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HANDBOOK OF

INTEGRALEQUATIONS

1998 by CRC Press LLC

Page iv

Andrei D. Polyanin

and

Alexander V. Manzhirov

HANDBOOK OF

INTEGRALEQUATIONS

Library of Congress Cataloging-in-Publication Data

Polyanin, A. D. (Andrei Dmitrievich)Handbook of integral equations/Andrei D. Polyanin, Alexander

V. Manzhirov.p. cm.

Includes bibliographical references (p. - ) and index.ISBN 0-8493-2876-4 (alk. paper)1. Integral equationsHandbooks, manuals, etc. I. Manzhirov. A.

V. (Aleksandr Vladimirovich) II. Title.QA431.P65 1998 98-10762

515.45dc21 CIP

This book contains information obtained from authentic and highly regarded sources. Reprinted material is quoted withpermission, and sources are indicated. A wide variety of references are listed. Reasonable efforts have been made to publishreliable data and information, but the author and the publisher cannot assume responsibility for the validity of all materialsor for the consequences of their use.

Neither this book nor any part may be reproduced or transmitted in any form or by any means, electronic or mechanical,including photocopying, microfilming, and recording, or by any information storage or retrieval system, without priorpermission in writing from the publisher.

The consent of CRC Press LLC does not extend to copying for general distribution, for promotion, for creating new works,or for resale. Specific permission must be obtained in writing from CRC Press LLC for such copying.

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Visit the CRC Press Web site at www.crcpress.com


No claim to original U.S. Government worksInternational Standard Book Number 0-8493-2876-4

Library of Congress Card Number 98-10762Printed in the United States of America 3 4 5 6 7 8 9 0

Printed on acid-free paper

http://www.crcpress.com

ANNOTATION

More than 2100 integral equations with solutions are given in the first part of the book. A lotof new exact solutions to linear and nonlinear equations are included. Special attention is paid toequations of general form, which depend on arbitrary functions. The other equations contain oneor more free parameters (it is the readers option to fix these parameters). Totally, the number ofequations described is an order of magnitude greater than in any other book available.

A number of integral equations are considered which are encountered in various fields ofmechanics and theoretical physics (elasticity, plasticity, hydrodynamics, heat and mass transfer,electrodynamics, etc.).

The second part of the book presents exact, approximate analytical and numerical methodsfor solving linear and nonlinear integral equations. Apart from the classical methods, some newmethods are also described. Each section provides examples of applications to specific equations.

The handbook has no analogs in the world literature and is intended for a wide audienceof researchers, college and university teachers, engineers, and students in the various fields ofmathematics, mechanics, physics, chemistry, and queuing theory.

Page iii


FOREWORD

Integral equations are encountered in various fields of science and numerous applications (inelasticity, plasticity, heat and mass transfer, oscillation theory, fluid dynamics, filtration theory,electrostatics, electrodynamics, biomechanics, game theory, control, queuing theory, electrical en-gineering, economics, medicine, etc.).

Exact (closed-form) solutions of integral equations play an important role in the proper un-derstanding of qualitative features of many phenomena and processes in various areas of naturalscience. Lots of equations of physics, chemistry and biology contain functions or parameters whichare obtained from experiments and hence are not strictly fixed. Therefore, it is expedient to choosethe structure of these functions so that it would be easier to analyze and solve the equation. As apossible selection criterion, one may adopt the requirement that the model integral equation admit asolution in a closed form. Exact solutions can be used to verify the consistency and estimate errorsof various numerical, asymptotic, and approximate methods.

More than 2100 integral equations and their solutions are given in the first part of the book(Chapters 16). A lot of new exact solutions to linear and nonlinear equations are included. Specialattention is paid to equations of general form, which depend on arbitrary functions. The otherequations contain one or more free parameters (the book actually deals with families of integralequations); it is the readers option to fix these parameters. Totally, the number of equationsdescribed in this handbook is an order of magnitude greater than in any other book currentlyavailable.

The second part of the book (Chapters 714) presents exact, approximate analytical, and numer-ical 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 apronounced preference to practical aspects of the matter; that is, to methods that allow effectivelyconstructing the solution. For the readers better understanding of the methods, each section issupplied with examples of specific equations. Some sections may be used by lecturers of collegesand universities as a basis for courses on integral equations and mathematical physics equations forgraduate and postgraduate students.

