 (t) = ν , g2 (x) = xγ + c, t +b B . and h2 (t) = δ t +d Solution: y(x) = f (x) + λ[A1 (xµ + a) + A2 (xγ + c)], where A1 and A2 are the constants determined by the formulas presented in 4.9.18.

319

4.1. EQUATIONS WHOSE KERNELS CONTAIN POWER-LAW FUNCTIONS

4.1-7. Singular Equations. In this subsection, all singular integrals are understood in the sense of the Cauchy principal value. 66.

Ay(x) +

B



π

1

–1

y(t) dt t–x

= f (x),

–1 < x < 1.

Here A and B are real numbers such that B ≠ 0, A ± B ≠ 0, and A2 + B 2 = 1. 1◦ . The solution bounded at the endpoints: B 1 g(x) f (t) dt , y(x) = Af (x) – π –1 g(t) t – x

g(x) = (1 + x)α (1 – x)1–α ,

(1)

where α is the solution of the trigonometric equation A + B cot(πα) = 0

(2)

1

on the interval 0 < α < 1. This solution y(x) exists if and only if –1

f (t) dt = 0. g(t)



2 . The solution bounded at the endpoint x = 1 and unbounded at the endpoint x = –1: B 1 g(x) f (t) dt , g(x) = (1 + x)α (1 – x)–α , y(x) = Af (x) – (3) π –1 g(t) t – x where α is the solution of the trigonometric equation (2) on the interval –1 < α < 0. 3◦ . The solution unbounded at the endpoints: B 1 g(x) f (t) dt + Cg(x), y(x) = Af (x) – π –1 g(t) t – x

g(x) = (1 + x)α (1 – x)–1–α ,

(4)

where C is an arbitrary constant and α is the solution of the trigonometric equation (2) on the interval –1 < α < 0. References: N. I. Muskhelishvili (1992), I. K. Lifanov, L. N. Poltavskii, and G. M. Vainikko (2004, pp. 6–7).

67.



y(x) – λ

y(t) dt

–∞

t–x

= f (x).   ∞ f (t) dt 1 . f (x) + λ y(x) = 1 + π 2 λ2 –∞ t – x

Solution:

Reference: M. L. Krasnov (1975).

68.



 1 1 – y(t) dt = f (x), 0 < x < 1. y(x) – λ t – x x + t – 2xt 0 Tricomi’s equation. Solution:     1 α t (1 – x)α 1 1 C(1 – x)β 1 y(x) = – f (t) dt + f (x) + , α α 1 + λ2 π 2 t – x x + t – 2xt x1+β 0 x (1 – t) βπ 2 = λπ (–2 < β < 0), α = arctan(λπ) (–1 < α < 1), tan π 2

1

where C is an arbitrary constant. References: P. P. Zabreyko, A. I. Koshelev, et al. (1975), F. G. Tricomi (1985).

320

LINEAR EQUATIONS OF THE SECOND KIND WITH CONSTANT LIMITS OF INTEGRATION

69.

1



1+t



2 n+2



1



1

y(t) dt = f (x), 0 < x < 1. 1+x t – x 1 – xt Tricomi–Gellerstedt equation. 1 In the class of functions y(x) for which integrals –1 |y(x)| ln |x ± 1| dx are finite the unique solution of the equation has the form y(x) + λ



–1

    1  1 1 1 1 – t2 n+2 1 y(x) = – f (t) dt . f (x) – λ 1 + λ2 π 2 1 – x2 t – x 1 – xt –1 Reference: S. G. Mikhlin (1967).

4.2. Equations Whose Kernels Contain Exponential Functions 4.2-1. Kernels Containing Exponential Functions. 1.

b

y(x) – λ

(eβx + eβt )y(t) dt = f (x).

a

The characteristic values of the equation: λ1,2 =

eβb – eβa ±



β 1 2 β(b

. – a)(e2βb – e2βa )

1◦ . Solution with λ ≠ λ1,2 : y(x) = f (x) + λ(A1 eβx + A2 ), where the constants A1 and A2 are given by 

f1 – λ f1 ∆β – (b – a)f2 f2 – λ(f2 ∆β – f1 ∆2β )

  A1 = 2 2 , A2 = 2 2 , λ ∆β – (b – a)∆2β – 2λ∆β + 1 λ ∆β – (b – a)∆2β – 2λ∆β + 1 b b 1 f1 = f (x) dx, f2 = f (x)eβx dx, ∆β = (eβb – eβa ). β a a 2◦ . Solution with λ = λ1 ≠ λ2 and f1 = f2 = 0: y(x) = f (x) + Cy1 (x),

y1 (x) = e

$ βx

e2βb – e2βa , 2β(b – a)

+

where C is an arbitrary constant and y1 (x) is an eigenfunction of the equation corresponding to the characteristic value λ1 . 3◦ . Solution with λ = λ2 ≠ λ1 and f1 = f2 = 0: y(x) = f (x) + Cy2 (x),

y2 (x) = e

$ βx



e2βb – e2βa , 2β(b – a)

where C is an arbitrary constant and y2 (x) is an eigenfunction of the equation corresponding to the characteristic value λ2 . 4◦ . The equation has no multiple characteristic values.

321

4.2. EQUATIONS WHOSE KERNELS CONTAIN EXPONENTIAL FUNCTIONS

2.

b

y(x) – λ

(eβx – eβt )y(t) dt = f (x).

a

The characteristic values of the equation: β λ1,2 = ±  . (eβb – eβa )2 – 12 β(b – a)(e2βb – e2βa ) 1◦ . Solution with λ ≠ λ1,2 : y(x) = f (x) + λ(A1 eβx + A2 ), where the constants A1 and A2 are given by 

f1 + λ f1 ∆β – (b – a)f2 –f2 + λ(f2 ∆β – f1 ∆2β )   , A2 = 2 , A1 = 2 λ (b – a)∆2β – ∆2β + 1 λ (b – a)∆2β – ∆2β + 1 b b 1 f1 = f (x) dx, f2 = f (x)eβx dx, ∆β = (eβb – eβa ). β a a 2◦ . Solution with λ = λ1 ≠ λ2 and f1 = f2 = 0: 1 – λ1 ∆β , λ1 (b – a)

y1 (x) = eβx +

y(x) = f (x) + Cy1 (x),

where C is an arbitrary constant and y1 (x) is an eigenfunction of the equation corresponding to the characteristic value λ1 . 3◦ . The solution with λ = λ2 ≠ λ1 and f1 = f2 = 0 is given by the formulas of item 2◦ in which one must replace λ1 and y1 (x) by λ2 and y2 (x), respectively.

3.

4◦ . The equation has no multiple characteristic values. b y(x) – λ (Aeβx + Beβt )y(t) dt = f (x). a

The characteristic values of the equation:  (A + B)∆β ± (A – B)2 ∆2β + 4AB(b – a)∆2β 

, λ1,2 = 2AB ∆2β – (b – a)∆2β

∆β =

1 βb βa (e – e ). β

1◦ . Solution with λ ≠ λ1,2 : y(x) = f (x) + λ(A1 eβx + A2 ), where the constants A1 and A2 are given by



Af1 – ABλ f1 ∆β – (b – a)f2

 , A1 = ABλ2 ∆2β – (b – a)∆2β – (A + B)λ∆β + 1

A2 =

Bf – ABλ(f2 ∆β – f1 ∆2β )

22  , ∆β – (b – a)∆2β – (A + B)λ∆β + 1 b b f1 = f (x) dx, f2 = f (x)eβx dx. ABλ2

a

a

2◦ . Solution with λ = λ1 ≠ λ2 and f1 = f2 = 0: y(x) = f (x) + Cy1 (x),

y1 (x) = eβx +

1 – Aλ1 ∆β , A(b – a)λ1

where C is an arbitrary constant and y1 (x) is an eigenfunction of the equation corresponding to the characteristic value λ1 .

322

LINEAR EQUATIONS OF THE SECOND KIND WITH CONSTANT LIMITS OF INTEGRATION

3◦ . The solution with λ = λ2 ≠ λ1 and f1 = f2 = 0 is given by the formulas of item 2◦ in which one must replace λ1 and y1 (x) by λ2 and y2 (x), respectively. 4◦ . Solution with λ = λ1,2 = λ∗ and f1 = f2 = 0 provided that A ≠ ±B, where the characteristic 2 value λ∗ = is double: (A + B)∆β (A – B)∆β , 2A(b – a) where C is an arbitrary constant and y∗ (x) is an eigenfunction of the equation corresponding to λ∗ . The equation has no multiple characteristic values if A = ±B. b

β(x–t)  Ae y(x) – λ + B y(t) dt = f (x). y(x) = f (x) + Cy∗ (x),

4.

y∗ (x) = eβx –

a

This is a special case of equation 4.9.18 with g1 (x) = eβx , h1 (t) = Ae–βt , g2 (x) = 1, and h2 (t) = B. Solution: y(x) = f (x) + λ(A1 eβx + A2 ),

5.

where A1 and A2 are the constants determined by the formulas presented in 4.9.18. b

βx+µt  y(x) – λ Ae + Be(β+µ)t y(t) dt = f (x). a

This is a special case of equation 4.9.6 with g(x) = eβx and h(t) = eµt . Solution: y(x) = f (x) + λ(A1 eβx + A2 ),

6.

where A1 and A2 are the constants determined by the formulas presented in 4.9.6. b

α(x+t)  y(x) – λ Ae + Beβ(x+t) y(t) dt = f (x). a

This is a special case of equation 4.9.14 with g(x) = eαx and h(t) = eβt . Solution: y(x) = f (x) + λ(A1 eαx + A2 eβx ),

7.

where A1 and A2 are the constants determined by the formulas presented in 4.9.14. b  αx+βt  y(x) – λ Ae + Beβx+αt y(t) dt = f (x). a

This is a special case of equation 4.9.17 with g(x) = eαx and h(t) = eβt . Solution: y(x) = f (x) + λ(A1 eαx + A2 eβx ),

8.

where A1 and A2 are the constants determined by the formulas presented in 4.9.17. b

 y(x) – λ De(γ+µ)x + Eeνt+µx y(t) dt = f (x). a

This is a special case of equation 4.9.18 with g1 (x) = e(γ+µ)x , h1 (t) = D, g2 (x) = eµx , and h2 (t) = Eeνt . Solution: y(x) = f (x) + λ[A1 e(γ+µ)x + A2 eµx ], where A1 and A2 are the constants determined by the formulas presented in 4.9.18.

4.2. EQUATIONS WHOSE KERNELS CONTAIN EXPONENTIAL FUNCTIONS

9.

b

y(x) – λ

323

(Aeαx+βt + Beγx+δt )y(t) dt = f (x).

a

This is a special case of equation 4.9.18 with g1 (x) = eαx , h1 (t) = Aeβt , g2 (x) = eγx , and h2 (t) = Beδt . Solution: y(x) = f (x) + λ(A1 eαx + A2 eγx ), where A1 and A2 are the constants determined by the formulas presented in 4.9.18. 10.

b

y(x) – λ

 n

a

Ak e

γk (x–t)

 y(t) dt = f (x).

k=1

This is a special case of equation 4.9.20 with gk (x) = eγk x and hk (t) = Ak e–γk t . 11.

y(x) –

1



2



e–|x–t| y(t) dt = Aeµx ,

0 < µ < 1.

0

Solution:

 y(x) = C(1 + x) + Aµ–2 (µ2 – 1)eµx – µ + 1 ,

where C is an arbitrary constant. Reference: P. P. Zabreyko, A. I. Koshelev, et al. (1975).

12.



y(x) + λ

e–|x–t| y(t) dt = f (x).

0

Solution:



∞  √  exp – 1 + 2λ |x – t| f (t) dt y(x) = f (x) – √ 1 + 2λ 0   ∞

√  λ+1 √ + 1– exp – 1 + 2λ (x + t) f (t) dt, 1 + 2λ 0

λ

where λ > – 12 . Reference: F. D. Gakhov and Yu. I. Cherskii (1978).

13.



y(x) – λ

e–|x–t| y(t) dt = 0,

λ > 0.

–∞

The Lalesco–Picard equation. Solution:  √   √  ⎧ C exp x 1 – 2λ + C exp –x 1 – 2λ for 0 < λ < 12 , 1 2 ⎪ ⎨ y(x) = C1 + C2 x for λ = 12 , ⎪  √   √  ⎩ C1 cos x 2λ – 1 + C2 sin x 2λ – 1 for λ > 12 , where C1 and C2 are arbitrary constants. Reference: M. L. Krasnov, A. I. Kisilev, and G. I. Makarenko (1971).

324

LINEAR EQUATIONS OF THE SECOND KIND WITH CONSTANT LIMITS OF INTEGRATION

14.



y(x) + λ

e–|x–t| y(t) dt = f (x).

–∞

1◦ . Solution with λ > – 12 : λ y(x) = f (x) – √ 1 + 2λ ◦

2 . If λ ≤

– 12 ,





 √  exp – 1 + 2λ |x – t| f (t) dt.

–∞

for the equation to be solvable the conditions ∞ ∞ f (x) cos(ax) dx = 0, f (x) sin(ax) dx = 0, –∞

–∞

√ where a = –1 – 2λ, must be satisfied. In this case, the solution has the form a2 + 1 ∞ y(x) = f (x) – sin(at)f (x + t) dt, (–∞ < x < ∞). 2a 0 In the class of solutions not belonging to L2 (–∞, ∞), the homogeneous equation (with f (x) ≡ 0) has a nontrivial solution. In this case, the general solution of the corresponding nonhomogeneous equation with λ ≤ – 21 has the form a2 + 1 ∞ y(x) = C1 sin(ax) + C2 cos(ax) + f (x) – sin(a|x – t|)f (t) dt. 4a –∞ Reference: F. D. Gakhov and Yu. I. Cherskii (1978).

15.

y(x) + A

b

eλ|x–t| y(t) dt = f (x).

a

This is a special case of equation 4.9.37 with g(t) = A. 1◦ . The function y = y(x) obeys the following second-order linear nonhomogeneous ordinary differential equation with constant coefficients:   yxx + λ(2A – λ)y = fxx (x) – λ2 f (x).

(1)

The boundary conditions for (1) have the form (see 4.9.37) yx (a) + λy(a) = fx (a) + λf (a), yx (b) – λy(b) = fx (b) – λf (b).

(2)

Equation (1) under the boundary conditions (2) determines the solution of the original integral equation. 2◦ . For λ(2A – λ) < 0, the general solution of equation (1) is given by 2Aλ x y(x) = C1 cosh(kx) + C2 sinh(kx) + f (x) – sinh[k(x – t)] f (t) dt, k a  k = λ(λ – 2A), where C1 and C2 are arbitrary constants. For λ(2A – λ) > 0, the general solution of equation (1) is given by 2Aλ x y(x) = C1 cos(kx) + C2 sin(kx) + f (x) – sin[k(x – t)] f (t) dt, k a  k = λ(2A – λ). For λ = 2A, the general solution of equation (1) is given by x y(x) = C1 + C2 x + f (x) – 4A2 (x – t)f (t) dt. a

The constants C1 and C2 in solutions (3)–(5) are determined by conditions (2).

(3)

(4)

(5)

325

4.2. EQUATIONS WHOSE KERNELS CONTAIN EXPONENTIAL FUNCTIONS

3◦ . In the special case a = 0 and λ(2A – λ) > 0, the solution of the integral equation is given by formula (4) with A(kIc – λIs ) A(kIc – λIs ) λ , C2 = – , (λ – A) sin µ – k cos µ k (λ – A) sin µ – k cos µ b b  k = λ(2A – λ), µ = bk, Is = sin[k(b – t)]f (t) dt, Ic = cos[k(b – t)]f (t) dt. C1 =

0

16.

b

y(x) +

 n

a

0

 Ak exp(λk |x – t|) y(t) dt = f (x),

–∞ < a < b < ∞.

k=1

1◦ . Let us remove the modulus in the kth summand of the integrand: b x b Ik (x) = exp(λk |x – t|)y(t) dt = exp[λk (x – t)]y(t) dt + exp[λk (t – x)]y(t) dt. (1) a

a

x

Differentiating (1) with respect to x twice yields x  Ik = λk exp[λk (x – t)]y(t) dt – λk a

Ik

= 2λk y(x) +

λ2k

b

exp[λk (t – x)]y(t) dt,

x



x

exp[λk (x – t)]y(t) dt + a

(2)

b

λ2k

exp[λk (t – x)]y(t) dt, x

where the primes denote the derivatives with respect to x. By comparing formulas (1) and (2), we find the relation between Ik and Ik : Ik = 2λk y(x) + λ2k Ik ,

Ik = Ik (x).

(3)

2◦ . With the aid of (1), the integral equation can be rewritten in the form y(x) +

n 

Ak Ik = f (x).

(4)

k=1

Differentiating (4) with respect to x twice and taking into account (3), we find that  yxx (x) + σn y(x) +

n 

 Ak λ2k Ik = fxx (x),

n 

Ak λk .

(5)

 Ak (λ2k – λ2n )Ik = fxx (x) – λ2n f (x).

(6)

k=1

σn = 2

k=1

Eliminating the integral In from (4) and (5) yields  yxx (x) + (σn – λ2n )y(x) +

n–1  k=1

Differentiating (6) with respect to x twice and eliminating In–1 from the resulting equation with the aid of (6), we obtain a similar equation whose left-hand side is a second-order linear n–2  differential operator (acting on y) with constant coefficients plus the sum Bk Ik . If we k=1

successively eliminate In–2 , In–3 , . . . , with the aid of double differentiation, then we finally arrive at a linear nonhomogeneous ordinary differential equation of order 2n with constant coefficients. 3◦ . The boundary conditions for y(x) can be found by setting x = a in the integral equation and all its derivatives. (Alternatively, these conditions can be found by setting x = a and x = b in the integral equation and all its derivatives obtained by means of double differentiation.)

326

LINEAR EQUATIONS OF THE SECOND KIND WITH CONSTANT LIMITS OF INTEGRATION

4.2-2. Kernels Containing Power-Law and Exponential Functions. 17.

b

y(x) – λ

(x – t)eγt y(t) dt = f (x).

a

This is a special case of equation 4.9.8 with A = 0, B = 1, and h(t) = eγt . 18.

b

y(x) – λ

(x – t)eγx y(t) dt = f (x).

a

This is a special case of equation 4.9.10 with A = 0, B = 1, and h(x) = eγx . 19.

b

y(x) – λ

(x – t)eγx+µt y(t) dt = f (x).

a

This is a special case of equation 4.9.18 with g1 (x) = xeγx , h1 (t) = eµt , g2 (x) = eγx , and h2 (t) = –teµt . 20.

b

y(x) – λ

[A + (Bx + Ct)eγx ]y(t) dt = f (x).

a

This is a special case of equation 4.9.11 with h(x) = eγx . 21.

b

y(x) – λ

(x2 + t2 )eγ(x+t) y(t) dt = f (x).

0

This is a special case of equation 4.9.15 with g(x) = x2 eγx and h(t) = eγt . 22.

b

y(x) – λ

(x2 – t2 )eγ(x–t) y(t) dt = f (x).

0

This is a special case of equation 4.9.18 with g1 (x) = x2 eγx , h1 (t) = e–γt , g2 (x) = eγx , and h2 (t) = –t2 e–γt . 23.

b

y(x) – λ

(Axn + Btn )eαx+βt y(t) dt = f (x),

n = 1, 2, . . .