For the convenience of a wide audience with different mathematical backgrounds, the authorstried to do their best, wherever possible, to avoid special terminology. Therefore, some of the methodsare outlined in a schematic and somewhat simplified manner, with necessary references made tobooks where these methods are considered in more detail. For some nonlinear equations, onlysolutions of the simplest form are given. The book does not cover two-, three- and multidimensionalintegral equations.

The handbook consists of chapters, sections and subsections. Equations and formulas arenumbered separately in each section. The equations within a section are arranged in increasingorder of complexity. The extensive table of contents provides rapid access to the desired equations.

For the readers convenience, the main material is followed by a number of supplements, wheresome properties of elementary and special functions are described, tables of indefinite and definiteintegrals are given, as well as tables of Laplace, Mellin, and other transforms, which are used in thebook.

The first and second parts of the book, just as many sections, were written so that they could beread independently from each other. This allows the reader to quickly get to the heart of the matter.

Page v


We would like to express our deep gratitude to Rolf Sulanke and Alexei Zhurov for fruitfuldiscussions and valuable remarks. We also appreciate the help of Vladimir Nazaikinskii andAlexander Shtern in translating the second part of this book, and are thankful to Inna Shingareva forher 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. PolyaninA. V. Manzhirov

Page vi


SOME REMARKS AND NOTATION

1. In Chapters 111 and 14, in the original integral equations, the independent variable isdenoted 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:

f x =df

dx, f xx =

d2f

dx2, f xxx =

d3f

dx3, f xxxx =

d4f

dx4, and f (n)x =

dnf

dxnfor n 5.

Occasionally, we use the similar notation for partial derivatives of a function of two variables,

for example, K x(x, t) =

xK(x, t).

3. In some cases, we use the operator notation[f (x)

d

dx

]ng(x), which is defined recursively by

[f (x)

d

dx

]ng(x) = f (x)

d

dx

{[f (x)

d

dx

]n1g(x)

}.

4. It is indicated in the beginning of Chapters 16 that f = f (x), g = g(x), K = K(x), etc. arearbitrary 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 realarguments;*

(b) if the solution contains derivatives of these functions, then the functions are assumed to besufficiently differentiable;**

(c) if the solution contains integrals with these functions (in combination with other functions), thenthe integrals are supposed to converge;

(d) the free parameters A, B, etc. may assume any real values for which the expressions occurring

in the equation and the solution make sense (for example, if a solution contains a factorA

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 acomplex quantity z.

6. In the first part of the book (Chapters 16) when referencing a particular equation, we use anotation 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 16);indicates the literature used in the preparation of the text in specific equations (Chapters 16) orsections (Chapters 714).

* 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.

Page vii


AUTHORS

Andrei D. Polyanin, D.Sc., Ph.D., is a noted scientist of broad interests, who works in variousfields of mathematics, mechanics, and chemical engineering science.

A. D. Polyanin graduated from the Department of Mechanics andMathematics of the Moscow State University in 1974. He receivedhis Ph.D. degree in 1981 and D.Sc. degree in 1986 at the Institute forProblems in Mechanics of the Russian (former USSR) Academy ofSciences. Since 1975, A. D. Polyanin has been a member of the staffof the Institute for Problems in Mechanics of the Russian Academy ofSciences.

Professor Polyanin has made important contributions to developingnew exact and approximate analytical methods of the theory of differ-ential equations, mathematical physics, integral equations, engineeringmathematics, nonlinear mechanics, theory of heat and mass transfer,and chemical hydrodynamics. He obtained exact solutions for sev-eral thousands of ordinary differential, partial differential, and integralequations.

Professor Polyanin is an author of 17 books in English, Russian, German, and Bulgarian. Hispublications also include more than 110 research papers and three patents. One of his most significantbooks is A. D. Polyanin and V. F. Zaitsev, Handbook of Exact Solutions for Ordinary DifferentialEquations, CRC Press, 1995.

In 1991, A. D. Polyanin was awarded a Chaplygin Prize of the USSR Academy of Sciences forhis research in mechanics.

Alexander V. Manzhirov, D.Sc., Ph.D., is a prominent scientist in the fields of mechanics andapplied mathematics, integral equations, and their applications.