0

This is a special case of equation 4.9.18 with g1 (x) = xn eαx , h1 (t) = Aeβt , g2 (x) = eαx , and h2 (t) = Btn eβt . 24.

b

y(x) – λ a

 n

νk αk x+βk t

Ak t e

 y(t) dt = f (x),

n = 1, 2, . . .

k=1

This is a special case of equation 4.9.20 with gk (x) = eαk x and hk (t) = Ak tνk eβk t . 25.

b

y(x) – λ a

 n

νk αk x+βk t

Ak x e

 y(t) dt = f (x),

n = 1, 2, . . .

k=1

This is a special case of equation 4.9.20 with gk (x) = Ak xνk eαk x and hk (t) = eβk t . 26.

b

y(x) – λ

(x – t)n eγ(x–t) y(t) dt = f (x),

a

This is a special case of equation 4.9.20.

n = 1, 2, . . .

4.3. EQUATIONS WHOSE KERNELS CONTAIN HYPERBOLIC FUNCTIONS

27.

b

y(x) – λ

(x – t)n eαx+βt y(t) dt = f (x),

327

n = 1, 2, . . .

a

This is a special case of equation 4.9.20. 28.

b

y(x) – λ

(Ax + Bt)n eαx+βt y(t) dt = f (x),

n = 1, 2, . . .

a

This is a special case of equation 4.9.20. 29.

y(x) + A

b

teλ|x–t| y(t) dt = f (x).

a

This is a special case of equation 4.9.37 with g(t) = At. The solution of the integral equation can be written via the Bessel functions (or modified Bessel functions) of order 1/3. 30.



y(x) +

(a + b|x – t|) exp(–|x – t|)y(t) dt = f (x). 0

Let the biquadratic polynomial P (k) = k 4 + 2(a – b + 1)k 2 + 2a + 2b + 1 have no real roots and let k = α + iβ be a root of the equation P (k) = 0 such that α > 0 and β > 0. In this case, the solution has the form ∞ y(x) = f (x) + ρ exp(–β|x – t|) cos(θ + α|x – t|)f (t) dt 0 [α + (β – 1)2 ]2 ∞ exp[–β(x + t)] cos[α(x – t)]f (t) dt + 4α2 β 0 ∞ R + exp[–β(x + t)] cos[ψ + α(x + t)]f (t) dt, 4α2 0 where the parameters ρ, θ, R, and ψ are determined from the system of algebraic equations obtained by separating real and imaginary parts in the relations ρeiθ =

µ , β – iα

Reiψ =

(β – 1 – iα)4 . 8α2 (β – iα)

Reference: F. D. Gakhov and Yu. I. Cherskii (1978).

4.3. Equations Whose Kernels Contain Hyperbolic Functions 4.3-1. Kernels Containing Hyperbolic Cosine. 1.

b

y(x) – λ

cosh(βx)y(t) dt = f (x). a

This is a special case of equation 4.9.1 with g(x) = cosh(βx) and h(t) = 1. 2.

b

cosh(βt)y(t) dt = f (x).

y(x) – λ a

This is a special case of equation 4.9.1 with g(x) = 1 and h(t) = cosh(βt).

328

LINEAR EQUATIONS OF THE SECOND KIND WITH CONSTANT LIMITS OF INTEGRATION

3.

b

y(x) – λ

cosh[β(x – t)]y(t) dt = f (x). a

This is a special case of equation 4.9.13 with g(x) = cosh(βx) and h(t) = sinh(βt). Solution: 

y(x) = f (x) + λ A1 cosh(βx) + A2 sinh(βx) , where A1 and A2 are the constants determined by the formulas presented in 4.9.13. 4.

b

y(x) – λ

cosh[β(x + t)]y(t) dt = f (x). a

This is a special case of equation 4.9.12 with g(x) = cosh(βx) and h(t) = sinh(βt). Solution:

 y(x) = f (x) + λ A1 cosh(βx) + A2 sinh(βx) , where A1 and A2 are the constants determined by the formulas presented in 4.9.12. 5.

b

y(x) – λ a

 n

Ak cosh[βk (x – t)] y(t) dt = f (x),

n = 1, 2, . . .

k=1

This is a special case of equation 4.9.20. 6.

b

y(x) – λ a

cosh(βx) cosh(βt)

y(t) dt = f (x).

This is a special case of equation 4.9.1 with g(x) = cosh(βx) and h(t) = 7.

b

y(x) – λ a

cosh(βt) cosh(βx)

y(t) dt = f (x).

This is a special case of equation 4.9.1 with g(x) = 8.

b

y(x) – λ

1 . cosh(βt)

1 and h(t) = cosh(βt). cosh(βx)

coshk (βx) coshm (µt)y(t) dt = f (x).

a

This is a special case of equation 4.9.1 with g(x) = coshk (βx) and h(t) = coshm (µt). 9.

b

y(x) – λ

tk coshm (βx)y(t) dt = f (x).

a

This is a special case of equation 4.9.1 with g(x) = coshm (βx) and h(t) = tk . 10.

b

y(x) – λ

xk coshm (βt)y(t) dt = f (x).

a

This is a special case of equation 4.9.1 with g(x) = xk and h(t) = coshm (βt). 11.

b

[A + B(x – t) cosh(βx)]y(t) dt = f (x).

y(x) – λ a

This is a special case of equation 4.9.10 with h(x) = cosh(βx).

329

4.3. EQUATIONS WHOSE KERNELS CONTAIN HYPERBOLIC FUNCTIONS

12.

b

y(x) – λ

[A + B(x – t) cosh(βt)]y(t) dt = f (x). a

This is a special case of equation 4.9.8 with h(t) = cosh(βt). 13.



y(x) + λ –∞

y(t) dt cosh[b(x – t)]

= f (x).

Solution with b > π|λ|: 2λb y(x) = f (x) – √ 2 b – π 2 λ2





–∞

sinh[2k(x – t)] f (t) dt, sinh[2b(x – t)]

k=

 πλ  b arccos . π b

Reference: F. D. Gakhov and Yu. I. Cherskii (1978).

4.3-2. Kernels Containing Hyperbolic Sine. 14.

b

y(x) – λ

sinh(βx)y(t) dt = f (x). a

This is a special case of equation 4.9.1 with g(x) = sinh(βx) and h(t) = 1. 15.

b

sinh(βt)y(t) dt = f (x).

y(x) – λ a

This is a special case of equation 4.9.1 with g(x) = 1 and h(t) = sinh(βt). 16.

b

sinh[β(x – t)]y(t) dt = f (x).

y(x) – λ a

This is a special case of equation 4.9.16 with g(x) = sinh(βx) and h(t) = cosh(βt). Solution:

 y(x) = f (x) + λ A1 sinh(βx) + A2 cosh(βx) , where A1 and A2 are the constants determined by the formulas presented in 4.9.16. 17.

b

sinh[β(x + t)]y(t) dt = f (x).

y(x) – λ a

This is a special case of equation 4.9.15 with g(x) = sinh(βx) and h(t) = cosh(βt). Solution:

 y(x) = f (x) + λ A1 sinh(βx) + A2 cosh(βx) , where A1 and A2 are the constants determined by the formulas presented in 4.9.15. 18.

b

y(x) – λ a

 n

Ak sinh[βk (x – t)] y(t) dt = f (x),

n = 1, 2, . . .

k=1

This is a special case of equation 4.9.20. 19.

b

y(x) – λ a

sinh(βx) y(t) dt = f (x). sinh(βt)

This is a special case of equation 4.9.1 with g(x) = sinh(βx) and h(t) =

1 . sinh(βt)

330

LINEAR EQUATIONS OF THE SECOND KIND WITH CONSTANT LIMITS OF INTEGRATION

20.

b

y(x) – λ a

sinh(βt) sinh(βx)

y(t) dt = f (x).

This is a special case of equation 4.9.1 with g(x) =

b

1 and h(t) = sinh (βt). sinh(βx)

sinhk (βx) sinhm (µt)y(t) dt = f (x).

21.

y(x) – λ

22.

This is a special case of equation 4.9.1 with g(x) = sinhk (βx) and h(t) = sinhm (µt). b y(x) – λ tk sinhm (βx)y(t) dt = f (x).

a

a

23.

This is a special case of equation 4.9.1 with g(x) = sinhm (βx) and h(t) = tk . b y(x) – λ xk sinhm (βt)y(t) dt = f (x).

24.

This is a special case of equation 4.9.1 with g(x) = xk and h(t) = sinhm (βt). b y(x) – λ [A + B(x – t) sinh(βt)]y(t) dt = f (x).

25.

This is a special case of equation 4.9.8 with h(t) = sinh(βt). b y(x) – λ [A + B(x – t) sinh(βx)]y(t) dt = f (x).

26.

This is a special case of equation 4.9.10 with h(x) = sinh(βx). b y(x) + A sinh(λ|x – t|)y(t) dt = f (x).

a

a

a

a

This is a special case of equation 4.9.38 with g(t) = A. 1◦ . The function y = y(x) obeys the following second-order linear nonhomogeneous ordinary differential equation with constant coefficients:   yxx + λ(2A – λ)y = fxx (x) – λ2 f (x).

(1)

The boundary conditions for (1) have the form (see 4.9.38) sinh[λ(b – a)]ϕx (b) – λ cosh[λ(b – a)]ϕ(b) = λϕ(a),

ϕ(x) = y(x) – f (x). (2) sinh[λ(b – a)]ϕx (a) + λ cosh[λ(b – a)]ϕ(a) = –λϕ(b), Equation (1) under the boundary conditions (2) determines the solution of the original integral equation. 2◦ . For λ(2A – λ) = –k 2 < 0, the general solution of equation (1) is given by 2Aλ x sinh[k(x – t)]f (t) dt, y(x) = C1 cosh(kx) + C2 sinh(kx) + f (x) – k a where C1 and C2 are arbitrary constants. For λ(2A – λ) = k 2 > 0, the general solution of equation (1) is given by 2Aλ x sin[k(x – t)]f (t) dt. y(x) = C1 cos(kx) + C2 sin(kx) + f (x) – k a For λ = 2A, the general solution of equation (1) is given by x (x – t)f (t) dt. y(x) = C1 + C2 x + f (x) – 4A2 a

The constants C1 and C2 in solutions (3)–(5) are determined by conditions (2).

(3)

(4)

(5)

331

4.3. EQUATIONS WHOSE KERNELS CONTAIN HYPERBOLIC FUNCTIONS

27.

y(x) + A

b

t sinh(λ|x – t|)y(t) dt = f (x).

a

28.

This is a special case of equation 4.9.38 with g(t) = At. The solution of the integral equation can be written via the Bessel functions (or modified Bessel functions) of order 1/3. b y(x) + A sinh3 (λ|x – t|)y(t) dt = f (x). a

Using the formula sinh3 β = 14 sinh 3β – 34 sinh β, we arrive at an equation of the form 4.3.29 with n = 2: b

1  3 y(x) + 4 A sinh(3λ|x – t|) – 4 A sinh(λ|x – t|) y(t) dt = f (x). a

29.

y(x) +

n b  a

 Ak sinh(λk |x – t|) y(t) dt = f (x),

–∞ < a < b < ∞.

k=1

1◦ . Let us remove the modulus in the kth summand of the integrand: b x b sinh(λk |x – t|)y(t) dt = sinh[λk (x – t)]y(t) dt + sinh[λk (t – x)]y(t) dt. (1) Ik (x) = a

a

x

Differentiating (1) with respect to x twice yields x  cosh[λk (x – t)]y(t) dt – λk Ik = λk a

Ik

= 2λk y(x) +

λ2k

b

cosh[λk (t – x)]y(t) dt, x



x

sinh[λk (x – t)]y(t) dt + a

(2)

b

λ2k

sinh[λk (t – x)]y(t) dt, x

where the primes denote the derivatives with respect to x. By comparing formulas (1) and (2), we find the relation between Ik and Ik : Ik = 2λk y(x) + λ2k Ik ,

Ik = Ik (x).

(3)



2 . With the aid of (1), the integral equation can be rewritten in the form n  Ak Ik = f (x). y(x) +

(4)

k=1

Differentiating (4) with respect to x twice and taking into account (3), we find that n n     yxx (x) + σn y(x) + Ak λ2k Ik = fxx (x), σn = 2 Ak λk . k=1

(5)

k=1

Eliminating the integral In from (4) and (5) yields  (x) + (σn – λ2n )y(x) + yxx

n–1 

 Ak (λ2k – λ2n )Ik = fxx (x) – λ2n f (x).

(6)

k=1

Differentiating (6) with respect to x twice and eliminating In–1 from the resulting equation with the aid of (6), we obtain a similar equation whose left-hand side is a second-order linear n–2  differential operator (acting on y) with constant coefficients plus the sum Bk Ik . If we k=1

successively eliminate In–2 , In–3 , . . . , with the aid of double differentiation, then we finally arrive at a linear nonhomogeneous ordinary differential equation of order 2n with constant coefficients. 3◦ . The boundary conditions for y(x) can be found by setting x = a in the integral equation and its derivatives. (Alternatively, these conditions can be found by setting x = a and x = b in the integral equation and all its derivatives obtained by means of double differentiation.)

332

LINEAR EQUATIONS OF THE SECOND KIND WITH CONSTANT LIMITS OF INTEGRATION

4.3-3. Kernels Containing Hyperbolic Tangent. 30.

b

y(x) – λ

tanh(βx)y(t) dt = f (x). a

This is a special case of equation 4.9.1 with g(x) = tanh(βx) and h(t) = 1. 31.

b

y(x) – λ

tanh(βt)y(t) dt = f (x). a

This is a special case of equation 4.9.1 with g(x) = 1 and h(t) = tanh(βt). 32.

b

y(x) – λ

[A tanh(βx) + B tanh(βt)]y(t) dt = f (x). a

This is a special case of equation 4.9.4 with g(x) = tanh(βx). 33.

b

y(x) – λ a

tanh(βx) y(t) dt = f (x). tanh(βt)

This is a special case of equation 4.9.1 with g(x) = tanh(βx) and h(t) = 34.

b

y(x) – λ a

tanh(βt) tanh(βx)

y(t) dt = f (x).

This is a special case of equation 4.9.1 with g(x) = 35.

b

y(x) – λ

1 . tanh(βt)

1 and h(t) = tanh(βt). tanh(βx)

tanhk (βx) tanhm (µt)y(t) dt = f (x).

a

This is a special case of equation 4.9.1 with g(x) = tanhk (βx) and h(t) = tanhm (µt). 36.

b

y(x) – λ

tk tanhm (βx)y(t) dt = f (x).

a

This is a special case of equation 4.9.1 with g(x) = tanhm (βx) and h(t) = tk . 37.

b

y(x) – λ

xk tanhm (βt)y(t) dt = f (x).

a

This is a special case of equation 4.9.1 with g(x) = xk and h(t) = tanhm (βt). 38.

b

y(x) – λ

[A + B(x – t) tanh(βt)]y(t) dt = f (x). a

This is a special case of equation 4.9.8 with h(t) = tanh(βt). 39.

b

y(x) – λ

[A + B(x – t) tanh(βx)]y(t) dt = f (x). a

This is a special case of equation 4.9.10 with h(x) = tanh(βx).

333

4.3. EQUATIONS WHOSE KERNELS CONTAIN HYPERBOLIC FUNCTIONS

4.3-4. Kernels Containing Hyperbolic Cotangent. 40.

b

y(x) – λ

coth(βx)y(t) dt = f (x). a

This is a special case of equation 4.9.1 with g(x) = coth(βx) and h(t) = 1. 41.

b

y(x) – λ

coth(βt)y(t) dt = f (x). a

This is a special case of equation 4.9.1 with g(x) = 1 and h(t) = coth(βt). 42.

b

y(x) – λ

[A coth(βx) + B coth(βt)]y(t) dt = f (x). a

This is a special case of equation 4.9.4 with g(x) = coth(βx). 43.

b

y(x) – λ a

coth(βx) y(t) dt = f (x). coth(βt)

This is a special case of equation 4.9.1 with g(x) = coth(βx) and h(t) = 44.

b

y(x) – λ a

coth(βt) coth(βx)

y(t) dt = f (x).

This is a special case of equation 4.9.1 with g(x) = 45.

b

y(x) – λ

1 . coth(βt)

1 and h(t) = coth(βt). coth(βx)

cothk (βx) cothm (µt)y(t) dt = f (x).

a

This is a special case of equation 4.9.1 with g(x) = cothk (βx) and h(t) = cothm (µt). 46.

b

y(x) – λ

tk cothm (βx)y(t) dt = f (x).

a

This is a special case of equation 4.9.1 with g(x) = cothm (βx) and h(t) = tk . 47.

b

y(x) – λ

xk cothm (βt)y(t) dt = f (x).

a

This is a special case of equation 4.9.1 with g(x) = xk and h(t) = cothm (βt). 48.

b

y(x) – λ

[A + B(x – t) coth(βt)]y(t) dt = f (x). a

This is a special case of equation 4.9.8 with h(t) = coth(βt). 49.

b

y(x) – λ

[A + B(x – t) coth(βx)]y(t) dt = f (x). a

This is a special case of equation 4.9.10 with h(x) = coth(βx).

334

LINEAR EQUATIONS OF THE SECOND KIND WITH CONSTANT LIMITS OF INTEGRATION

4.3-5. Kernels Containing Combination of Hyperbolic Functions. 50.

b

y(x) – λ

coshk (βx) sinhm (µt)y(t) dt = f (x).

a

This is a special case of equation 4.9.1 with g(x) = coshk (βx) and h(t) = sinhm (µt). 51.

b

y(x) – λ

[A sinh(αx) cosh(βt) + B sinh(γx) cosh(δt)]y(t) dt = f (x). a

This is a special case of equation 4.9.18 with g1 (x) = sinh(αx), h1 (t) = A cosh(βt), g2 (x) = sinh(γx), and h2 (t) = B cosh(δt). 52.

b

y(x) – λ

tanhk (γx) cothm (µt)y(t) dt = f (x).

a

This is a special case of equation 4.9.1 with g(x) = tanhk (γx) and h(t) = cothm (µt). 53.

b

y(x) – λ

[A tanh(αx) coth(βt) + B tanh(γx) coth(δt)]y(t) dt = f (x). a

This is a special case of equation 4.9.18 with g1 (x) = tanh(αx), h1 (t) = A coth(βt), g2 (x) = tanh(γx), and h2 (t) = B coth(δt).

4.4. Equations Whose Kernels Contain Logarithmic Functions 4.4-1. Kernels Containing Logarithmic Functions. 1.

b

y(x) – λ

ln(γx)y(t) dt = f (x). a

This is a special case of equation 4.9.1 with g(x) = ln(γx) and h(t) = 1. 2.

b

y(x) – λ

ln(γt)y(t) dt = f (x). a

This is a special case of equation 4.9.1 with g(x) = 1 and h(t) = ln(γt). 3.

b

y(x) – λ

(ln x – ln t)y(t) dt = f (x). a

This is a special case of equation 4.9.3 with g(x) = ln x. 4.

b

y(x) – λ a

ln(γx) ln(γt)

y(t) dt = f (x).

This is a special case of equation 4.9.1 with g(x) = ln(γx) and h(t) = 5.

b

y(x) – λ a

ln(γt) ln(γx)

1 . ln(γt)

y(t) dt = f (x).

This is a special case of equation 4.9.1 with g(x) =

1 and h(t) = ln(γt). ln(γx)

4.5. EQUATIONS WHOSE KERNELS CONTAIN TRIGONOMETRIC FUNCTIONS

6.

b

y(x) – λ

335

lnk (γx) lnm (µt)y(t) dt = f (x).

a

This is a special case of equation 4.9.1 with g(x) = lnk (γx) and h(t) = lnm (µt).

4.4-2. Kernels Containing Power-Law and Logarithmic Functions. 7.

b

y(x) – λ

tk lnm (γx)y(t) dt = f (x).

a

This is a special case of equation 4.9.1 with g(x) = lnm (γx) and h(t) = tk . 8.

b

y(x) – λ

xk lnm (γt)y(t) dt = f (x).

a

This is a special case of equation 4.9.1 with g(x) = xk and h(t) = lnm (γt). 9.

b

[A + B(x – t) ln(γt)]y(t) dt = f (x).

y(x) – λ a

This is a special case of equation 4.9.8 with h(t) = ln(γt). 10.

b

[A + B(x – t) ln(γx)]y(t) dt = f (x).

y(x) – λ a

This is a special case of equation 4.9.10 with h(x) = ln(γx). 11.

b

[A + (Bx + Ct) ln(γt)]y(t) dt = f (x).

y(x) – λ a

This is a special case of equation 4.9.9 with h(t) = ln(γt). 12.

b

[A + (Bx + Ct) ln(γx)]y(t) dt = f (x).

y(x) – λ a

This is a special case of equation 4.9.11 with h(x) = ln(γx). 13.

b

y(x) – λ

[Atn lnm (βx) + Bxk lnl (γt)]y(t) dt = f (x).

a

This is a special case of equation 4.9.18 with g1 (x) = lnm (βx), h1 (t) = Atn , g2 (x) = xk , and h2 (t) = B lnl (γt).

4.5. Equations Whose Kernels Contain Trigonometric Functions 4.5-1. Kernels Containing Cosine. 1.

b

y(x) – λ

cos(βx)y(t) dt = f (x). a

This is a special case of equation 4.9.1 with g(x) = cos(βx) and h(t) = 1.

336

LINEAR EQUATIONS OF THE SECOND KIND WITH CONSTANT LIMITS OF INTEGRATION

2.

b

y(x) – λ

cos(βt)y(t) dt = f (x). a

3.

This is a special case of equation 4.9.1 with g(x) = 1 and h(t) = cos(βt). b cos[β(x – t)]y(t) dt = f (x). y(x) – λ a

This is a special case of equation 4.9.12 with g(x) = cos(βx) and h(t) = sin(βt). Solution:

 y(x) = f (x) + λ A1 cos(βx) + A2 sin(βx) ,

4.

where A1 and A2 are the constants determined by the formulas presented in 4.9.12. b cos[β(x + t)]y(t) dt = f (x). y(x) – λ a

This is a special case of equation 4.9.13 with g(x) = cos(βx) and h(t) = sin(βt). Solution:

 y(x) = f (x) + λ A1 cos(βx) + A2 sin(βx) ,

5.

where A1 and A2 are the constants determined by the formulas presented in 4.9.13. ∞ cos(xt)y(t) dt = 0. y(x) – λ 0

 Characteristic values: λ = ± 2/π. For the characteristic values, the integral equation has infinitely many linearly independent eigenfunctions.  Eigenfunctions for λ = + 2/π have the form  ∞ 2 f (t) cos(xt) dt, (1) y+ (x) = f (x) + π 0 where f = f (x) is any continuous  function absolutely integrable on the interval [0, ∞). Eigenfunctions for λ = – 2/π have the form  ∞ 2 f (t) cos(xt) dt, y– (x) = f (x) – π 0 where f = f (x) is any continuous function absolutely integrable on the interval [0, ∞). In particular, from (1) and (2) with f (x) = e–ax we obtain   2 2 a –ax , + for λ = + y+ (x) = e 2 2 π a +x π   2 2 a , y– (x) = e–ax – for λ = – 2 2 π a +x π where a is any positive number. Reference: M. L. Krasnov, A. I. Kisilev, and G. I. Makarenko (1971).

6.

y(x) – λ



cos(xt)y(t) dt = f (x). 0

Solution:  where λ ≠ ± 2/π.

λ f (x) y(x) = π 2 + 1– 2λ 1 – π2 λ2





cos(xt)f (t) dt, 0

Reference: M. L. Krasnov, A. I. Kisilev, and G. I. Makarenko (1971).

(2)

337

4.5. EQUATIONS WHOSE KERNELS CONTAIN TRIGONOMETRIC FUNCTIONS

7.

b

y(x) – λ a

 n

Ak cos[βk (x – t)] y(t) dt = f (x),

n = 1, 2, . . .

k=1

This equation can be reduced to a special case of equation 4.9.20; the formula cos[β(x – t)] = cos(βx) cos(βt) + sin(βx) sin(βt) must be used. 8.

b

y(x) – λ a

cos(βx) y(t) dt = f (x). cos(βt)

This is a special case of equation 4.9.1 with g(x) = cos(βx) and h(t) = 9.

b

y(x) – λ a

cos(βt) cos(βx)

y(t) dt = f (x).

This is a special case of equation 4.9.1 with g(x) = 10.

b

y(x) – λ

1 . cos(βt)

1 and h(t) = cos(βt). cos(βx)

cosk (βx) cosm (µt)y(t) dt = f (x).

a

This is a special case of equation 4.9.1 with g(x) = cosk (βx) and h(t) = cosm (µt). 11.

b

y(x) – λ

tk cosm (βx)y(t) dt = f (x).

a

This is a special case of equation 4.9.1 with g(x) = cosm (βx) and h(t) = tk . 12.

b

y(x) – λ

xk cosm (βt)y(t) dt = f (x).

a

This is a special case of equation 4.9.1 with g(x) = xk and h(t) = cosm (βt). 13.

b

[A + B(x – t) cos(βx)]y(t) dt = f (x).

y(x) – λ a

This is a special case of equation 4.9.10 with h(x) = cos(βx). 14.

b

[A + B(x – t) cos(βt)]y(t) dt = f (x).

y(x) – λ a

This is a special case of equation 4.9.8 with h(t) = cos(βt). 4.5-2. Kernels Containing Sine. 15.

b

y(x) – λ

sin(βx)y(t) dt = f (x). a

This is a special case of equation 4.9.1 with g(x) = sin(βx) and h(t) = 1. 16.

b

sin(βt)y(t) dt = f (x).

y(x) – λ a

This is a special case of equation 4.9.1 with g(x) = 1 and h(t) = sin(βt).

338

LINEAR EQUATIONS OF THE SECOND KIND WITH CONSTANT LIMITS OF INTEGRATION

17.

b

y(x) – λ

sin[β(x – t)]y(t) dt = f (x). a

This is a special case of equation 4.9.16 with g(x) = sin(βx) and h(t) = cos(βt). Solution:

 y(x) = f (x) + λ A1 sin(βx) + A2 cos(βx) , where A1 and A2 are the constants determined by the formulas presented in 4.9.16. 18.

b

sin[β(x + t)]y(t) dt = f (x).

y(x) – λ a

This is a special case of equation 4.9.15 with g(x) = sin(βx) and h(t) = cos(βt). Solution:

 y(x) = f (x) + λ A1 sin(βx) + A2 cos(βx) , where A1 and A2 are the constants determined by the formulas presented in 4.9.15. Example. Let us consider the case of a = 0, b = π, β = 1 in detail. 1◦ . Solution for λ ≠ ±2/π:

(1)

y(x) = f (x) + λA sin x + λB cos x, where A=

f1 + 1–

1 πλf2 2 1 2 2 π λ 4

,

B=

1 πλf1 + f2 2 1 – 14 π 2 λ2

,



π

f1 =

f (t) cos t dt,

π

f2 =

0

f (t) sin t dt.

(2)

0

2◦ . Characteristic values and normed eigenfunctions of the homogeneous equation for f (x) ≡ 0 are given by the formulas 2 1 λ1 = – , y1 (x) = √ (sin x – cos x); π π 2 1 λ2 = , y2 (x) = √ (sin x + cos x). π π 3◦ . If λ = –2/π and f1 = f2 (values of f1 and f2 can be found using formulas of Item 1◦ ). In this case the solution can be obtained with the help of formula (1) in which B = f1 – A where A is an arbitrary constant. If λ = 2/π and f1 = –f2 then the solution can be found using formula (1) in which B = A – f1 where A is an arbitrary constant. 4◦ . If λ = –2/π and f1 ≠ f2 or λ = 2/π and f1 ≠ –f2 , then the equation under consideration has no solutions.

19.

y(x) – λ



sin(xt)y(t) dt = 0. 0

 Characteristic values: λ = ± 2/π. For the characteristic values, the integral equation has infinitely many linearly independent eigenfunctions.  Eigenfunctions for λ = + 2/π have the form  y+ (x) = f (x) +

2 π





f (t) sin(xt) dt, 0

where f = f (x) is any continuous  function absolutely integrable on the interval [0, ∞). Eigenfunctions for λ = – 2/π have the form  y– (x) = f (x) –

2 π





f (t) sin(xt) dt, 0

where f = f (x) is any continuous function absolutely integrable on the interval [0, ∞). Reference: M. L. Krasnov, A. I. Kisilev, and G. I. Makarenko (1971).

339

4.5. EQUATIONS WHOSE KERNELS CONTAIN TRIGONOMETRIC FUNCTIONS

20.



sin(xt)y(t) dt = f (x).

y(x) – λ 0

Solution:

 where λ ≠ ± 2/π.

f (x) λ y(x) = π 2 + 1– 2λ 1 – π2 λ2





sin(xt)f (t) dt, 0

References: M. L. Krasnov, A. I. Kisilev, and G. I. Makarenko (1971), F. D. Gakhov and Yu. I. Cherskii (1978).

21.

b

y(x) – λ a

 n

Ak sin[βk (x – t)] y(t) dt = f (x),

n = 1, 2, . . .

k=1

This equation can be reduced to a special case of equation 4.9.20; the formula sin[β(x – t)] = sin(βx) cos(βt) – sin(βt) cos(βx) must be used. 22.

b

y(x) – λ a

sin(βx) sin(βt)

y(t) dt = f (x).

This is a special case of equation 4.9.1 with g(x) = sin(βx) and h(t) = 23.

b

y(x) – λ a

sin(βt) sin(βx)

y(t) dt = f (x).

This is a special case of equation 4.9.1 with g(x) = 24.

b

y(x) – λ

1 . sin(βt)

1 and h(t) = sin (βt). sin(βx)

sink (βx) sinm (µt)y(t) dt = f (x).

a

This is a special case of equation 4.9.1 with g(x) = sink (βx) and h(t) = sinm (µt). 25.

b

y(x) – λ

tk sinm (βx)y(t) dt = f (x).

a

This is a special case of equation 4.9.1 with g(x) = sinm (βx) and h(t) = tk . 26.

b

y(x) – λ

xk sinm (βt)y(t) dt = f (x).

a

This is a special case of equation 4.9.1 with g(x) = xk and h(t) = sinm (βt). 27.

b

y(x) – λ

[A + B(x – t) sin(βt)]y(t) dt = f (x). a

This is a special case of equation 4.9.8 with h(t) = sin(βt). 28.

b

y(x) – λ

[A + B(x – t) sin(βx)]y(t) dt = f (x). a

This is a special case of equation 4.9.10 with h(x) = sin(βx).

340 29.

LINEAR EQUATIONS OF THE SECOND KIND WITH CONSTANT LIMITS OF INTEGRATION

y(x) + A

b

sin(λ|x – t|)y(t) dt = f (x).

a

This is a special case of equation 4.9.39 with g(t) = A. 1◦ . The function y = y(x) obeys the following second-order linear nonhomogeneous ordinary differential equation with constant coefficients:   yxx + λ(2A + λ)y = fxx (x) + λ2 f (x).

(1)

The boundary conditions for (1) have the form (see 4.9.39) sin[λ(b – a)]ϕx (b) – λ cos[λ(b – a)]ϕ(b) = λϕ(a),

ϕ(x) = y(x) – f (x).

sin[λ(b – a)]ϕx (a) + λ cos[λ(b – a)]ϕ(a) = –λϕ(b),

(2)

Equation (1) under the boundary conditions (2) determines the solution of the original integral equation. 2◦ . For λ(2A + λ) = –k 2 < 0, the general solution of equation (1) is given by y(x) = C1 cosh(kx) + C2 sinh(kx) + f (x) –

2Aλ k



x

sinh[k(x – t)] f (t) dt,

(3)

a

where C1 and C2 are arbitrary constants. For λ(2A + λ) = k 2 > 0, the general solution of equation (1) is given by 2Aλ y(x) = C1 cos(kx) + C2 sin(kx) + f (x) – k



x

sin[k(x – t)] f (t) dt.

(4)

a

For λ = 2A, the general solution of equation (1) is given by 2

y(x) = C1 + C2 x + f (x) + 4A

x

(x – t)f (t) dt.

(5)

a

The constants C1 and C2 in solutions (3)–(5) are determined by conditions (2). 30.

y(x) + A

b

t sin(λ|x – t|)y(t) dt = f (x).

a

This is a special case of equation 4.9.39 with g(t) = At. The solution of the integral equation can be written via the Bessel functions (or modified Bessel functions) of order 1/3. 31.

y(x) + A

b

sin3 (λ|x – t|)y(t) dt = f (x).

a

Using the formula sin3 β = – 14 sin 3β + with n = 2: y(x) + a

b

3 4

sin β, we arrive at an equation of the form 4.5.32

 – 14 A sin(3λ|x – t|) + 34 A sin(λ|x – t|) y(t) dt = f (x).

341

4.5. EQUATIONS WHOSE KERNELS CONTAIN TRIGONOMETRIC FUNCTIONS

32.

b

y(x) +

 n

a

 Ak sin(λk |x – t|) y(t) dt = f (x),

–∞ < a < b < ∞.

k=1

1◦ . Let us remove the modulus in the kth summand of the integrand: b x b Ik (x) = sin(λk |x – t|)y(t) dt = sin[λk (x – t)]y(t) dt + sin[λk (t – x)]y(t) dt. (1) a

a

x

Differentiating (1) with respect to x twice yields x  Ik = λk cos[λk (x – t)]y(t) dt – λk a

Ik

= 2λk y(x) –



b

cos[λk (t – x)]y(t) dt,

x



x

λ2k

sin[λk (x – t)]y(t) dt – a

(2)

b

λ2k

sin[λk (t – x)]y(t) dt, x

where the primes denote the derivatives with respect to x. By comparing formulas (1) and (2), we find the relation between Ik and Ik : Ik = 2λk y(x) – λ2k Ik ,

Ik = Ik (x).

(3)

2◦ . With the aid of (1), the integral equation can be rewritten in the form y(x) +

n 

Ak Ik = f (x).

(4)

k=1

Differentiating (4) with respect to x twice and taking into account (3), we find that  yxx (x) + σn y(x) –

n 

 Ak λ2k Ik = fxx (x),

n 

Ak λk .

(5)

 Ak (λ2n – λ2k )Ik = fxx (x) + λ2n f (x).

(6)

k=1

σn = 2

k=1

Eliminating the integral In from (4) and (5) yields  yxx (x) + (σn + λ2n )y(x) +

n–1  k=1

Differentiating (6) with respect to x twice and eliminating In–1 from the resulting equation with the aid of (6), we obtain a similar equation whose left-hand side is a second-order linear n–2  differential operator (acting on y) with constant coefficients plus the sum Bk Ik . If we k=1

successively eliminate In–2 , In–3 , . . . , with the aid of double differentiation, then we finally arrive at a linear nonhomogeneous ordinary differential equation of order 2n with constant coefficients.

33.

3◦ . The boundary conditions for y(x) can be found by setting x = a in the integral equation and all its derivatives. (Alternatively, these conditions can be found by setting x = a and x = b in the integral equation and all its derivatives obtained by means of double differentiation.) ∞ sin(x – t) y(x) – λ y(t) dt = f (x). x–t –∞ Solution:  ∞ sin(x – t) 2 λ f (t) dt, λ≠ . y(x) = f (x) + √ π 2π – πλ –∞ x – t Reference: F. D. Gakhov and Yu. I. Cherskii (1978).

342

LINEAR EQUATIONS OF THE SECOND KIND WITH CONSTANT LIMITS OF INTEGRATION

4.5-3. Kernels Containing Tangent. 34.

b

y(x) – λ

tan(βx)y(t) dt = f (x). a

This is a special case of equation 4.9.1 with g(x) = tan(βx) and h(t) = 1. 35.

b

y(x) – λ

tan(βt)y(t) dt = f (x). a

This is a special case of equation 4.9.1 with g(x) = 1 and h(t) = tan(βt). 36.

b

y(x) – λ

[A tan(βx) + B tan(βt)]y(t) dt = f (x). a

This is a special case of equation 4.9.4 with g(x) = tan(βx). 37.

b

y(x) – λ a

tan(βx) y(t) dt = f (x). tan(βt)

This is a special case of equation 4.9.1 with g(x) = tan(βx) and h(t) = 38.

b

y(x) – λ a

tan(βt) tan(βx)

y(t) dt = f (x).

This is a special case of equation 4.9.1 with g(x) = 39.

b

y(x) – λ

1 . tan(βt)

1 and h(t) = tan(βt). tan(βx)

tank (βx) tanm (µt)y(t) dt = f (x).

a

This is a special case of equation 4.9.1 with g(x) = tank (βx) and h(t) = tanm (µt). 40.

b

y(x) – λ

tk tanm (βx)y(t) dt = f (x).

a

This is a special case of equation 4.9.1 with g(x) = tanm (βx) and h(t) = tk . 41.

b

y(x) – λ

xk tanm (βt)y(t) dt = f (x).

a

This is a special case of equation 4.9.1 with g(x) = xk and h(t) = tanm (βt). 42.

b

y(x) – λ

[A + B(x – t) tan(βt)]y(t) dt = f (x). a

This is a special case of equation 4.9.8 with h(t) = tan(βt). 43.

b

y(x) – λ

[A + B(x – t) tan(βx)]y(t) dt = f (x). a

This is a special case of equation 4.9.10 with h(x) = tan(βx).

343

4.5. EQUATIONS WHOSE KERNELS CONTAIN TRIGONOMETRIC FUNCTIONS

4.5-4. Kernels Containing Cotangent. 44.

b

y(x) – λ

cot(βx)y(t) dt = f (x). a

This is a special case of equation 4.9.1 with g(x) = cot(βx) and h(t) = 1. 45.

b

y(x) – λ

cot(βt)y(t) dt = f (x). a

This is a special case of equation 4.9.1 with g(x) = 1 and h(t) = cot(βt). 46.

b

y(x) – λ

[A cot(βx) + B cot(βt)]y(t) dt = f (x). a

This is a special case of equation 4.9.4 with g(x) = cot(βx). 47.

b

y(x) – λ a

cot(βx) y(t) dt = f (x). cot(βt)

This is a special case of equation 4.9.1 with g(x) = cot(βx) and h(t) = 48.

b

y(x) – λ a

cot(βt) cot(βx)

y(t) dt = f (x).

This is a special case of equation 4.9.1 with g(x) = 49.

b

y(x) – λ

1 . cot(βt)

1 and h(t) = cot(βt). cot(βx)

cotk (βx) cotm (µt)y(t) dt = f (x).

a

This is a special case of equation 4.9.1 with g(x) = cotk (βx) and h(t) = cotm (µt). 50.

b

y(x) – λ

tk cotm (βx)y(t) dt = f (x).

a

This is a special case of equation 4.9.1 with g(x) = cotm (βx) and h(t) = tk . 51.

b

y(x) – λ

xk cotm (βt)y(t) dt = f (x).

a

This is a special case of equation 4.9.1 with g(x) = xk and h(t) = cotm (βt). 52.

b

y(x) – λ

[A + B(x – t) cot(βt)]y(t) dt = f (x). a

This is a special case of equation 4.9.8 with h(t) = cot(βt). 53.

b

y(x) – λ

[A + B(x – t) cot(βx)]y(t) dt = f (x). a

This is a special case of equation 4.9.10 with h(x) = cot(βx).