After graduating from the Department of Mechanics and Mathemat-ics of the Rostov State University in 1979, A. V. Manzhirov attended apostgraduate course at the Moscow Institute of Civil Engineering. Hereceived his Ph.D. degree in 1983 at the Moscow Institute of ElectronicEngineering Industry and his D.Sc. degree in 1993 at the Institute forProblems in Mechanics of the Russian (former USSR) Academy ofSciences. Since 1983, A. V. Manzhirov has been a member of the staffof the Institute for Problems in Mechanics of the Russian Academyof Sciences. He is also a Professor of Mathematics at the BaumanMoscow State Technical University and a Professor of Mathematicsat the Moscow State Academy of Engineering and Computer Science.Professor Manzhirov is a member of the editorial board of the jour-nal Mechanics of Solids and a member of the European MechanicsSociety (EUROMECH).

Professor Manzhirov has made important contributions to new mathematical methods for solvingproblems in the fields of integral equations, mechanics of solids with accretion, contact mechanics,and the theory of viscoelasticity and creep. He is an author of 3 books, 60 scientific publications,and two patents.

Page ix


CONTENTS

Annotation

Foreword

Some Remarks and Notation

Part I. Exact Solutions of Integral Equations1. Linear Equations of the First Kind With Variable Limit of Integration

1.1. Equations Whose Kernels Contain Power-Law Functions1.1-1. Kernels Linear in the Arguments x and t1.1-2. Kernels Quadratic in the Arguments x and t1.1-3. Kernels Cubic in the Arguments x and t1.1-4. Kernels Containing Higher-Order Polynomials in x and t1.1-5. Kernels Containing Rational Functions1.1-6. Kernels Containing Square Roots1.1-7. Kernels Containing Arbitrary Powers

1.2. Equations Whose Kernels Contain Exponential Functions1.2-1. Kernels Containing Exponential Functions1.2-2. Kernels Containing Power-Law and Exponential Functions

1.3. Equations Whose Kernels Contain Hyperbolic Functions1.3-1. Kernels Containing Hyperbolic Cosine1.3-2. Kernels Containing Hyperbolic Sine1.3-3. Kernels Containing Hyperbolic Tangent1.3-4. Kernels Containing Hyperbolic Cotangent1.3-5. Kernels Containing Combinations of Hyperbolic Functions

1.4. Equations Whose Kernels Contain Logarithmic Functions1.4-1. Kernels Containing Logarithmic Functions1.4-2. Kernels Containing Power-Law and Logarithmic Functions

1.5. Equations Whose Kernels Contain Trigonometric Functions1.5-1. Kernels Containing Cosine1.5-2. Kernels Containing Sine1.5-3. Kernels Containing Tangent1.5-4. Kernels Containing Cotangent1.5-5. Kernels Containing Combinations of Trigonometric Functions

1.6. Equations Whose Kernels Contain Inverse Trigonometric Functions1.6-1. Kernels Containing Arccosine1.6-2. Kernels Containing Arcsine1.6-3. Kernels Containing Arctangent1.6-4. Kernels Containing Arccotangent

Page ix


1.7. Equations Whose Kernels Contain Combinations of Elementary Functions1.7-1. Kernels Containing Exponential and Hyperbolic Functions1.7-2. Kernels Containing Exponential and Logarithmic Functions1.7-3. Kernels Containing Exponential and Trigonometric Functions1.7-4. Kernels Containing Hyperbolic and Logarithmic Functions1.7-5. Kernels Containing Hyperbolic and Trigonometric Functions1.7-6. Kernels Containing Logarithmic and Trigonometric Functions

1.8. Equations Whose Kernels Contain Special Functions1.8-1. Kernels Containing Bessel Functions1.8-2. Kernels Containing Modified Bessel Functions1.8-3. Kernels Containing Associated Legendre Functions1.8-4. Kernels Containing Hypergeometric Functions

1.9. Equations Whose Kernels Contain Arbitrary Functions1.9-1. 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

1.10. Some Formulas and Transformations

2. Linear Equations of the Second Kind With Variable Limit of Integration2.1. Equations Whose Kernels Contain Power-Law Functions

2.1-1. Kernels Linear in the Arguments x and t2.1-2. Kernels Quadratic in the Arguments x and t2.1-3. Kernels Cubic in the Arguments x and t2.1-4. Kernels Containing Higher-Order Polynomials in x and t2.1-5. Kernels Containing Rational Functions2.1-6. Kernels Containing Square Roots and Fractional Powers2.1-7. Kernels Containing Arbitrary Powers