344

LINEAR EQUATIONS OF THE SECOND KIND WITH CONSTANT LIMITS OF INTEGRATION

4.5-5. Kernels Containing Combinations of Trigonometric Functions. 54.

b

cosk (βx) sinm (µt)y(t) dt = f (x).

y(x) – λ a

This is a special case of equation 4.9.1 with g(x) = cosk (βx) and h(t) = sinm (µt). 55.

b

[A sin(αx) cos(βt) + B sin(γx) cos(δt)]y(t) dt = f (x).

y(x) – λ a

This is a special case of equation 4.9.18 with g1(x) = sin(αx), h1 (t) = A cos(βt), g2 (x) = sin(γx), and h2 (t) = B cos(δt). 56.

b

tank (γx) cotm (µt)y(t) dt = f (x).

y(x) – λ a

This is a special case of equation 4.9.1 with g(x) = tank (γx) and h(t) = cotm (µt). 57.

b

[A tan(αx) cot(βt) + B tan(γx) cot(δt)]y(t) dt = f (x).

y(x) – λ a

This is a special case of equation 4.9.18 with g1(x) = tan(αx), h1 (t) = A cot(βt), g2 (x) = tan(γx), and h2 (t) = B cot(δt). 4.5-6. Singular Equation. 58.

B 2π

Ay(x) –





cot 0

t – x y(t) dt = f (x), 2

0 ≤ x ≤ 2π.

Here the integral is understood in the sense of the Cauchy principal value. Without loss of generality we may assume that A2 + B 2 = 1. Solution: y(x) = Af (x) +

B 2π

0



2π t–x B2 f (t) dt + cot f (t) dt. 2 2πA 0

Reference: I. K. Lifanov (1996).

4.6. Equations Whose Kernels Contain Inverse Trigonometric Functions 4.6-1. Kernels Containing Arccosine. 1.

b

y(x) – λ

arccos(βx)y(t) dt = f (x). a

This is a special case of equation 4.9.1 with g(x) = arccos(βx) and h(t) = 1. 2.

b

arccos(βt)y(t) dt = f (x).

y(x) – λ a

This is a special case of equation 4.9.1 with g(x) = 1 and h(t) = arccos(βt).

345

4.6. EQUATIONS WHOSE KERNELS CONTAIN INVERSE TRIGONOMETRIC FUNCTIONS

3.

b

y(x) – λ a

arccos(βx) arccos(βt)

y(t) dt = f (x).

This is a special case of equation 4.9.1 with g(x) = arccos(βx) and h(t) = 4.

b

y(x) – λ a

arccos(βt) arccos(βx)

y(t) dt = f (x).

This is a special case of equation 4.9.1 with g(x) = 5.

b

y(x) – λ

1 . arccos(βt)

1 and h(t) = arccos(βt). arccos(βx)

arccosk (βx) arccosm (µt)y(t) dt = f (x).

a

This is a special case of equation 4.9.1 with g(x) = arccosk (βx) and h(t) = arccosm (µt). 6.

b

y(x) – λ

tk arccosm (βx)y(t) dt = f (x).

a

This is a special case of equation 4.9.1 with g(x) = arccosm (βx) and h(t) = tk . 7.

b

y(x) – λ

xk arccosm (βt)y(t) dt = f (x).

a

This is a special case of equation 4.9.1 with g(x) = xk and h(t) = arccosm (βt). 8.

b

[A + B(x – t) arccos(βx)]y(t) dt = f (x).

y(x) – λ a

This is a special case of equation 4.9.10 with h(x) = arccos(βx). 9.

b

[A + B(x – t) arccos(βt)]y(t) dt = f (x).

y(x) – λ a

This is a special case of equation 4.9.8 with h(t) = arccos(βt).

4.6-2. Kernels Containing Arcsine. 10.

b

y(x) – λ

arcsin(βx)y(t) dt = f (x). a

This is a special case of equation 4.9.1 with g(x) = arcsin(βx) and h(t) = 1. 11.

b

arcsin(βt)y(t) dt = f (x).

y(x) – λ a

This is a special case of equation 4.9.1 with g(x) = 1 and h(t) = arcsin(βt). 12.

b

y(x) – λ a

arcsin(βx) y(t) dt = f (x). arcsin(βt)

This is a special case of equation 4.9.1 with g(x) = arcsin(βx) and h(t) =

1 . arcsin(βt)

346

LINEAR EQUATIONS OF THE SECOND KIND WITH CONSTANT LIMITS OF INTEGRATION

13.

b

y(x) – λ a

arcsin(βt) arcsin(βx)

y(t) dt = f (x).

This is a special case of equation 4.9.1 with g(x) = 14.

b

y(x) – λ

1 and h(t) = arcsin (βt). arcsin(βx)

arcsink (βx) arcsinm (µt)y(t) dt = f (x).

a

This is a special case of equation 4.9.1 with g(x) = arcsink (βx) and h(t) = arcsinm (µt). 15.

b

y(x) – λ

tk arcsinm (βx)y(t) dt = f (x).

a

This is a special case of equation 4.9.1 with g(x) = arcsinm (βx) and h(t) = tk . 16.

b

y(x) – λ

xk arcsinm (βt)y(t) dt = f (x).

a

This is a special case of equation 4.9.1 with g(x) = xk and h(t) = arcsinm (βt). 17.

b

y(x) – λ

[A + B(x – t) arcsin(βt)]y(t) dt = f (x). a

This is a special case of equation 4.9.8 with h(t) = arcsin(βt). 18.

b

[A + B(x – t) arcsin(βx)]y(t) dt = f (x).

y(x) – λ a

This is a special case of equation 4.9.10 with h(x) = arcsin(βx).

4.6-3. Kernels Containing Arctangent. 19.

b

y(x) – λ

arctan(βx)y(t) dt = f (x). a

This is a special case of equation 4.9.1 with g(x) = arctan(βx) and h(t) = 1. 20.

b

arctan(βt)y(t) dt = f (x).

y(x) – λ a

This is a special case of equation 4.9.1 with g(x) = 1 and h(t) = arctan(βt). 21.

b

[A arctan(βx) + B arctan(βt)]y(t) dt = f (x).

y(x) – λ a

This is a special case of equation 4.9.4 with g(x) = arctan(βx). 22.

b

y(x) – λ a

arctan(βx) arctan(βt)

y(t) dt = f (x).

This is a special case of equation 4.9.1 with g(x) = arctan(βx) and h(t) =

1 . arctan(βt)

4.6. EQUATIONS WHOSE KERNELS CONTAIN INVERSE TRIGONOMETRIC FUNCTIONS

23.

b

y(x) – λ a

arctan(βt) arctan(βx)

y(t) dt = f (x).

This is a special case of equation 4.9.1 with g(x) = 24.

b

y(x) – λ

1 and h(t) = arctan(βt). arctan(βx)

arctank (βx) arctanm (µt)y(t) dt = f (x).

a

This is a special case of equation 4.9.1 with g(x) = arctank (βx) and h(t) = arctanm (µt). 25.

b

y(x) – λ

tk arctanm (βx)y(t) dt = f (x).

a

This is a special case of equation 4.9.1 with g(x) = arctanm (βx) and h(t) = tk . 26.

b

y(x) – λ

xk arctanm (βt)y(t) dt = f (x).

a

This is a special case of equation 4.9.1 with g(x) = xk and h(t) = arctanm (βt). 27.

b

y(x) – λ

[A + B(x – t) arctan(βt)]y(t) dt = f (x). a

This is a special case of equation 4.9.8 with h(t) = arctan(βt). 28.

b

[A + B(x – t) arctan(βx)]y(t) dt = f (x).

y(x) – λ a

This is a special case of equation 4.9.10 with h(x) = arctan(βx).

4.6-4. Kernels Containing Arccotangent. 29.

b

y(x) – λ

arccot(βx)y(t) dt = f (x). a

This is a special case of equation 4.9.1 with g(x) = arccot(βx) and h(t) = 1. 30.

b

arccot(βt)y(t) dt = f (x).

y(x) – λ a

This is a special case of equation 4.9.1 with g(x) = 1 and h(t) = arccot(βt). 31.

b

[A arccot(βx) + B arccot(βt)]y(t) dt = f (x).

y(x) – λ a

This is a special case of equation 4.9.4 with g(x) = arccot(βx). 32.

b

y(x) – λ a

arccot(βx) arccot(βt)

y(t) dt = f (x).

This is a special case of equation 4.9.1 with g(x) = arccot(βx) and h(t) =

1 . arccot(βt)

347

348

LINEAR EQUATIONS OF THE SECOND KIND WITH CONSTANT LIMITS OF INTEGRATION

33.

b

y(x) – λ a

arccot(βt) arccot(βx)

y(t) dt = f (x).

This is a special case of equation 4.9.1 with g(x) = 34.

b

y(x) – λ

1 and h(t) = arccot(βt). arccot(βx)

arccotk (βx) arccotm (µt)y(t) dt = f (x).

a

This is a special case of equation 4.9.1 with g(x) = arccotk (βx) and h(t) = arccotm (µt). 35.

b

y(x) – λ

tk arccotm (βx)y(t) dt = f (x).

a

This is a special case of equation 4.9.1 with g(x) = arccotm (βx) and h(t) = tk . 36.

b

y(x) – λ

xk arccotm (βt)y(t) dt = f (x).

a

This is a special case of equation 4.9.1 with g(x) = xk and h(t) = arccotm (βt). 37.

b

[A + B(x – t) arccot(βt)]y(t) dt = f (x).

y(x) – λ a

This is a special case of equation 4.9.8 with h(t) = arccot(βt). 38.

b

[A + B(x – t) arccot(βx)]y(t) dt = f (x).

y(x) – λ a

This is a special case of equation 4.9.10 with h(x) = arccot(βx).

4.7. Equations Whose Kernels Contain Combinations of Elementary Functions 4.7-1. Kernels Containing Exponential and Hyperbolic Functions. 1.

b

y(x) – λ

eµ(x–t) cosh[β(x – t)]y(t) dt = f (x).

a

This is a special case of equation 4.9.18 with g1 (x) = eµx cosh(βx), h1 (t) = e–µt cosh(βt), g2 (x) = eµx sinh(βx), and h2 (t) = –e–µt sinh(βt). 2.

b

y(x) – λ

eµ(x–t) sinh[β(x – t)]y(t) dt = f (x).

a

This is a special case of equation 4.9.18 with g1 (x) = eµx sinh(βx), h1 (t) = e–µt cosh(βt), g2 (x) = eµx cosh(βx), and h2 (t) = –e–µt sinh(βt). 3.

b

y(x) – λ

teµ(x–t) sinh[β(x – t)]y(t) dt = f (x).

a

This is a special case of equation 4.9.18 with g1 (x) = eµx sinh(βx), h1 (t) = te–µt cosh(βt), g2 (x) = eµx cosh(βx), and h2 (t) = –te–µt sinh(βt).

4.7. EQUATIONS WHOSE KERNELS CONTAIN COMBINATIONS OF ELEMENTARY FUNCTIONS

349

4.7-2. Kernels Containing Exponential and Logarithmic Functions. 4.

b

y(x) – λ

eµt ln(βx)y(t) dt = f (x).

a

This is a special case of equation 4.9.1 with g(x) = ln(βx) and h(t) = eµt . 5.

b

y(x) – λ

eµx ln(βt)y(t) dt = f (x).

a

This is a special case of equation 4.9.1 with g(x) = eµx and h(t) = ln(βt). 6.

b

y(x) – λ

eµ(x–t) ln(βx)y(t) dt = f (x).

a

This is a special case of equation 4.9.1 with g(x) = eµx ln(βx) and h(t) = e–µt . 7.

b

y(x) – λ

eµ(x–t) ln(βt)y(t) dt = f (x).

a

This is a special case of equation 4.9.1 with g(x) = eµx and h(t) = e–µt ln(βt). 8.

b

y(x) – λ

eµ(x–t) (ln x – ln t)y(t) dt = f (x).

a

This is a special case of equation 4.9.18 with g1 (x) = eµx ln x, h1 (t) = e–µt , g2 (x) = eµx , and h2 (t) = –e–µt ln t. 9.



 x  exp –a ln y(t) dt = f (x). 2a t t 0 Solution with a > 0, b > 0, and x > 0:  x  a2 – b2 ∞ 1 exp –b ln f (t) dt. y(x) = f (x) + 2b t t 0 y(x) +

b2 – a2



1

Reference: F. D. Gakhov and Yu. I. Cherskii (1978).

4.7-3. Kernels Containing Exponential and Trigonometric Functions. 10.

b

y(x) – λ

eµt cos(βx)y(t) dt = f (x).

a

This is a special case of equation 4.9.1 with g(x) = cos(βx) and h(t) = eµt . 11.

b

y(x) – λ

eµx cos(βt)y(t) dt = f (x).

a

12.

This is a special case of equation 4.9.1 with g(x) = eµx and h(t) = cos(βt). ∞ y(x) – λ eµ(x–t) cos(xt)y(t) dt = f (x). 0

Solution: f (x) λ y(x) = π 2 + 1– 2λ 1 – π2 λ2

0



eµ(x–t) cos(xt)f (t) dt,

 λ ≠ ± 2/π.

350

LINEAR EQUATIONS OF THE SECOND KIND WITH CONSTANT LIMITS OF INTEGRATION

13.

b

y(x) – λ

eµ(x–t) cos[β(x – t)]y(t) dt = f (x).

a

This is a special case of equation 4.9.18 with g1 (x) = eµx cos(βx), h1 (t) = e–µt cos(βt), g2 (x) = eµx sin(βx), and h2 (t) = e–µt sin(βt). 14.

b

y(x) – λ

eµt sin(βx)y(t) dt = f (x).

a

This is a special case of equation 4.9.1 with g(x) = sin(βx) and h(t) = eµt . 15.

b

y(x) – λ

eµx sin(βt)y(t) dt = f (x).

a

This is a special case of equation 4.9.1 with g(x) = eµx and h(t) = sin(βt). 16.



y(x) – λ

eµ(x–t) sin(xt)y(t) dt = f (x).

0

Solution: y(x) = 17.

b

y(x) – λ

λ f (x) + 1 – π2 λ2 1 – π2 λ2





eµ(x–t) sin(xt)f (t) dt,

 λ ≠ ± 2/π.

0

eµ(x–t) sin[β(x – t)]y(t) dt = f (x).

a

This is a special case of equation 4.9.18 with g1 (x) = eµx sin(βx), h1 (t) = e–µt cos(βt), g2 (x) = eµx cos(βx), and h2 (t) = –e–µt sin(βt). 18.

b

e

y(x) – λ a

µ(x–t)

 n

Ak sin[βk (x – t)] y(t) dt = f (x),

n = 1, 2, . . .

k=1

This is a special case of equation 4.9.20. 19.

b

y(x) – λ

teµ(x–t) sin[β(x – t)]y(t) dt = f (x).

a

This is a special case of equation 4.9.18 with g1 (x) = eµx sin(βx), h1 (t) = te–µt cos(βt), g2 (x) = eµx cos(βx), and h2 (t) = –te–µt sin(βt). 20.

b

y(x) – λ

xeµ(x–t) sin[β(x – t)]y(t) dt = f (x).

a

This is a special case of equation 4.9.18 with g1 (x) = xeµx sin(βx), h1 (t) = e–µt cos(βt), g2 (x) = xeµx cos(βx), and h2 (t) = –e–µt sin(βt). 21.

b

y(x) – λ

eµt tan(βx)y(t) dt = f (x).

a

This is a special case of equation 4.9.1 with g(x) = tan(βx) and h(t) = eµt . 22.

b

y(x) – λ

eµx tan(βt)y(t) dt = f (x).

a

This is a special case of equation 4.9.1 with g(x) = eµx and h(t) = tan(βt).

4.7. EQUATIONS WHOSE KERNELS CONTAIN COMBINATIONS OF ELEMENTARY FUNCTIONS

23.

b

y(x) – λ

351

eµ(x–t) [tan(βx) – tan(βt)]y(t) dt = f (x).

a

This is a special case of equation 4.9.18 with g1 (x) = eµx tan(βx), h1 (t) = e–µt , g2 (x) = eµx , and h2 (t) = –e–µt tan(βt). 24.

b

y(x) – λ

eµt cot(βx)y(t) dt = f (x).

a

This is a special case of equation 4.9.1 with g(x) = cot(βx) and h(t) = eµt . 25.

b

y(x) – λ

eµx cot(βt)y(t) dt = f (x).

a

This is a special case of equation 4.9.1 with g(x) = eµx and h(t) = cot(βt). 4.7-4. Kernels Containing Hyperbolic and Logarithmic Functions. 26.

b

y(x) – λ

coshk (βx) lnm (µt)y(t) dt = f (x).

a

This is a special case of equation 4.9.1 with g(x) = coshk (βx) and h(t) = lnm (µt). 27.

b

y(x) – λ

coshk (βt) lnm (µx)y(t) dt = f (x).

a

This is a special case of equation 4.9.1 with g(x) = lnm (µx) and h(t) = coshk (βt). 28.

b

y(x) – λ

sinhk (βx) lnm (µt)y(t) dt = f (x).

a

This is a special case of equation 4.9.1 with g(x) = sinhk (βx) and h(t) = lnm (µt). 29.

b

y(x) – λ

sinhk (βt) lnm (µx)y(t) dt = f (x).

a

This is a special case of equation 4.9.1 with g(x) = lnm (µx) and h(t) = sinhk (βt). 30.

b

y(x) – λ

tanhk (βx) lnm (µt)y(t) dt = f (x).

a

This is a special case of equation 4.9.1 with g(x) = tanhk (βx) and h(t) = lnm (µt). 31.

b

y(x) – λ

tanhk (βt) lnm (µx)y(t) dt = f (x).

a

This is a special case of equation 4.9.1 with g(x) = lnm (µx) and h(t) = tanhk (βt). 32.

b

y(x) – λ

cothk (βx) lnm (µt)y(t) dt = f (x).

a

This is a special case of equation 4.9.1 with g(x) = cothk (βx) and h(t) = lnm (µt). 33.

b

y(x) – λ

cothk (βt) lnm (µx)y(t) dt = f (x).

a

This is a special case of equation 4.9.1 with g(x) = lnm (µx) and h(t) = cothk (βt).

352

LINEAR EQUATIONS OF THE SECOND KIND WITH CONSTANT LIMITS OF INTEGRATION

4.7-5. Kernels Containing Hyperbolic and Trigonometric Functions. 34.

b

y(x) – λ

coshk (βx) cosm (µt)y(t) dt = f (x).

a

This is a special case of equation 4.9.1 with g(x) = coshk (βx) and h(t) = cosm (µt). 35.

b

y(x) – λ

coshk (βt) cosm (µx)y(t) dt = f (x).

a

This is a special case of equation 4.9.1 with g(x) = cosm (µx) and h(t) = coshk (βt). 36.

b

y(x) – λ

coshk (βx) sinm (µt)y(t) dt = f (x).

a

This is a special case of equation 4.9.1 with g(x) = coshk (βx) and h(t) = sinm (µt). 37.

b

y(x) – λ

coshk (βt) sinm (µx)y(t) dt = f (x).

a

This is a special case of equation 4.9.1 with g(x) = sinm (µx) and h(t) = coshk (βt). 38.

b

y(x) – λ

sinhk (βx) cosm (µt)y(t) dt = f (x).

a

This is a special case of equation 4.9.1 with g(x) = sinhk (βx) and h(t) = cosm (µt). 39.

b

y(x) – λ

sinhk (βt) cosm (µx)y(t) dt = f (x).

a

This is a special case of equation 4.9.1 with g(x) = cosm (µx) and h(t) = sinhk (βt). 40.

b

y(x) – λ

sinhk (βx) sinm (µt)y(t) dt = f (x).

a

This is a special case of equation 4.9.1 with g(x) = sinhk (βx) and h(t) = sinm (µt). 41.

b

y(x) – λ

sinhk (βt) sinm (µx)y(t) dt = f (x).

a

This is a special case of equation 4.9.1 with g(x) = sinm (µx) and h(t) = sinhk (βt). 42.

b

y(x) – λ

tanhk (βx) cosm (µt)y(t) dt = f (x).

a

This is a special case of equation 4.9.1 with g(x) = tanhk (βx) and h(t) = cosm (µt). 43.

b

y(x) – λ

tanhk (βt) cosm (µx)y(t) dt = f (x).

a

This is a special case of equation 4.9.1 with g(x) = cosm (µx) and h(t) = tanhk (βt). 44.

b

y(x) – λ

tanhk (βx) sinm (µt)y(t) dt = f (x).

a

This is a special case of equation 4.9.1 with g(x) = tanhk (βx) and h(t) = sinm (µt).