2.3. Equations Whose Kernels Contain Hyperbolic Functions2.3-1. Kernels Containing Hyperbolic Cosine2.3-2. Kernels Containing Hyperbolic Sine2.3-3. Kernels Containing Hyperbolic Tangent2.3-4. Kernels Containing Hyperbolic Cotangent2.3-5. Kernels Containing Combinations of Hyperbolic Functions


2.5. Equations Whose Kernels Contain Trigonometric Functions2.5-1. Kernels Containing Cosine2.5-2. Kernels Containing Sine2.5-3. Kernels Containing Tangent2.5-4. Kernels Containing Cotangent2.5-5. Kernels Containing Combinations of Trigonometric Functions


Page x


2.7. Equations Whose Kernels Contain Combinations of Elementary Functions2.7-1. Kernels Containing Exponential and Hyperbolic Functions2.7-2. Kernels Containing Exponential and Logarithmic Functions3.7-3. Kernels Containing Exponential and Trigonometric Functions2.7-4. Kernels Containing Hyperbolic and Logarithmic Functions2.7-5. Kernels Containing Hyperbolic and Trigonometric Functions2.7-6. Kernels Containing Logarithmic and Trigonometric Functions

2.8. Equations Whose Kernels Contain Special Functions2.8-1. Kernels Containing Bessel Functions2.8-2. Kernels Containing Modified Bessel Functions

2.9. Equations Whose Kernels Contain Arbitrary Functions2.9-1. 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


3. Linear Equation of the First Kind With Constant Limits of Integration

3.1. Equations Whose Kernels Contain Power-Law Functions3.1-1. Kernels Linear in the Arguments x and t3.1-2. Kernels Quadratic in the Arguments x and t3.1-3. Kernels Containing Integer Powers of x and t or Rational Functions3.1-4. Kernels Containing Square Roots3.1-5. Kernels Containing Arbitrary Powers3.1-6. Equation Containing the Unknown Function of a Complicated Argument3.1-7. Singular Equations


3.3. Equations Whose Kernels Contain Hyperbolic Functions3.3-1. Kernels Containing Hyperbolic Cosine3.3-2. Kernels Containing Hyperbolic Sine3.3-3. Kernels Containing Hyperbolic Tangent3.3-4. Kernels Containing Hyperbolic Cotangent

3.4. Equations Whose Kernels Contain Logarithmic Functions3.4-1. Kernels Containing Logarithmic Functions3.4-2. Kernels Containing Power-Law and Logarithmic Functions3.4-3. An Equation Containing the Unknown Function of a Complicated Argument

3.5. Equations Whose Kernels Contain Trigonometric Functions3.5-1. Kernels Containing Cosine3.5-2. Kernels Containing Sine3.5-3. Kernels Containing Tangent3.5-4. Kernels Containing Cotangent3.5-5. Kernels Containing a Combination of Trigonometric Functions3.5-6. Equations Containing the Unknown Function of a Complicated Argument3.5-7. A Singular Equation

3.6. Equations Whose Kernels Contain Combinations of Elementary Functions3.6-1. Kernels Containing Hyperbolic and Logarithmic Functions3.6-2. Kernels Containing Logarithmic and Trigonometric Functions

Page xi


3.7. Equations Whose Kernels Contain Special Functions3.7-1. Kernels Containing Bessel Functions3.7-2. Kernels Containing Modified Bessel Functions3.7-3. Other Kernels

3.8. Equations Whose Kernels Contain Arbitrary Functions3.8-1. Equations With Degenerate Kernel3.8-2. Equations Containing Modulus3.8-3. Equations With Difference Kernel: K(x, t) = K(x t)3.8-4. Other Equations of the Form

baK(x, t)y(t) dt = F (x)

3.8-5. Equations of the Form b

aK(x, t)y( ) dt = F (x)

4. Linear Equations of the Second Kind With Constant Limits of Integration

4.1. Equations Whose Kernels Contain Power-Law Functions4.1-1. Kernels Linear in the Arguments x and t4.1-2. Kernels Quadratic in the Arguments x and t4.1-3. Kernels Cubic in the Arguments x and t4.1-4. Kernels Containing Higher-Order Polynomials in x and t4.1-5. Kernels Containing Rational Functions4.1-6. Kernels Containing Arbitrary Powers4.1-7. Singular Equations