4.8. EQUATIONS WHOSE KERNELS CONTAIN SPECIAL FUNCTIONS

45.

b

y(x) – λ

tanhk (βt) sinm (µx)y(t) dt = f (x).

a

This is a special case of equation 4.9.1 with g(x) = sinm (µx) and h(t) = tanhk (βt). 4.7-6. Kernels Containing Logarithmic and Trigonometric Functions. 46.

b

y(x) – λ

cosk (βx) lnm (µt)y(t) dt = f (x).

a

47.

This is a special case of equation 4.9.1 with g(x) = cosk (βx) and h(t) = lnm (µt). b cosk (βt) lnm (µx)y(t) dt = f (x). y(x) – λ a

48.

This is a special case of equation 4.9.1 with g(x) = lnm (µx) and h(t) = cosk (βt). b sink (βx) lnm (µt)y(t) dt = f (x). y(x) – λ

49.

This is a special case of equation 4.9.1 with g(x) = sink (βx) and h(t) = lnm (µt). b sink (βt) lnm (µx)y(t) dt = f (x). y(x) – λ

50.

This is a special case of equation 4.9.1 with g(x) = lnm (µx) and h(t) = sink (βt). b tank (βx) lnm (µt)y(t) dt = f (x). y(x) – λ

51.

This is a special case of equation 4.9.1 with g(x) = tank (βx) and h(t) = lnm (µt). b tank (βt) lnm (µx)y(t) dt = f (x). y(x) – λ

52.

This is a special case of equation 4.9.1 with g(x) = lnm (µx) and h(t) = tank (βt). b cotk (βx) lnm (µt)y(t) dt = f (x). y(x) – λ

53.

This is a special case of equation 4.9.1 with g(x) = cotk (βx) and h(t) = lnm (µt). b cotk (βt) lnm (µx)y(t) dt = f (x). y(x) – λ

a

a

a

a

a

a

This is a special case of equation 4.9.1 with g(x) = lnm (µx) and h(t) = cotk (βt).

4.8. Equations Whose Kernels Contain Special Functions 4.8-1. Kernels Containing Bessel Functions. 1.

b

y(x) – λ a

Jν (βx)y(t) dt = f (x).

This is a special case of equation 4.9.1 with g(x) = Jν (βx) and h(t) = 1.

353

354

LINEAR EQUATIONS OF THE SECOND KIND WITH CONSTANT LIMITS OF INTEGRATION

2.

b

y(x) – λ

Jν (βt)y(t) dt = f (x).

a

This is a special case of equation 4.9.1 with g(x) = 1 and h(t) = Jν (βt). 3.



tJν (xt)y(t) dt = 0,

y(x) + λ

ν > –1.

0

Characteristic values: λ = ±1. For the characteristic values, the integral equation has infinitely many linearly independent eigenfunctions. Eigenfunctions for λ = +1 have the form ∞ y+ (x) = f (x) – tJν (xt)f (t) dt, 0

where f = f (x) is an arbitrary function. Eigenfunctions for λ = –1 have the form y– (x) = f (x) +



tJν (xt)f (t) dt,

0

where f = f (x) is an arbitrary function. 4.



y(x) + λ

tJν (xt)y(t) dt = f (x),

ν > –1.

0

Solution:

λ f (x) – y(x) = 1 – λ2 1 – λ2

5.



y(x) + λ





tJν (xt)f (t) dt,

λ ≠ ±1.

0

 √  Jν 2 xt y(t) dt = f (x).

0

By setting x = 12 z 2 , t = 12 τ 2 , y(x) = Y (z), and f (x) = F (z), we arrive at an equation of the form 4.8.4: ∞ Y (z) + λ τ Jν (zτ )Y (τ ) dτ = F (z). 0

6.

b

y(x) – λ a

[A + B(x – t)Jν (βt)]y(t) dt = f (x).

This is a special case of equation 4.9.8 with h(t) = Jν (βt). 7.

b

y(x) – λ a

[A + B(x – t)Jν (βx)]y(t) dt = f (x).

This is a special case of equation 4.9.10 with h(x) = Jν (βx). 8.

b

y(x) – λ a

[AJµ (αx) + BJν (βt)]y(t) dt = f (x).

This is a special case of equation 4.9.5 with g(x) = AJµ (αx) and h(t) = BJν (βt). 9.

b

y(x) – λ a

[AJµ (x)Jν (t) + BJν (x)Jµ (t)]y(t) dt = f (x).

This is a special case of equation 4.9.17 with g(x) = Jµ (x) and h(t) = Jν (t).

4.8. EQUATIONS WHOSE KERNELS CONTAIN SPECIAL FUNCTIONS

10.

b

y(x) – λ a

Yν (βx)y(t) dt = f (x).

This is a special case of equation 4.9.1 with g(x) = Yν (βx) and h(t) = 1. 11.

b

y(x) – λ a

Yν (βt)y(t) dt = f (x).

This is a special case of equation 4.9.1 with g(x) = 1 and h(t) = Yν (βt). 12.

b

y(x) – λ a

[A + B(x – t)Yν (βt)]y(t) dt = f (x).

This is a special case of equation 4.9.8 with h(t) = Yν (βt). 13.

b

y(x) – λ a

[A + B(x – t)Yν (βx)]y(t) dt = f (x).

This is a special case of equation 4.9.10 with h(x) = Yν (βx). 14.

b

y(x) – λ a

[AYµ (αx) + BYν (βt)]y(t) dt = f (x).

This is a special case of equation 4.9.5 with g(x) = AYµ (αx) and h(t) = BYν (βt). 15.

b

y(x) – λ a

[AYµ (x)Yµ (t) + BYν (x)Yν (t)]y(t) dt = f (x).

This is a special case of equation 4.9.14 with g(x) = Yµ (x) and h(t) = Yν (t). 16.

b

y(x) – λ a

[AYµ (x)Yν (t) + BYν (x)Yµ (t)]y(t) dt = f (x).

This is a special case of equation 4.9.17 with g(x) = Yµ (x) and h(t) = Yν (t). 4.8-2. Kernels Containing Modified Bessel Functions. 17.

b

y(x) – λ a

Iν (βx)y(t) dt = f (x).

This is a special case of equation 4.9.1 with g(x) = Iν (βx) and h(t) = 1. 18.

b

y(x) – λ a

Iν (βt)y(t) dt = f (x).

This is a special case of equation 4.9.1 with g(x) = 1 and h(t) = Iν (βt). 19.

b

y(x) – λ a

[A + B(x – t)Iν (βt)]y(t) dt = f (x).

This is a special case of equation 4.9.8 with h(t) = Iν (βt). 20.

b

y(x) – λ a

[A + B(x – t)Iν (βx)]y(t) dt = f (x).

This is a special case of equation 4.9.10 with h(x) = Iν (βx).

355

356

LINEAR EQUATIONS OF THE SECOND KIND WITH CONSTANT LIMITS OF INTEGRATION

21.

b

y(x) – λ a

[AIµ (αx) + BIν (βt)]y(t) dt = f (x).

This is a special case of equation 4.9.5 with g(x) = AIµ (αx) and h(t) = BIν (βt). 22.

b

y(x) – λ a

[AIµ (x)Iµ (t) + BIν (x)Iν (t)]y(t) dt = f (x).

This is a special case of equation 4.9.14 with g(x) = Iµ (x) and h(t) = Iν (t). 23.

b

y(x) – λ a

[AIµ (x)Iν (t) + BIν (x)Iµ (t)]y(t) dt = f (x).

This is a special case of equation 4.9.17 with g(x) = Iµ (x) and h(t) = Iν (t). 24.

b

y(x) – λ a

Kν (βx)y(t) dt = f (x).

This is a special case of equation 4.9.1 with g(x) = Kν (βx) and h(t) = 1. 25.

b

y(x) – λ a

Kν (βt)y(t) dt = f (x).

This is a special case of equation 4.9.1 with g(x) = 1 and h(t) = Kν (βt). 26.

b

y(x) – λ a

[A + B(x – t)Kν (βt)]y(t) dt = f (x).

This is a special case of equation 4.9.8 with h(t) = Kν (βt). 27.

b

y(x) – λ a

[A + B(x – t)Kν (βx)]y(t) dt = f (x).

This is a special case of equation 4.9.10 with h(x) = Kν (βx). 28.

b

y(x) – λ a

[AKµ (αx) + BKν (βt)]y(t) dt = f (x).

This is a special case of equation 4.9.5 with g(x) = AKµ (αx) and h(t) = BKν (βt). 29.

b

y(x) – λ a

[AKµ (x)Kµ (t) + BKν (x)Kν (t)]y(t) dt = f (x).

This is a special case of equation 4.9.14 with g(x) = Kµ (x) and h(t) = Kν (t). 30.

b

y(x) – λ a

[AKµ (x)Kν (t) + BKν (x)Kµ (t)]y(t) dt = f (x).

This is a special case of equation 4.9.17 with g(x) = Kµ (x) and h(t) = Kν (t).

357

4.9. EQUATIONS WHOSE KERNELS CONTAIN ARBITRARY FUNCTIONS

4.9. Equations Whose Kernels Contain Arbitrary Functions 4.9-1. Equations with Degenerate Kernel: K(x, t) = g1 (x)h1 (t) + · · · + gn (x)hn (t). 1.

b

y(x) – λ

g(x)h(t)y(t) dt = f (x). a

1◦ . Assume that λ ≠ Solution:



–1

b a

g(t)h(t) dt

y(x) = f (x) + λkg(x),

.

where k =

 1–λ

–1

b

g(t)h(t) dt a

2◦ . Assume that λ = b

For



h(t)f (t) dt. a

–1

b a

b

g(t)h(t) dt

.

h(t)f (t) dt = 0, the solution has the form

a

y = f (x) + Cg(x), where C is an arbitrary constant. b

For h(t)f (t) dt ≠ 0, there is no solution. a The limits of integration may take the values a = –∞ and/or b = ∞, provided that the corresponding improper integral converges. 2.

b

y(x) – λ

[g(x) + g(t)]y(t) dt = f (x). a

The characteristic values of the equation: λ1 =

1 √ , g1 + (b – a)g2

where

g1 =

λ2 =

1 √ , g1 – (b – a)g2



b

g(x) dx,

b

g 2 (x) dx.

g2 =

a

a

1◦ . Solution with λ ≠ λ1,2 : y(x) = f (x) + λ[A1 g(x) + A2 ], where the constants A1 and A2 are given by A1 =

f1 – λ[f1 g1 – (b – a)f2 ] f2 – λ(f2 g1 – f1 g2 ) , A2 = 2 , 2 – (b – a)g2 ]λ – 2g1 λ + 1 [g1 – (b – a)g2 ]λ2 – 2g1 λ + 1 b b f1 = f (x) dx, f2 = f (x)g(x) dx.

[g12

a

a



2 . Solution with λ = λ1 ≠ λ2 and f1 = f2 = 0: y(x) = f (x) + Cy1 (x),

 y1 (x) = g(x) +

g2 , b–a

where C is an arbitrary constant and y1 (x) is an eigenfunction of the equation corresponding to the characteristic value λ1 .

358

LINEAR EQUATIONS OF THE SECOND KIND WITH CONSTANT LIMITS OF INTEGRATION

3◦ . Solution with λ = λ2 ≠ λ1 and f1 = f2 = 0:  y(x) = f (x) + Cy2 (x),

y2 (x) = g(x) –

g2 , b–a

where C is an arbitrary constant and y2 (x) is an eigenfunction of the equation corresponding to the characteristic value λ2 . 4◦ . The equation has no multiple characteristic values. 3.

b

[g(x) – g(t)]y(t) dt = f (x).

y(x) – λ a

The characteristic values of the equation: 1 , λ1 =  2 g1 – (b – a)g2 where





b

g(x) dx,

g1 =

1 λ2 = –  2 , g1 – (b – a)g2 b

g 2 (x) dx.

g2 =

a

a

1◦ . Solution with λ ≠ λ1,2 : y(x) = f (x) + λ[A1 g(x) + A2 ], where the constants A1 and A2 are given by A1 =

f1 + λ[f1 g1 – (b – a)f2 ] –f2 + λ(f2 g1 – f1 g2 ) , A2 = , [(b – a)g2 – g12 ]λ2 + 1 [(b – a)g2 – g12 ]λ2 + 1 b b f1 = f (x) dx, f2 = f (x)g(x) dx. a

a

2◦ . Solution with λ = λ1 ≠ λ2 and f1 = f2 = 0: y(x) = f (x) + Cy1 (x),

y1 (x) = g(x) +

1 – λ1 g1 , λ1 (b – a)

where C is an arbitrary constant and y1 (x) is an eigenfunction of the equation corresponding to the characteristic value λ1 . 3◦ . The solution with λ = λ2 ≠ λ1 and f1 = f2 = 0 is given by the formulas of item 2◦ in which one must replace λ1 and y1 (x) by λ2 and y2 (x), respectively. 4◦ . The equation has no multiple characteristic values. 4.

b

[Ag(x) + Bg(t)]y(t) dt = f (x).

y(x) – λ a

The characteristic values of the equation: λ1,2

 (A + B)g1 ± (A – B)2 g12 + 4AB(b – a)g2 , = 2AB[g12 – (b – a)g2 ]

4.9. EQUATIONS WHOSE KERNELS CONTAIN ARBITRARY FUNCTIONS

where

g1 =



b

g(x) dx,

b

g 2 (x) dx.

g2 =

a

1◦ . Solution with λ ≠ λ1,2 :

359

a

y(x) = f (x) + λ[A1 g(x) + A2 ], where the constants A1 and A2 are given by Af1 – λAB[f1 g1 – (b – a)f2] Bf2 – λAB(f2 g1 – f1 g2 ) , A2 = , A1 = AB[g12 – (b – a)g2]λ2 – (A + B)g1 λ + 1 AB[g12 – (b – a)g2]λ2 – (A + B)g1 λ + 1 b b f1 = f (x) dx, f2 = f (x)g(x) dx. a

2◦ . Solution with λ = λ1 ≠ λ2 and f1 = f2 = 0:

a

1 – λ1 Ag1 , λ1 A(b – a) where C is an arbitrary constant and y1 (x) is an eigenfunction of the equation corresponding to the characteristic value λ1 . y1 (x) = g(x) +

y(x) = f (x) + Cy1 (x),

3◦ . The solution with λ = λ2 ≠ λ1 and f1 = f2 = 0 is given by the formulas of item 2◦ in which one must replace λ1 and y1 (x) by λ2 and y2 (x), respectively.

5.

4◦ . Solution with λ = λ1,2 = λ∗ and f1 = f2 = 0 provided that A ≠ ±B, where the characteristic 2 value λ∗ = is double: (A + B)g1 (A – B)g1 . y∗ (x) = g(x) – y(x) = f (x) + Cy∗ (x), 2A(b – a) Here C is an arbitrary constant and y∗ (x) is an eigenfunction of the equation corresponding to λ∗ . The equation has no multiple characteristic values if A = ±B. b y(x) – λ [g(x) + h(t)]y(t) dt = f (x). a

The characteristic values of the equation:

 s1 + s3 ± (s1 – s3 )2 + 4(b – a)s2 , 2[s1 s3 – (b – a)s2 ]

λ1,2 = where

s1 =



b

g(x) dx, a

s2 =



b

g(x)h(x) dx,

s3 =

a

1◦ . Solution with λ ≠ λ1,2 :

b

h(x) dx. a

y(x) = f (x) + λ[A1 g(x) + A2 ], where the constants A1 and A2 are given by f1 – λ[f1 s3 – (b – a)f2 ] f2 – λ(f2 s1 – f1 s2 ) , A2 = , A1 = [s1 s3 – (b – a)s2 ]λ2 – (s1 + s3 )λ + 1 [s1 s3 – (b – a)s2 ]λ2 – (s1 + s3 )λ + 1 b b f1 = f (x) dx, f2 = f (x)h(x) dx. a

2◦ . Solution with λ = λ1 ≠ λ2 and f1 = f2 = 0:

a

1 – λ1 s1 , λ1 (b – a) where C is an arbitrary constant and y1 (x) is an eigenfunction of the equation corresponding to the characteristic value λ1 . y(x) = f (x) + Cy1 (x),

y1 (x) = g(x) +

360

LINEAR EQUATIONS OF THE SECOND KIND WITH CONSTANT LIMITS OF INTEGRATION

3◦ . The solution with λ = λ2 ≠ λ1 and f1 = f2 = 0 is given by the formulas of item 2◦ in which one must replace λ1 and y1 (x) by λ2 and y2 (x), respectively. 4◦ . Solution with λ = λ1,2 = λ∗ and f1 = f2 = 0 provided that s1 ≠ ±s3 , where the characteristic 2 value λ∗ = is double: s1 + s3 s1 – s3 . y(x) = f (x) + Cy∗ (x), y∗ (x) = g(x) – 2(b – a) Here C is an arbitrary constant and y∗ (x) is an eigenfunction of the equation corresponding to λ∗ . The equation has no multiple characteristic values if s1 = ±s3 . 6.

b

[Ag(x) + Bg(t)]h(t) y(t) dt = f (x).

y(x) – λ a

The characteristic values of the equation: λ1,2 where



 (A + B)s1 ± (A – B)2 s12 + 4ABs0 s2 , = 2AB(s12 – s0 s2 )

b

s0 =

h(x) dx, a



b

s1 =

g(x)h(x) dx, a

b

g 2 (x)h(x) dx.

s2 = a

1◦ . Solution with λ ≠ λ1,2 : y(x) = f (x) + λ[A1 g(x) + A2 ], where the constants A1 and A2 are given by A1 =

Af1 – ABλ(f1 s1 – f2 s0 ) Bf2 – ABλ(f2 s1 – f1 s2 ) , A2 = , AB(s12 – s0 s2 )λ2 – (A + B)s1 λ + 1 AB(s12 – s0 s2 )λ2 – (A + B)s1 λ + 1 b b f1 = f (x)h(x) dx, f2 = f (x)g(x)h(x) dx. a

a



2 . Solution with λ = λ1 ≠ λ2 and f1 = f2 = 0: y(x) = f (x) + Cy1 (x),

y1 (x) = g(x) +

1 – λ1 As1 , λ1 As0

where C is an arbitrary constant and y1 (x) is an eigenfunction of the equation corresponding to the characteristic value λ1 . 3◦ . The solution with λ = λ2 ≠ λ1 and f1 = f2 = 0 is given by the formulas of item 2◦ in which one must replace λ1 and y1 (x) by λ2 and y2 (x), respectively. 4◦ . Solution with λ = λ1,2 = λ∗ and f1 = f2 = 0 provided that A ≠ ±B, where the characteristic 2 value λ∗ = is double: (A + B)s1 y(x) = f (x) + Cy∗ (x), where C is an arbitrary constant and y∗ (x) = g(x) –

(A – B)s1 2As0

is an eigenfunction of the equation corresponding to λ∗ . The equation has no multiple characteristic values if A = ±B.