4.3. Equations Whose Kernels Contain Hyperbolic Functions4.3-1. Kernels Containing Hyperbolic Cosine4.3-2. Kernels Containing Hyperbolic Sine4.3-3. Kernels Containing Hyperbolic Tangent4.3-4. Kernels Containing Hyperbolic Cotangent4.3-5. Kernels Containing Combination of Hyperbolic Functions


4.5. Equations Whose Kernels Contain Trigonometric Functions4.5-1. Kernels Containing Cosine4.5-2. Kernels Containing Sine4.5-3. Kernels Containing Tangent4.5-4. Kernels Containing Cotangent4.5-5. Kernels Containing Combinations of Trigonometric Functions4.5-6. A Singular Equation


4.7. Equations Whose Kernels Contain Combinations of Elementary Functions4.7-1. Kernels Containing Exponential and Hyperbolic Functions4.7-2. Kernels Containing Exponential and Logarithmic Functions4.7-3. Kernels Containing Exponential and Trigonometric Functions4.7-4. Kernels Containing Hyperbolic and Logarithmic Functions

Page xii


4.7-5. Kernels Containing Hyperbolic and Trigonometric Functions4.7-6. Kernels Containing Logarithmic and Trigonometric Functions

4.8. Equations Whose Kernels Contain Special Functions4.8-1. Kernels Containing Bessel Functions 4.8-2. Kernels Containing Modified Bessel Functions

4.9. Equations Whose Kernels Contain Arbitrary Functions4.9-1. 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)4.9-3. Other Equations of the Form y(x) +

baK(x, t)y(t) dt = F (x)

4.9-4. Equations of the Form y(x) + b

aK(x, t)y( ) dt = F (x)


5. Nonlinear Equations With Variable Limit of Integration

5.1. Equations With Quadratic Nonlinearity That Contain Arbitrary Parameters5.1-1. Equations of the Form

x0 y(t)y(x t) dt = F (x)

5.1-2. Equations of the Form x

0 K(x, t)y(t)y(x t) dt = F (x)5.1-3. Equations of the Form

x0 G( ) dt = F (x)

5.1-4. Equations of the Form y(x) + x

aK(x, t)y2(t) dt = F (x)


aK(x, t)y(t)y(x t) dt = F (x)

5.2. Equations With Quadratic Nonlinearity That Contain Arbitrary Functions5.2-1. Equations of the Form

xaG( ) dt = F (x)




aG( ) dt = F (x)

5.3. Equations With Power-Law Nonlinearity 5.3-1. Equations Containing Arbitrary Parameters 5.3-2. Equations Containing Arbitrary Functions

5.4. Equations With Exponential Nonlinearity5.4-1. Equations Containing Arbitrary Parameters5.4-2. Equations Containing Arbitrary Functions

5.5. Equations With Hyperbolic Nonlinearity5.5-1. Integrands With Nonlinearity of the Form cosh[y(t)]5.5-2. Integrands With Nonlinearity of the Form sinh[y(t)]5.5-3. Integrands With Nonlinearity of the Form tanh[y(t)]5.5-4. Integrands With Nonlinearity of the Form coth[y(t)]

5.6. Equations With Logarithmic Nonlinearity5.6-1. Integrands Containing Power-Law Functions of x and t5.6-2. Integrands Containing Exponential Functions of x and t5.6-3. Other Integrands

5.7. Equations With Trigonometric Nonlinearity5.7-1. Integrands With Nonlinearity of the Form cos[y(t)]5.7-2. Integrands With Nonlinearity of the Form sin[y(t)]5.7-3. Integrands With Nonlinearity of the Form tan[y(t)]5.7-4. Integrands With Nonlinearity of the Form cot[ y(t)]

5.8. Equations With Nonlinearity of General Form5.8-1. Equations of the Form

xaG( ) dt = F (x)


aK(x, t)G

(y(t)

)dt = F (x)


aK(x, t)G

(t, y(t)

)dt = F (x)

5.8-4. Other Equations

Page xiii


6. Nonlinear Equations With Constant Limits of Integration

6.1. Equations With Quadratic Nonlinearity That Contain Arbitrary Parameters


aK(t)y(x)y(t) dt = F (x)


aG( ) dt = F (x)




aK(x, t)y(x)y(t) dt = F (x)


aG( ) dt = F (x)