361

4.9. EQUATIONS WHOSE KERNELS CONTAIN ARBITRARY FUNCTIONS

7.

b

y(x) – λ

[Ag(x) + Bg(t) + C]h(t) y(t) dt = f (x). a

The characteristic values of the equation:

λ1,2 =

(A + B)s1 + Cs0 ±

where

 (A – B)2 s12 + 2(A + B)Cs1 s0 + C 2 s02 + 4ABs0 s2 2AB(s12 – s0 s2 )





b

s0 =

h(x) dx,

s1 =

a



b

g(x)h(x) dx,

b

g 2 (x)h(x) dx.

s2 =

a

,

a

1◦ . Solution with λ ≠ λ1,2 : y(x) = f (x) + λ[A1 g(x) + A2 ], where the constants A1 and A2 are given by Af1 – ABλ(f1 s1 – f2 s0 ) , AB(s12 – s0 s2 )λ2 – [(A + B)s1 + Cs0 ]λ + 1 C1 f1 + Bf2 – ABλ(f2 s1 – f1 s2 ) A2 = , AB(s12 – s0 s2 )λ2 – [(A + B)s1 + Cs0 ]λ + 1 b b f1 = f (x)h(x) dx, f2 = f (x)g(x)h(x) dx. A1 =

a

a

2◦ . Solution with λ = λ1 ≠ λ2 and f1 = f2 = 0: % 1 (x), y(x) = f (x) + Cy

y1 (x) = g(x) +

1 – λ1 As1 , λ1 As0

% is an arbitrary constant and y1 (x) is an eigenfunction of the equation corresponding where C to the characteristic value λ1 . 3◦ . The solution with λ = λ2 ≠ λ1 and f1 = f2 = 0 is given by the formulas of item 2◦ in which one must replace λ1 and y1 (x) by λ2 and y2 (x), respectively. 4◦ . Solution with λ = λ1,2 = λ∗ and f1 = f2 = 0 provided that (A ± B)s1 ± Cs0 =≠ 0, where 2 the characteristic value λ∗ = is double: (A + B)s1 + Cs0 % ∗ (x), y(x) = f (x) + Cy % is an arbitrary constant and where C y∗ (x) = g(x) –

(A – B)s1 – Cs0 2As0

is an eigenfunction of the equation corresponding to λ∗ . The equation has no multiple characteristic values if (A ± B)s1 ± Cs0 = 0.

362

LINEAR EQUATIONS OF THE SECOND KIND WITH CONSTANT LIMITS OF INTEGRATION

8.

b

y(x) – λ

[A + B(x – t)h(t)]y(t) dt = f (x). a

The characteristic values of the equation:

λ1,2

 A(b – a) ± [A(b – a) – 2Bh1 ]2 + 2Bh0 [A(b2 – a2 ) – 2Bh2 ]   , = B A(b – a)[2h1 – (b + a)h0 ] – 2B(h21 – h0 h2 )

where





b

h(x) dx,

h0 =



b

h1 =

xh(x) dx,

a

b

x2 h(x) dx.

h2 =

a

a

1◦ . Solution with λ ≠ λ1,2 : y(x) = f (x) + λ(A1 + A2 x), where the constants A1 and A2 are given by

 f1 – λ B(f1 h1 + f2 h2 ) – 12 Af2 (b2 – a2 ) 

 , A1 =  B A(b – a) h1 – 12 (b + a)h0 – B(h21 – h0 h2 ) λ2 + A(b – a)λ + 1 f – λ[A(b – a)f2 – B(f1 h0 + f2 h1 )] 

2  , A2 =  B A(b – a) h1 – 12 (b + a)h0 – B(h21 – h0 h2 ) λ2 + A(b – a)λ + 1 b b b f (x) dx – B xf (x)h(x) dx, f2 = B f (x)h(x) dx. f1 = A a

a

a

2◦ . Solution with λ = λ1 ≠ λ2 and f1 = f2 = 0: y(x) = f (x) + Cy1 (x),

y1 (x) = 1 +

2 – 2λ1 [A(b – a) – Bh1 ] x, λ1 [A(b2 – a2 ) – 2Bh2 ]

where C is an arbitrary constant, and y1 (x) is an eigenfunction of the equation corresponding to the characteristic value λ1 . 3◦ . The solution with λ = λ2 ≠ λ1 and f1 = f2 = 0 is given by the formulas of item 2◦ in which one must replace λ1 and y1 (x) by λ2 and y2 (x), respectively. 4◦ . Solution with λ = λ1,2 = λ∗ and f1 = f2 = 0 provided that A ≠ 0 or 2Bh1 – A(b – a) ≠ 0, 2 is double: where the characteristic value λ∗ = A(b – a) y(x) = f (x) + Cy∗ (x), where C is an arbitrary constant, and y∗ (x) = 1 –

A(b – a) – 2Bh1 x A(b2 – a2 ) – 2Bh2

is an eigenfunction of the equation corresponding to λ∗ . The equation has no multiple characteristic values if A = 0 or 2Bh1 – A(b – a) = 0.

4.9. EQUATIONS WHOSE KERNELS CONTAIN ARBITRARY FUNCTIONS

9.

363

b

y(x) – λ

[A + (Bx + Ct)h(t)]y(t) dt = f (x). a

The characteristic values of the equation: λ1,2

√ A(b – a) + (C + B)h1 ± D , =  B A(b – a)[2h1 – (b + a)h0 ] + 2C(h21 – h0 h2 )

D = [A(b – a) + (C – B)h1 ]2 + 2Bh0 [A(b2 – a2 ) + 2Ch2 ], where





b

h(x) dx,

h0 = a



b

h1 =

xh(x) dx, a

b

x2 h(x) dx.

h2 = a



1 . Solution with λ ≠ λ1,2 : y(x) = f (x) + λ(A1 + A2 x), where the constants A1 and A2 are given by  

A1 = ∆–1 f1 – λ Bf1 h1 – Cf2 h2 – 12 A(b2 – a2 )f2 ,  

A2 = ∆–1 f2 – λ A(b – a)f2 – Bf1 h0 + Cf2 h1 , 

  ∆ = B A(b – a) h1 – 12 (b + a)h0 + C(h21 – h0 h2 ) λ2 + [A(b – a) + (B + C)h1 ]λ + 1, b b b f (x) dx + C xf (x)h(x) dx, f2 = B f (x)h(x) dx. f1 = A a

a

a

2◦ . Solution with λ = λ1 ≠ λ2 and f1 = f2 = 0: % 1 (x), y(x) = f (x) + Cy

y1 (x) = 1 +

2 – 2λ1 [A(b – a) + Ch1 ] x, λ1 [A(b2 – a2 ) + 2Ch2 ]

% is an arbitrary constant and y1 (x) is an eigenfunction of the equation corresponding where C to the characteristic value λ1 . 3◦ . The solution with λ = λ2 ≠ λ1 and f1 = f2 = 0 is given by the formulas of item 2◦ in which one must replace λ1 and y1 (x) by λ2 and y2 (x), respectively. 4◦ . Solution with λ = λ1,2 = λ∗ and f1 = f2 = 0 provided that ±A(b – a) + (B ± C)h1 ≠ 0, 2 where the characteristic value λ∗ = is double: A(b – a) + (B + C)h1 % ∗ (x), y(x) = f (x) + Cy % is an arbitrary constant and where C y∗ (x) = 1 –

A(b – a) + (C – B)h1 x A(b2 – a2 ) + 2Ch2

is an eigenfunction of the equation corresponding to λ∗ . The equation has no multiple characteristic values if ±A(b – a) + (B ± C)h1 = 0.

364

LINEAR EQUATIONS OF THE SECOND KIND WITH CONSTANT LIMITS OF INTEGRATION

10.

b

y(x) – λ

[A + B(x – t)h(x)]y(t) dt = f (x). a

The characteristic values of the equation:

λ1,2

 A(b – a) ± [A(b – a) + 2Bh1 ]2 – 4Bh0 [A(b – a) + Bh2 ] , = 2B{h0 [A(b – a) + Bh2 ] – h1 [A(b – a) + Bh1 ]}

where





b

h(x) dx,

h0 = a



b

h1 =

xh(x) dx, a

b

x2 h(x) dx.

h2 = a

1◦ . Solution with λ ≠ λ1,2 :

 y(x) = f (x) + λ AE1 + (BE1 x + E2 )h(x) , where the constants E1 and E2 are given by

 E1 = ∆–1 f1 + λB(f1 h1 – f2 h0 ) , 



 E2 = ∆–1 f2 – λf2 A(b – a) + Bh1 – λf1 A(b – a) + Bh2 , ∆ = B {h0 [A(b – a) + Bh2 ] – h1 [A(b – a) + Bh1 ]} λ2 – A(b – a)λ + 1, b b f (x) dx, f2 = xf (x) dx. f1 = a

a

2◦ . Solution with λ = λ1 ≠ λ2 and f1 = f2 = 0: y(x) = f (x) + Cy1 (x),

y1 (x) = A + Bxh(x) +

1 – λ1 [A(b – a) + Bh1 ] h(x), λ1 h0

where C is an arbitrary constant and y1 (x) is an eigenfunction of the equation corresponding to the characteristic value λ1 . 3◦ . The solution with λ = λ2 ≠ λ1 and f1 = f2 = 0 is given by the formulas of item 2◦ in which one must replace λ1 and y1 (x) by λ2 and y2 (x), respectively. 4◦ . Solution with λ = λ1,2 = λ∗ and f1 = f2 = 0 provided that A ≠ 0 or A(b – a) + 4Bh1 ≠ 0, 2 is double: where the characteristic value λ∗ = A(b – a) y(x) = f (x) + Cy∗ (x), where C is an arbitrary constant and y∗ (x) = A + Bxh(x) –

A(b – a) + 2Bh1 h(x) 2h0

is an eigenfunction of the equation corresponding to λ∗ . The equation has no multiple characteristic values if A = 0 or A(b – a) + 4Bh1 = 0.

4.9. EQUATIONS WHOSE KERNELS CONTAIN ARBITRARY FUNCTIONS

11.

365

b

y(x) – λ

[A + (Bx + Ct)h(x)]y(t) dt = f (x). a

The characteristic values of the equation: √ A(b – a) + (B + C)h1 ± D , 2C{h1 [A(b – a) + Bh1 ] – h0 [A(b – a) + Bh2 ]} D = [A(b – a) + (B – C)h1 ]2 + 4Ch0 [A(b – a) + Bh2 ],

λ1,2 =

where





b

h(x) dx,

h0 = a



b

h1 =

xh(x) dx, a

b

x2 h(x) dx.

h2 = a

1◦ . Solution with λ ≠ λ1,2 : 

y(x) = f (x) + λ AE1 + (BE1 x + E2 )h(x) , where the constants E1 and E2 are given by E1 = ∆–1 [f1 – λC(f1 h1 – f2 h0 )], 



 E2 = C∆–1 f2 – λf2 A(b – a) + Bh1 – λf1 A(b – a) + Bh2 , 

 ∆ = C h1 [A(b – a) + Bh1 ] – h0 A(b – a) + Bh2 λ2 – [A(b – a) + (B + C)h1 ]λ + 1, b b f (x) dx, f2 = xf (x) dx. f1 = a

a

2◦ . Solution with λ = λ1 ≠ λ2 and f1 = f2 = 0: % 1 (x), y(x) = f (x) + Cy

y1 (x) = A + Bxh(x) +

1 – λ1 [A(b – a) + Bh1 ] h(x), λ1 h0

% is an arbitrary constant and y1 (x) is an eigenfunction of the equation corresponding where C to the characteristic value λ1 . 3◦ . The solution with λ = λ2 ≠ λ1 and f1 = f2 = 0 is given by the formulas of item 2◦ in which one must replace λ1 and y1 (x) by λ2 and y2 (x), respectively. 4◦ . Solution with λ = λ1,2 = λ∗ and f1 = f2 = 0 provided that A(b – a) + (B ± C)h1 ≠ 0, where 2 the characteristic value λ∗ = is double: A(b – a) + (B + C)h1 % ∗ (x), y(x) = f (x) + Cy % is an arbitrary constant and where C y∗ (x) = A + Bxh(x) –

A(b – a) + (B – C)h1 h(x) 2h0

is an eigenfunction of the equation corresponding to λ∗ . The equation has no multiple characteristic values if A(b – a) + (B ± C)h1 = 0.

366

LINEAR EQUATIONS OF THE SECOND KIND WITH CONSTANT LIMITS OF INTEGRATION

12.

b

y(x) – λ

[g(x)g(t) + h(x)h(t)]y(t) dt = f (x). a

The characteristic values of the equation:  s1 + s3 + (s1 – s3 )2 + 4s22 λ1 = , 2(s1 s3 – s22 ) where





b

g 2 (x) dx,

s1 =



b

s2 =

a

 s1 + s3 – (s1 – s3 )2 + 4s22 λ2 = , 2(s1 s3 – s22 )

g(x)h(x) dx,

b

h2 (x) dx.

s3 =

a

a

1◦ . Solution with λ ≠ λ1,2 : y(x) = f (x) + λ[A1 g(x) + A2 h(x)], where the constants A1 and A2 are given by A1 =

f1 – λ(f1 s3 – f2 s2 ) , (s1 s3 – s22 )λ2 – (s1 + s3 )λ + 1 b f1 = f (x)g(x) dx,

f2 – λ(f2 s1 – f1 s2 ) , (s1 s3 – s22 )λ2 – (s1 + s3 )λ + 1 b f2 = f (x)h(x) dx. A2 =

a

a



2 . Solution with λ = λ1 ≠ λ2 and f1 = f2 = 0: y(x) = f (x) + Cy1 (x),

y1 (x) = g(x) + (1 – λ1 s1 )h(x)/(λ1 s2 ),

where C is an arbitrary constant and y1 (x) is an eigenfunction of the equation corresponding to the characteristic value λ1 . 3◦ . The solution with λ = λ2 ≠ λ1 and f1 = f2 = 0 is given by the formulas of item 2◦ in which one must replace λ1 and y1 (x) by λ2 and y2 (x), respectively. 4◦ . Solution with λ = λ1,2 = λ∗ and f1 = f2 = 0 provided that s1 ≠ ±s3 , where the characteristic value λ∗ = 1/s1 is double: %2 h(x), %1 g(x) + C y(x) = f (x) + C %1 and C %2 are arbitrary constants. where C The equation has no multiple characteristic values if s1 = ±s3 . 13.

b

y(x) – λ

[g(x)g(t) – h(x)h(t)]y(t) dt = f (x). a

The characteristic values of the equation:  s1 – s3 + (s1 + s3 )2 – 4s22 λ1 = , 2(s22 – s1 s3 ) where





b

g 2 (x) dx,

s1 = a

s2 =

 s1 – s3 – (s1 + s3 )2 – 4s22 λ2 = , 2(s22 – s1 s3 )

b

g(x)h(x) dx, a

1◦ . Solution with λ ≠ λ1,2 : y(x) = f (x) + λ[A1 g(x) + A2 h(x)],

b

h2 (x) dx.

s3 = a

4.9. EQUATIONS WHOSE KERNELS CONTAIN ARBITRARY FUNCTIONS

367

where the constants A1 and A2 are given by A1 =

f1 + λ(f1 s3 – f2 s2 ) , (s22 – s1 s3 )λ2 – (s1 – s3 )λ + 1 b f1 = f (x)g(x) dx,

–f2 + λ(f2 s1 – f1 s2 ) , (s22 – s1 s3 )λ2 – (s1 – s3 )λ + 1 b f2 = f (x)h(x) dx. A2 =

a

a



2 . Solution with λ = λ1 ≠ λ2 and f1 = f2 = 0: y(x) = f (x) + Cy1 (x),

1 – λ1 s1 h(x), λ1 s2

y1 (x) = g(x) +

where C is an arbitrary constant and y1 (x) is an eigenfunction of the equation corresponding to the characteristic value λ1 . 3◦ . The solution with λ = λ2 ≠ λ1 and f1 = f2 = 0 is given by the formulas of item 2◦ in which one must replace λ1 and y1 (x) by λ2 and y2 (x), respectively. 4◦ . Solution with λ = λ1,2 = λ∗ and f1 = f2 = 0 provided that s1 ≠ ±s3 , where the characteristic 2 value λ∗ = is double: s1 – s3 y∗ (x) = g(x) –

y(x) = f (x) + Cy∗ (x),

s 1 + s3 h(x), 2s2

where C is an arbitrary constant and y∗ (x) is an eigenfunction of the equation corresponding to λ∗ . The equation has no multiple characteristic values if s1 = ±s3 . 14.

b

[Ag(x)g(t) + Bh(x)h(t)]y(t) dt = f (x).

y(x) – λ a

The characteristic values of the equation: λ1,2 where



 As1 + Bs3 ± (As1 – Bs3 )2 + 4ABs22 , = 2AB(s1 s3 – s22 )

b

g 2 (x) dx,

s1 = a



b

s2 =

g(x)h(x) dx, a

b

h2 (x) dx.

s3 = a

1◦ . Solution with λ ≠ λ1,2 : y(x) = f (x) + λ[A1 g(x) + A2 h(x)], where the constants A1 and A2 are given by A1 =

Af1 – λAB(f1 s3 – f2 s2 ) , AB(s1 s3 – s22 )λ2 – (As1 + Bs3 )λ + 1 b f1 = f (x)g(x) dx, a

Bf2 – λAB(f2 s1 – f1 s2 ) , AB(s1 s3 – s22 )λ2 – (As1 + Bs3 )λ + 1 b f2 = f (x)h(x) dx. A2 =

a



2 . Solution with λ = λ1 ≠ λ2 and f1 = f2 = 0: y(x) = f (x) + Cy1 (x),

y1 (x) = g(x) +

1 – λ1 As1 h(x), λ1 As2

where C is an arbitrary constant and y1 (x) is an eigenfunction of the equation corresponding to the characteristic value λ1 .

368

LINEAR EQUATIONS OF THE SECOND KIND WITH CONSTANT LIMITS OF INTEGRATION

3◦ . The solution with λ = λ2 ≠ λ1 and f1 = f2 = 0 is given by the formulas of item 2◦ in which one must replace λ1 and y1 (x) by λ2 and y2 (x), respectively. 4◦ . Solution with λ = λ1,2 = λ∗ and f1 = f2 = 0 provided that As1 ≠ ±Bs3 , where the 2 characteristic value λ∗ = is double: As1 + Bs3 y(x) = f (x) + Cy∗ (x), where C is an arbitrary constant and y∗ (x) = g(x) –

As1 – Bs3 h(x) 2As2

is an eigenfunction of the equation corresponding to λ∗ . The equation has no multiple characteristic values if As1 = ±Bs3 . 15.

b

y(x) – λ

[g(x)h(t) + h(x)g(t)]y(t) dt = f (x). a

The characteristic values of the equation: λ1 = where

s1 =

1 , √ s1 + s2 s3

λ2 =



b

h(x)g(x) dx,



b

h2 (x) dx,

s2 =

a

1 , √ s1 – s2 s3

a

b

g 2 (x) dx.

s3 = a

1◦ . Solution with λ ≠ λ1,2 : y(x) = f (x) + λ[A1 g(x) + A2 h(x)], where the constants A1 and A2 are given by A1 =

f1 – λ(f1 s1 – f2 s2 ) , (s12 – s2 s3 )λ2 – 2s1 λ + 1 b f1 = f (x)h(x) dx,

f2 – λ(f2 s1 – f1 s3 ) , (s12 – s2 s3 )λ2 – 2s1 λ + 1 b f2 = f (x)g(x) dx.

A2 =

a

a

2◦ . Solution with λ = λ1 ≠ λ2 and f1 = f2 = 0:  y(x) = f (x) + Cy1 (x),

y1 (x) = g(x) +

s3 h(x), s2

where C is an arbitrary constant and y1 (x) is an eigenfunction of the equation corresponding to the characteristic value λ1 . 3◦ . Solution with λ = λ2 ≠ λ1 and f1 = f2 = 0:  y(x) = f (x) + Cy2 (x),

y2 (x) = g(x) –

s3 h(x), s2

where C is an arbitrary constant and y2 (x) is an eigenfunction of the equation corresponding to the characteristic value λ2 . 4◦ . The equation has no multiple characteristic values.

369

4.9. EQUATIONS WHOSE KERNELS CONTAIN ARBITRARY FUNCTIONS

16.

b

y(x) – λ

[g(x)h(t) – h(x)g(t)]y(t) dt = f (x). a

The characteristic values of the equation: 1 λ1 =  , s12 – s2 s3 where



1 λ2 = –  , s12 – s2 s3



b

s1 =

h(x)g(x) dx,



b

h2 (x) dx,

s2 =

a

b

g 2 (x) dx.

s3 =

a

a

1◦ . Solution with λ ≠ λ1,2 : y(x) = f (x) + λ[A1 g(x) + A2 h(x)], where the constants A1 and A2 are given by f1 + λ(f1 s1 – f2 s2 ) –f2 + λ(f2 s1 – f1 s3 ) , A2 = , 2 2 (s2 s3 – s1 )λ + 1 (s2 s3 – s12 )λ2 + 1 b b f1 = f (x)h(x) dx, f2 = f (x)g(x) dx.

A1 =

a

a



2 . Solution with λ = λ1 ≠ λ2 and f1 = f2 = 0:  s12 – s2 s3 – s1 y1 (x) = g(x) + h(x), s2

y(x) = f (x) + Cy1 (x),

where C is an arbitrary constant and y1 (x) is an eigenfunction of the equation corresponding to the characteristic value λ1 . 3◦ . Solution with λ = λ2 ≠ λ1 and f1 = f2 = 0:  y(x) = f (x) + Cy2 (x),

y2 (x) = g(x) –

s12 – s2 s3 + s1 h(x), s2

where C is an arbitrary constant and y2 (x) is an eigenfunction of the equation corresponding to the characteristic value λ2 . 4◦ . The equation has no multiple characteristic values. 17.

b

y(x) – λ

[Ag(x)h(t) + Bh(x)g(t)]y(t) dt = f (x). a

The characteristic values of the equation: λ1,2 where

s1 =

 (A + B)s1 ± (A – B)2 s12 + 4ABs2 s3 , = 2AB(s12 – s2 s3 )

b

h(x)g(x) dx, a



b

h2 (x) dx,

s2 = a

1◦ . Solution with λ ≠ λ1,2 : y(x) = f (x) + λ[A1 g(x) + A2 h(x)],

b

g 2 (x) dx.

s3 = a

370

LINEAR EQUATIONS OF THE SECOND KIND WITH CONSTANT LIMITS OF INTEGRATION

where the constants A1 and A2 are given by Af1 – λAB(f1 s1 – f2 s2 ) , A1 = AB(s12 – s2 s3 )λ2 – (A + B)s1 λ + 1 b f1 = f (x)h(x) dx,

Bf2 – λAB(f2 s1 – f1 s3 ) , AB(s12 – s2 s3 )λ2 – (A + B)s1 λ + 1 b f2 = f (x)g(x) dx.

A2 =

a

a

2◦ . Solution with λ = λ1 ≠ λ2 and f1 = f2 = 0:

1 – λ1 As1 h(x), λ1 As2 where C is an arbitrary constant and y1 (x) is an eigenfunction of the equation corresponding to the characteristic value λ1 . y(x) = f (x) + Cy1 (x),

y1 (x) = g(x) +

3◦ . The solution with λ = λ2 ≠ λ1 and f1 = f2 = 0 is given by the formulas of item 2◦ in which one must replace λ1 and y1 (x) by λ2 and y2 (x), respectively.

18.

4◦ . Solution with λ = λ1,2 = λ∗ and f1 = f2 = 0 provided that A ≠ ±B, where the characteristic 2 value λ∗ = is double: (A + B)s1 (A – B)s1 y(x) = f (x) + Cy∗ (x), y∗ (x) = g(x) – h(x). 2As2 Here C is an arbitrary constant and y∗ (x) is an eigenfunction of the equation corresponding to λ∗ . The equation has no multiple characteristic values if A = ±B. b y(x) – λ [g1 (x)h1 (t) + g2 (x)h2 (t)]y(t) dt = f (x). a

The characteristic values of the equation λ1 and λ2 are given by  s11 + s22 ± (s11 – s22 )2 + 4s12 s21 , λ1,2 = 2(s11 s22 – s12 s21 ) provided that the integrals b b b b s11 = h1(x)g1(x) dx, s12 = h1(x)g2(x) dx, s21 = h2(x)g1(x) dx, s22 = h2(x)g2(x) dx a

a

a

a

are convergent. 1◦ . Solution with λ ≠ λ1,2 : y(x) = f (x) + λ[A1 g1 (x) + A2 g2 (x)], where the constants A1 and A2 are given by f1 – λ(f1 s22 – f2 s12 ) , A1 = (s11 s22 – s12 s21 )λ2 – (s11 + s22 )λ + 1 b f1 = f (x)h1 (x) dx, a

2◦ . Solution with λ = λ1 ≠ λ2 and f1 = f2 = 0:

f2 – λ(f2 s11 – f1 s21 ) , (s11 s22 – s12 s21 )λ2 – (s11 + s22 )λ + 1 b f2 = f (x)h2 (x) dx.

A2 =

a

y(x) = f (x) + Cy1 (x), where C is an arbitrary constant and y1 (x) is an eigenfunction of the equation corresponding to the characteristic value λ1 : 1 – λ1 s11 λ1 s21 y1 (x) = g1 (x) + g2 (x) = g1 (x) + g2 (x). λ1 s12 1 – λ1 s22 3◦ . The solution with λ = λ2 ≠ λ1 and f1 = f2 = 0 is given by the formulas of item 2◦ in which one must replace λ1 and y1 (x) by λ2 and y2 (x), respectively.

371

4.9. EQUATIONS WHOSE KERNELS CONTAIN ARBITRARY FUNCTIONS

4◦ . Solution with λ = λ1,2 = λ∗ and f1 = f2 = 0 provided that s11 ≠ ±s22 , where the characteristic 2 value λ∗ = is double: s11 + s22 y(x) = f (x) + Cy∗ (x), where C is an arbitrary constant and y∗ (x) = g1 (x) –

s11 – s22 g2 (x) 2s12

is an eigenfunction of the equation corresponding to λ∗ . The equation has no multiple characteristic values if s11 = ±s22 . 19.

b

y(x) – λ

[g(x) + h(t)]m y(t) dt = f (x),

m = 1, 2, . . .

a k m–k h (t), and This is a special case of equation 4.9.20 with gk (x) = g k (x), hk (t) = Cm k = 1, . . . , m. Solution: m  Ak g k (x), y(x) = f (x) + λ k=0

where the Ak are constants that can be determined from 4.9.20. 20.

b

y(x) – λ a

 n

 gk (x)hk (t) y(t) dt = f (x),

n = 2, 3, . . .

k=1

The characteristic values of the integral equation (counting the multiplicity, we have exactly n of them) are the roots of the algebraic equation ∆(λ) = 0, where 1 – λs11 –λs21 ∆(λ) = –λs31 .. . –λsn1

–λs12 1 – λs22 –λs32 .. .

–λs13 –λs23 1 – λs33 .. .

··· ··· ··· .. .

–λs1n –λs2n –λs3n .. .



–λs · · · 1 – λsnn n3 –1 s11 – λ s12 s13 –1 s21 s – λ s23 22 –1 s s s 31 32 33 – λ = (–λ)n .. .. .. . . . sn1 sn2 sn3

–λsn2

and the integrals

smk =

hm (x)gk (x) dx; a

are assumed to be convergent.

b

m, k = 1, . . . , n,

, · · · snn – λ–1 ··· ··· ··· .. .

s1n s2n s3n .. .

372

LINEAR EQUATIONS OF THE SECOND KIND WITH CONSTANT LIMITS OF INTEGRATION

Solution with regular λ: y(x) = f (x) + λ

n 

Ak gk (x),

k=1

where the constants Ak form the solution of the following system of algebraic equations: b n  smk Ak = fm , fm = f (x)hm (x) dx, m = 1, . . . , n. Am – λ a

k=1

The Ak can be calculated by Cramer’s rule: Ak = ∆k (λ)/∆(λ), –λs1n 1 – λs11 · · · –λs1k–1 f1 –λs1k+1 · · · · · · –λs2k–1 f2 –λs2k+1 · · · –λs2n –λs21 ∆k (λ) = . ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ –λsn1 · · · –λsnk–1 fn –λsnk+1 · · · 1 – λsnn For solutions of the equation in the case in which λ is a characteristic value, see Subsection 13.2-2.

where

Reference: S. G. Mikhlin (1960).

4.9-2. Equations with Difference Kernel: K(x, t) = K(x – t). 21.

π

y(x) = λ

K(x – t)y(t) dt,

K(x) = K(–x).

–π

Characteristic values: λn =

1 , πan

an =

1 π



π

K(x) cos(nx) dx (n = 0, 1, 2, . . . ). –π

The corresponding eigenfunctions are y0 (x) = 1,

yn(1) (x) = cos(nx),

yn(2) (x) = sin(nx)

(n = 1, 2, . . . ).

For each value λn with n ≠ 0, there are two corresponding linearly independent eigenfunctions yn(1) (x) and yn(2) (x). Reference: M. L. Krasnov, A. I. Kisilev, and G. I. Makarenko (1971).

22.



y(x) +

K(x – t)y(t) dt = Aeλx .

–∞

Solution: y(x) = 23.

A λx e , 1+q





q=

K(x)e–λx dx.

–∞



K(x – t)y(t) dt = A cos(λx) + B sin(λx).

y(x) + –∞

Solution: AIc + BIs BIc – AIs y(x) = cos(λx) + sin(λx), Ic2 + Is2 Ic2 + Is2 ∞ ∞ Ic = 1 + K(z) cos(λz) dz, Is = K(z) sin(λz) dz. –∞

–∞

4.9. EQUATIONS WHOSE KERNELS CONTAIN ARBITRARY FUNCTIONS

24.

373



K(x – t)y(t) dt = f (x).

y(x) – –∞

Here –∞ < x < ∞, f (x) ∈ L1 (–∞, ∞), and K(x) ∈ L1 (–∞, ∞). For the integral equation to be solvable (in L1 ), it is necessary and sufficient that √ % ≠ 0, –∞ < u < ∞, 1 – 2π K(u) % where K(u) =

√1 2π

∞ –∞

(1)

K(x)e–iux dx is the Fourier transform of K(x). In this case, the

equation has a unique solution, which is given by ∞ R(x – t)f (t) dt, y(x) = f (x) + 1 R(x) = √ 2π



–∞ ∞

% R(u) =

iux % du, R(u)e

–∞

% K(u) . √ % 1 – 2π K(u)

Reference: V. A. Ditkin and A. P. Prudnikov (1965).

25.

y(x) –



K(x – t)y(t) dt = f (x). 0

The Wiener–Hopf equation of the second kind.* Here 0 ≤ x < ∞, K(x) ∈ L1 (–∞, ∞), f (x) ∈ L1 (0, ∞), and y(x) ∈ L1 (0, ∞). For the integral equation to be solvable, it is necessary and sufficient that ˇ Ω(u) = 1 – K(u) ≠ 0,

–∞ < u < ∞,

(1)



ˇ where K(u) = K(x)eiux dx is the Fourier transform (in the asymmetric form) of K(x). –∞ In this case, the index of the equation can be introduced, ν = –ind Ω(u) = – 1◦ . Solution with ν = 0:

y(x) = f (x) +

∞ 1 arg Ω(u) –∞ . 2π



R(x, t)f (t) dt, 0



where R(x, t) = R+ (x – t) + R– (t – x) +



R+ (x – s)R– (t – s) ds, 0

and the functions R+ (x) and R– (x) satisfy the conditions R+ (x) = 0 and R– (x) = 0 for x < 0 and are uniquely defined by their Fourier transforms as follows:   ∞ ∞ ln Ω(t) 1 1 ±iut dt . 1+ R± (t)e dt = exp – ln Ω(u) ∓ 2 2πi –∞ t – u 0 Alternatively, R+ (x) and R– (x) can be obtained by constructing the solutions of the equations ∞ R+ (x) + K(x – t)R+ (t) dt = K(x), 0 ≤ x ≤ ∞, 0∞ R– (x) + K(t – x)R– (t) dt = K(–x), 0 ≤ x ≤ ∞. 0

* A comprehensive discussion of this equation is given in Subsection 13.10-1, Section 13.11, and Section 13.12.

374

LINEAR EQUATIONS OF THE SECOND KIND WITH CONSTANT LIMITS OF INTEGRATION

2◦ . Solution with ν > 0: y(x) = f (x) +

ν 

m–1 –x

Cm x

e



+

  ν  m–1 –t R (x, t) f (t) + Cm t e dt, ◦

0

m=1

m=1

where the Cm are arbitrary constants, ◦

R (x, t) =

R+(0) (x

– t) +



R–(1) (t



– x) +

R+(0) (x – s)R–(1) (t – s) ds,

0

and the functions R+(0) (x) and R–(1) (x) are uniquely defined by their Fourier transforms: ν    ∞ ∞ u–i (1) (0) 1+ 1+ R± (t)e±iut dt = R± (t)e±iut dt , u+i 0 0   ∞ ∞ ln Ω◦ (t) 1 1 (0) dt , R± (t)e±iut dt = exp – ln Ω◦ (u) ∓ 1+ 2 2πi –∞ t – u 0 Ω◦ (u)(u + i)ν = Ω(u)(u – i)ν . 3◦ . For ν < 0, the solution exists only if the conditions ∞ f (x)ψm (x) dx = 0, m = 1, 2, . . . , –ν, 0

are satisfied. Here ψ1 (x), . . . , ψν (x) is the system of linearly independent solutions of the transposed homogeneous equation ∞ ψ(x) – K(t – x)ψ(t) dt = 0. 0



Then y(x) = f (x) +



R∗ (x, t)f (t) dt,

0



where ∗

R (x, t) =

R+(1) (x

– t) +

R–(0) (t

– x) +



R+(1) (x – s)R–(0) (t – s) ds,

0

and the functions R+(1) (x) and R–(0) (x) are uniquely defined in item 2◦ by their Fourier transforms. References: V. I. Smirnov (1974), F. D. Gakhov and Yu. I. Cherskii (1978), I. M. Vinogradov (1979).

4.9-3. Other Equations of the Form y(x) + 26.

b a

K(x, t)y(t) dt = F (x).



y(x) –

K(x + t)y(t) dt = f (x). –∞

The Fourier transform is used to solve this equation. Solution: √ ∞ % % f (u) + 2π f%(–u)K(u) 1 y(x) = √ eiux du, √ % K(–u) % 2π –∞ 1 – 2π K(u) where ∞ ∞ 1 1 % f (x)e–iux dx, K(u) = √ K(x)e–iux dx. f%(u) = √ 2π –∞ 2π –∞ Reference: V. A. Ditkin and A. P. Prudnikov (1965).

4.9. EQUATIONS WHOSE KERNELS CONTAIN ARBITRARY FUNCTIONS

27.



y(x) +

375

eβt K(x + t)y(t) dt = Aeλx .

–∞

Solution: y(x) = 28.



y(x) +



eλx – k(λ)e–(β+λ)x , 1 – k(λ)k(–β – λ)



k(λ) =

K(x)e(λ+β)x dx.

–∞

[eβt K(x + t) + M (x – t)]y(t) dt = Aeλx .

–∞

Solution: y(x) = A

Ik (λ)epx – [1 + Im (p)]eλx , Ik (λ)Ik (p) – [1 + Im (λ)][1 + Im (p)]

where





K(z)e

Ik (λ) =

(β+λ)z

dz,

–∞

29.



Im (λ) =

p = –λ – β,

M (z)e–λz dz.

–∞



K(xt)y(t) dt = f (x).

y(x) – 0

The solution can be obtained with the aid of the inverse Mellin transform: y(x) =



1 2πi

c+i∞

c–i∞

% + K(s) % f%(1 – s) f(s) x–s ds, % % 1 – K(s)K(1 – s)

% stand for the Mellin transforms of the right-hand side and of the kernel of the where f% and K integral equation, ∞ ∞ s–1 % % f (s) = f (x)x dx, K(s) = K(x)xs–1 dx. 0

0

Reference: M. L. Krasnov, A. I. Kisilev, and G. I. Makarenko (1971).

30.



y(x) –

K(xt)tβ y(t) dt = Axλ .

0

Solution:

xλ + Iβ+λ x–β–λ–1 , y(x) = A 1 – Iβ+λ I–λ–1

Iµ =



K(ξ)ξ µ dξ.

0

It is assumed that all improper integrals are convergent. 31.

y(x) –



K(xt)tβ y(t) dt = f (x).

0

The solution can be obtained with the aid of the inverse Mellin transform as follows: 1 y(x) = 2πi



c+i∞

c–i∞

% f%(1 + β – s) f%(s) + K(s) x–s ds, % % + β – s) 1 – K(s)K(1

% stand for the Mellin transforms of the right-hand side and of the kernel of the where f% and K integral equation, f%(s) =

0



f (x)xs–1 dx,

% K(s) =

0



K(x)xs–1 dx.

376

LINEAR EQUATIONS OF THE SECOND KIND WITH CONSTANT LIMITS OF INTEGRATION

32.



y(x) –

g(xt)xλ tµ y(t) dt = f (x).

0

This equation can be rewritten in the form of equation 4.9.31 by setting K(z) = z λ g(z) and β = µ – λ. 33.



y(x) –

1 t

0

K

x t

y(t) dt = 0.

Eigenfunctions of this integral equation are determined by the roots of the following transcendental (algebraic) equation for the parameter λ:



K 0

1 z λ–1 dz = 1. z

(1)

1◦ . For a real simple root λn of equation (1), there is a corresponding eigenfunction yn (x) = xλn . 2◦ . For a real root λn of multiplicity r, there are corresponding r eigenfunctions yn1 (x) = xλn ,

yn2 (x) = xλn ln x,

...,

ynr (x) = xλn lnr–1 x.

3◦ . For a complex simple root λn = αn + iβn of equation (1), there is a corresponding pair of eigenfunctions yn(1) (x) = xαn cos(βn ln x),

yn(2) (x) = xαn sin(βn ln x).

4◦ . For a complex root λn = αn +iβn of multiplicity r, there are corresponding r eigenfunction pairs (1) yn1 (x) = xαn cos(βn ln x),

(2) (x) = xαn sin(βn ln x), yn1

(1) yn2 (x) = xαn ln x cos(βn ln x), ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅

(2) yn2 (x) = xαn ln x sin(βn ln x), ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅

(1) (x) = xαn lnr–1 x cos(βn ln x), ynr

(2) (x) = xαn lnr–1 x sin(βn ln x). ynr

The general solution is the linear combination (with arbitrary constants) of the eigenfunctions of the homogeneous integral equation. 34.



y(x) – 0

1 t

K

x t

y(t) dt = Axb .

A solution: A y(x) = xb , B





B =1–

K 0

1 ξ

ξ b–1 dξ.

It is assumed that the improper integral is convergent and B ≠ 0. The general solution of the integral equations is the sum of the above solution and the solution of the homogeneous equation 4.9.33.

4.9. EQUATIONS WHOSE KERNELS CONTAIN ARBITRARY FUNCTIONS



35.

377

1 x K y(t) dt = f (x). t t 0 The solution can be obtained with the aid of the inverse Mellin transform: c+i∞ % f (s) 1 y(x) = x–s ds, % 2πi c–i∞ 1 – K(s) ∞

y(x) –

% stand for the Mellin transforms of the right-hand side and the kernel of the where f% and K integral equation, ∞ ∞ s–1 % % f (s) = f (x)x dx, K(s) = K(x)xs–1 dx. 0

0

Example. For f (x) = Ae–λx and K(x) = 12 e–x , the solution of the integral equation has the form ⎧ 4A ⎪ ⎪ for λx > 1, ⎪ ⎨ (3 – 2C)(λx)3 y(x) = ∞  1 ⎪ ⎪ ⎪ for λx < 1. ⎩ –2A sk ψ(s ) (λx) k k=1 Here C = 0.5772 . . . is the Euler constant, ψ(z) = [ln Γ(z)]z is the logarithmic derivative of the gamma function, and the sk are the negative roots of the transcendental equation Γ(sk ) = 2, where Γ(z) is the gamma function. Reference: M. L. Krasnov, A. I. Kisilev, and G. I. Makarenko (1971).

36.

b

y(x) + a

|x – t|g(t)y(t) dt = f (x),

a ≤ x ≤ b.

1◦ . Let us remove the modulus in the integrand, b x (x – t)g(t)y(t) dt + (t – x)g(t)y(t) dt = f (x). y(x) + a

(1)

x

Differentiating (1) with respect to x yields x g(t)y(t) dt – yx (x) + a

b

g(t)y(t) dt = fx (x).

(2)

x

Differentiating (2), we arrive at a second-order ordinary differential equation for y = y(x),   + 2g(x)y = fxx (x). yxx

(3)



2 . Let us derive the boundary conditions for equation (3). We assume that the limits of integration satisfy the conditions –∞ < a < b < ∞. By setting x = a and x = b in (1), we obtain two consequences b y(a) + (t – a)g(t)y(t) dt = f (a), a (4) b y(b) +

(b – t)g(t)y(t) dt = f (b).

a  yxx

 and fxx and substitute the result into (4). Integrating Let us express g(x)y from (3) via by parts yields the desired boundary conditions for y(x),

y(a) + y(b) + (b – a)[fx (b) – yx (b)] = f (a) + f (b), y(a) + y(b) + (a – b)[fx (a) – yx (a)] = f (a) + f (b).

(5)

Note a useful consequence of (5), yx (a) + yx (b) = fx (a) + fx (b),

(6)

which can be used together with one of conditions (5). Equation (3) under the boundary conditions (5) determines the solution of the original integral equation. Conditions (5) make it possible to calculate the constants of integration that occur in the solution of the differential equation (3).

378

LINEAR EQUATIONS OF THE SECOND KIND WITH CONSTANT LIMITS OF INTEGRATION

37.

b

y(x) +

eλ|x–t| g(t)y(t) dt = f (x),

a ≤ x ≤ b.

a

1◦ . Let us remove the modulus in the integrand: x λ(x–t) y(x) + e g(t)y(t) dt + a

b

eλ(t–x) g(t)y(t) dt = f (x).

(1)

x

Differentiating (1) with respect to x twice yields x  (x) + 2λg(x)y(x) + λ2 eλ(x–t) g(t)y(t) dt + λ2 yxx a

b

 eλ(t–x) g(t)y(t) dt = fxx (x).

(2)

x

Eliminating the integral terms from (1) and (2), we arrive at a second-order ordinary differential equation for y = y(x),   yxx + 2λg(x)y – λ2 y = fxx (x) – λ2 f (x).

(3)

2◦ . Let us derive the boundary conditions for equation (3). We assume that the limits of integration satisfy the conditions –∞ < a < b < ∞. By setting x = a and x = b in (1), we obtain two consequences b y(a) + e–λa eλt g(t)y(t) dt = f (a), a (4) b y(b) + eλb

e–λt g(t)y(t) dt = f (b). a

Let us express g(x)y from (3) via by parts yields the conditions

 yxx

 and fxx and substitute the result into (4). Integrating

eλb ϕx (b) – eλa ϕx (a) = λeλa ϕ(a) + λeλb ϕ(b), e–λb ϕx (b) – e–λa ϕx (a) = λe–λa ϕ(a) + λe–λb ϕ(b),

ϕ(x) = y(x) – f (x).

Finally, after some manipulations, we arrive at the desired boundary conditions for y(x): ϕx (a) + λϕ(a) = 0,

ϕx (b) – λϕ(b) = 0;

ϕ(x) = y(x) – f (x).

(5)

Equation (3) under the boundary conditions (5) determines the solution of the original integral equation. Conditions (5) make it possible to calculate the constants of integration that occur in solving the differential equation (3). 38.

b

y(x) +

a ≤ x ≤ b.

sinh(λ|x – t|)g(t)y(t) dt = f (x), a

1◦ . Let us remove the modulus in the integrand: x sinh[λ(x – t)]g(t)y(t) dt + y(x) + a

b

sinh[λ(t – x)]g(t)y(t) dt = f (x).

(1)

x

Differentiating (1) with respect to x twice yields x  (x) + 2λg(x)y(x) + λ2 sinh[λ(x – t)]g(t)y(t) dt yxx a



b

+ λ2 x

 sinh[λ(t – x)]g(t)y(t) dt = fxx (x).

(2)

4.9. EQUATIONS WHOSE KERNELS CONTAIN ARBITRARY FUNCTIONS

379

Eliminating the integral terms from (1) and (2), we arrive at a second-order ordinary differential equation for y = y(x),   yxx + 2λg(x)y – λ2 y = fxx (x) – λ2 f (x).

(3)



2 . Let us derive the boundary conditions for equation (3). We assume that the limits of integration satisfy the conditions –∞ < a < b < ∞. By setting x = a and x = b in (1), we obtain two corollaries b y(a) + sinh[λ(t – a)]g(t)y(t) dt = f (a), a (4) b sinh[λ(b – t)]g(t)y(t) dt = f (b).

y(b) + a

  and fxx and substitute the result into (4). Integrating Let us express g(x)y from (3) via yxx by parts yields the desired boundary conditions for y(x),

sinh[λ(b – a)]ϕx (b) – λ cosh[λ(b – a)]ϕ(b) = λϕ(a), sinh[λ(b – a)]ϕx (a) + λ cosh[λ(b – a)]ϕ(a) = –λϕ(b);

39.

ϕ(x) = y(x) – f (x).

(5)

Equation (3) under the boundary conditions (5) determines the solution of the original integral equation. Conditions (5) make it possible to calculate the constants of integration that occur in solving the differential equation (3). b y(x) + sin(λ|x – t|)g(t)y(t) dt = f (x), a ≤ x ≤ b. a

1◦ . Let us remove the modulus in the integrand: x sin[λ(x – t)]g(t)y(t) dt + y(x) + a

b

sin[λ(t – x)]g(t)y(t) dt = f (x).

(1)

x

Differentiating (1) with respect to x twice yields x  (x) + 2λg(x)y(x) – λ2 sin[λ(x – t)]g(t)y(t) dt yxx

a b

– λ2

 sin[λ(t – x)]g(t)y(t) dt = fxx (x).

(2)

x

Eliminating the integral terms from (1) and (2), we arrive at a second-order ordinary differential equation for y = y(x),   + 2λg(x)y + λ2 y = fxx (x) + λ2 f (x). yxx

(3)



2 . Let us derive the boundary conditions for equation (3). We assume that the limits of integration satisfy the conditions –∞ < a < b < ∞. By setting x = a and x = b in (1), we obtain two consequences b y(a) + sin[λ(t – a)]g(t)y(t) dt = f (a), a (4) b sin[λ(b – t)]g(t)y(t) dt = f (b).

y(b) + a

  and fxx and substitute the result into (4). Integrating Let us express g(x)y from (3) via yxx by parts yields the desired boundary conditions for y(x),

sin[λ(b – a)]ϕx (b) – λ cos[λ(b – a)]ϕ(b) = λϕ(a), sin[λ(b – a)]ϕx (a) + λ cos[λ(b – a)]ϕ(a) = –λϕ(b);

ϕ(x) = y(x) – f (x).

(5)

Equation (3) under the boundary conditions (5) determines the solution of the original integral equation. Conditions (5) make it possible to calculate the constants of integration that occur in solving the differential equation (3).

380

LINEAR EQUATIONS OF THE SECOND KIND WITH CONSTANT LIMITS OF INTEGRATION

40.



y(x) +

–|x–t| λe + ϕ(x)ψ(t)]y(t) dt = f (x).

–∞

The solutions can be obtained by the methods described in Subsection 13.2-3; it must be taken into account that the truncated equation, with ϕ(x) = 0, coincides with equation 4.2.14. Solution: y(x) = Yf (x) + AYϕ (x), where

∞ 0

A=–

1+ 41.



∞  √  exp – 1 + 2λ |x – t| f (t) dt, Yf (x) = f (x) – √ 1 + 2λ –∞ ∞  √  λ Yϕ (x) = ϕ(x) – √ exp – 1 + 2λ |x – t| ϕ(t) dt, 1 + 2λ –∞

λ

ψ(t)Yf (t) dt

∞ 0

, ψ(t)Yϕ (t) dt

1 λ>– . 2



y(x) –

[λ sin(xt) + ϕ(x)ψ(t)]y(t) dt = f (x). 0

The solution can be obtained by the methods described in Subsection 13.2-3; it must be taken into account that the truncated equation, with ϕ(x) = 0, coincides with equation 4.5.20. Solution: y(x) = Yf (x) + AYϕ (x), where

∞ f (x) λ + sin(xt)f (t) dt, 1 – π2 λ2 1 – π2 λ2 0 ∞ ϕ(x) λ + sin(xt)ϕ(t) dt, Yϕ (x) = 1 – π2 λ2 1 – π2 λ2 0 ∞  ψ(t)Yf (t) dt 2 0 . , λ≠± A= ∞ π 1– ψ(t)Y (t) dt Yf (x) =

0

42.

y(x) –

ϕ



[λ cos(xt) + ϕ(x)ψ(t)]y(t) dt = f (x). 0

The solution can be obtained by the methods described in Subsection 13.2-3; it must be taken into account that the truncated equation, with ϕ(x) = 0, coincides with equation 4.5.6. Solution: y(x) = Yf (x) + AYϕ (x), where

∞ f (x) λ + cos(xt)f (t) dt, Yf (x) = 1 – π2 λ2 1 – π2 λ2 0 ∞ ϕ(x) λ + cos(xt)ϕ(t) dt, Yϕ (x) = 1 – π2 λ2 1 – π2 λ2 0 ∞  ψ(t)Yf (t) dt 2 0 . , λ≠± A= ∞ π 1– ψ(t)Y (t) dt 0

ϕ

4.9. EQUATIONS WHOSE KERNELS CONTAIN ARBITRARY FUNCTIONS

43.

381



[λtJν (xt) + ϕ(x)ψ(t)]y(t) dt = f (x),

y(x) +

ν > –1.

0

Here Jν (z) is the Bessel function of the first kind. The solution can be obtained by the methods described in Subsection 13.2-3; it must be taken into account that the truncated equation, with ϕ(x) = 0, coincides with equation 4.8.4. Solution: y(x) = Yf (x) + AYϕ (x), ∞ f (x) λ – tJν (xt)f (t) dt, Yf (x) = 1 – λ2 1 – λ2 0 ∞ ϕ(x) λ Yϕ (x) = – tJν (xt)ϕ(t) dt, 1 – λ2 1 – λ2 0

where

∞ 0

A=–

1+

4.9-4. Equations of the Form y(x) + 44.

b a

ψ(t)Yf (t) dt

∞ 0

,

λ ≠ ±1.

ψ(t)Yϕ (t) dt

K(x, t)y(· · ·) dt = F (x).

b

y(x) +

f (t)y(x – t) dt = 0. a

Eigenfunctions of this integral equation* are determined by the roots of the following characteristic (transcendental or algebraic) equation for µ: b f (t) exp(–µt) dt = –1. (1) a

1◦ . For a real (simple) root µk of equation (1), there is a corresponding eigenfunction yk (x) = exp(µk x). 2◦ . For a real root µk of multiplicity r, there are corresponding r eigenfunctions yk1 (x) = exp(µk x),

yk2 (x) = x exp(µk x),

...,

ykr (x) = xr–1 exp(µk x).

3◦ . For a complex (simple) root µk = αk + iβk of equation (1), there is a corresponding pair of eigenfunctions yk(1) (x) = exp(αk x) cos(βk x),

yk(2) (x) = exp(αk x) sin(βk x).

4◦ . For a complex root µk = αk + iβk of multiplicity r, there are corresponding r pairs of eigenfunctions (1) yk1 (x) = exp(αk x) cos(βk x),

(2) (x) = exp(αk x) sin(βk x), yk1

(1) yk2 (x) = x exp(αk x) cos(βk x),

(2) yk2 (x) = x exp(αk x) sin(βk x),

⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ (1) (x) ykr

= xr–1 exp(αk x) cos(βk x),

⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ (2) (x) = xr–1 exp(αk x) sin(βk x). ykr

The general solution is the linear combination (with arbitrary constants) of the eigenfunctions of the homogeneous integral equation. * In the equations below that contain y(x – t) in the integrand, the arguments can have, for example, the domain (a) –∞ < x < ∞, –∞ < t < ∞ for a = –∞ and b = ∞ or (b) a ≤ t ≤ b, –∞ ≤ x < ∞, for a and b such that –∞ < a < b < ∞. Case (b) is a special case of (a) if f (t) is nonzero only on the interval a ≤ t ≤ b.

382

LINEAR EQUATIONS OF THE SECOND KIND WITH CONSTANT LIMITS OF INTEGRATION

 For equations 4.9.45–4.9.50, only particular solutions are given. To obtain the general solution, one must add the particular solution to the general solution of the corresponding homogeneous equation 4.9.44. b 45. y(x) + f (t)y(x – t) dt = Ax + B. a

A solution: y(x) = px + q, where the coefficients p and q are given by

46.

A AI1 B p= , q= + , 1 + I0 (1 + I0 )2 1 + I0 b y(x) + f (t)y(x – t) dt = Aeλx .





b

I0 =

f (t) dt,

b

I1 =

a

tf (t) dt. a

a

A solution:

b A λx e , B =1+ f (t) exp(–λt) dt. B a The general solution of the integral equation is the sum of the specified particular solution and the general solution of the homogeneous equation 4.9.44. b f (t)y(x – t) dt = A sin(λx). y(x) + y(x) =

47.

a

A solution:

AIc AIs sin(λx) + 2 cos(λx), 2 + Is Ic + Is2 where the coefficients Ic and Is are given by b b Ic = 1 + f (t) cos(λt) dt, Is = f (t) sin(λt) dt. y(x) =

Ic2

a

48.

a

b

y(x) +

f (t)y(x – t) dt = A cos(λx). a

A solution:

AIc AIs sin(λx) + 2 cos(λx), 2 + Is Ic + Is2 where the coefficients Ic and Is are given by b b Ic = 1 + f (t) cos(λt) dt, Is = f (t) sin(λt) dt. y(x) = –

Ic2

a

49.

b

y(x) +

a

f (t)y(x – t) dt = eµx (A sin λx + B cos λx).

a

A solution: y(x) = eµx (p sin λx + q cos λx), where the coefficients p and q are given by p=

AIc – BIs , Ic2 + Is2

b

f (t)e–µt cos(λt) dt,

Ic = 1 + a

AIs + BIc , Ic2 + Is2 b Is = f (t)e–µt sin(λt) dt.

q=

a

4.9. EQUATIONS WHOSE KERNELS CONTAIN ARBITRARY FUNCTIONS

50.

383

b

y(x) +

f (t)y(x – t) dt = g(x). a

1◦ . For g(x) =

n 

Ak exp(λk x), the equation has a solution

k=1

y(x) =



n  Ak exp(λk x), Bk

b

Bk = 1 +

f (t) exp(–λk t) dt. a

k=1

2◦ . For polynomial right-hand side of the equation, g(x) =

n 

Ak xk , a solution has the form

k=0

y(x) =

n 

Bk xk ,

k=0

where the constants Bk can be found by the method of undetermined coefficients. n  3◦ . For g(x) = eλx Ak xk , a solution of the equation has the form k=0

y(x) = eλx

n 

Bk xk ,

k=0

where the constants Bk can be found by the method of undetermined coefficients. n  4◦ . For g(x) = Ak cos(λk x), a solution of the equation has the form k=1

y(x) =

n 

Bk cos(λk x) +

k=1

n 

Ck sin(λk x),

k=1

where the constants Bk and Ck can be found by the method of undetermined coefficients. n  5◦ . For g(x) = Ak sin(λk x), a solution of the equation has the form k=1

y(x) =

n 

Bk cos(λk x) +

k=1

n 

Ck sin(λk x),

k=1

where the constants Bk and Ck can be found by the method of undetermined coefficients. n  6◦ . For g(x) = cos(λx) Ak xk , a solution of the equation has the form k=0

y(x) = cos(λx)

n 

Bk xk + sin(λx)

k=0

n 

Ck xk ,

k=0

where the constants Bk and Ck can be found by the method of undetermined coefficients. n  7◦ . For g(x) = sin(λx) Ak xk , a solution of the equation has the form k=0

y(x) = cos(λx)

n  k=0

Bk xk + sin(λx)

n 

Ck xk ,

k=0

where the constants Bk and Ck can be found by the method of undetermined coefficients.

384

LINEAR EQUATIONS OF THE SECOND KIND WITH CONSTANT LIMITS OF INTEGRATION

8◦ . For g(x) = eµx

n 

Ak cos(λk x), a solution of the equation has the form

k=1

y(x) = eµx

n 

Bk cos(λk x) + eµx

k=1

n 

Ck sin(λk x),

k=1

where the constants Bk and Ck can be found by the method of undetermined coefficients. 9◦ . For g(x) = eµx

n 

Ak sin(λk x), a solution of the equation has the form

k=1

y(x) = eµx

n 

Bk cos(λk x) + eµx

k=1

n 

Ck sin(λk x),

k=1

where the constants Bk and Ck can be found by the method of undetermined coefficients. 10◦ . For g(x) = cos(λx)

n 

Ak exp(µk x), a solution of the equation has the form

k=1

y(x) = cos(λx)

n 

Bk exp(µk x) + sin(λx)

k=1

n 

Bk exp(µk x),

k=1

where the constants Bk and Ck can be found by the method of undetermined coefficients. 11◦ . For g(x) = sin(λx)

n 

Ak exp(µk x), a solution of the equation has the form

k=1

y(x) = cos(λx)

n 

Bk exp(µk x) + sin(λx)

k=1

n 

Bk exp(µk x),

k=1

where the constants Bk and Ck can be found by the method of undetermined coefficients. 51.

b

f (t)y(x + βt) dt = Ax + B.

y(x) + a

A solution:* y(x) = px + q, where A , 1 + I0

p= 52.

b

y(x) +

q=

B AI1 β – , 1 + I0 (1 + I0 )2

I0 =



b

f (t) dt,

I1 =