6.2. Equations With Quadratic Nonlinearity That Contain Arbitrary Functions


aG( ) dt = F (x)




a

Knm(x, t)yn(x)ym(t) dt = F (x), n + m 2


aG( ) dt = F (x)

6.3. Equations With Power-Law Nonlinearity


aG( ) dt = F (x)


aK(x, t)y(t) dt = F (x)


aG( ) dt = F (x)

6.4. Equations With Exponential Nonlinearity6.4-1. Integrands With Nonlinearity of the Form exp[y(t)]6.4-2. Other Integrands

6.5. Equations With Hyperbolic Nonlinearity6.5-1. Integrands With Nonlinearity of the Form cosh[y(t)]6.5-2. Integrands With Nonlinearity of the Form sinh[y(t)]6.5-3. Integrands With Nonlinearity of the Form tanh[y(t)]6.5-4. Integrands With Nonlinearity of the Form coth[y(t)]6.5-5. Other Integrands

6.6. Equations With Logarithmic Nonlinearity6.6-1. Integrands With Nonlinearity of the Form ln[y(t)]6.6-2. Other Integrands

6.7. Equations With Trigonometric Nonlinearity6.7-1. Integrands With Nonlinearity of the Form cos[y(t)]6.7-2. Integrands With Nonlinearity of the Form sin[y(t)]6.7-3. Integrands With Nonlinearity of the Form tan[y(t)]6.7-4. Integrands With Nonlinearity of the Form cot[ y(t)]6.7-5. Other Integrands

6.8. Equations With Nonlinearity of General Form


aG( ) dt = F (x)


aK(x, t)G

(y(t)

)dt = F (x)


aK(x, t)G

(t, y(t)

)dt = F (x)


aG

(x, t, y(t)

)dt = F (x)

6.8-5. Equations of the Form F(x, y(x)

) +

baG

(x, t, y(x), y(t)

)dt = 0

6.8-6. Other Equations

Page xiv


Part II. Methods for Solving Integral Equations7 Main Definitions and Formulas. Integral Transforms

7.1. Some Definitions, Remarks, and Formulas7.1-1. Some Definitions7.1-2. The Structure of Solutions to Linear Integral Equations7.1-3. Integral Transforms7.1-4. Residues. Calculation Formulas7.1-5. The Jordan Lemma

7.2. The Laplace Transform7.2-1. Definition. The Inversion Formula7.2-2. The Inverse Transforms of Rational Functions7.2-3. The Convolution Theorem for the Laplace Transform7.2-4. Limit Theorems7.2-5. Main Properties of the Laplace Transform7.2-6. The PostWidder Formula

7.3. The Mellin Transform7.3-1. Definition. The Inversion Formula7.3-2. Main Properties of the Mellin Transform7.3-3. The Relation Among the Mellin, Laplace, and Fourier Transforms

7.4. The Fourier Transform7.4-1. Definition. The Inversion Formula7.4-2. An Asymmetric Form of the Transform7.4-3. The Alternative Fourier Transform7.4-4. The Convolution Theorem for the Fourier Transform

7.5. The Fourier Sine and Cosine Transforms7.5-1. The Fourier Cosine Transform7.5-2. The Fourier Sine Transform

7.6. Other Integral Transforms7.6-1. The Hankel Transform7.6-2. The Meijer Transform7.6-3. The KontorovichLebedev Transform and Other Transforms

8. Methods for Solving Linear Equations of the Form x

aK(x, t)y(t) dt = f (x)

8.1. Volterra Equations of the First Kind8.1-1. Equations of the First Kind. Function and Kernel Classes8.1-2. Existence and Uniqueness of a Solution

8.2. Equations With Degenerate Kernel: K(x, t) = g1(x)h1(t) + + gn(x)hn(t)8.2-1. Equations With Kernel of the Form K(x, t) = g1(x)h1(t) + g2(x)h2(t)8.2-2. Equations With General Degenerate Kernel

8.3. Reduction of Volterra Equations of the 1st Kind to Volterra Equations of the 2nd Kind8.3-1. The First Method8.3-2. The Second Method

8.4. Equations With Difference Kernel: K(x, t) = K(x t)8.4-1. A Solution Method Based on the Laplace Transform8.4-2. The Case in Which the Transform of the Solution is a Rational Function8.4-3. Convolution Representation of a Solution8.4-4. Application of an Auxiliary Equation8.4-5. Reduction to Ordinary Differential Equations8.4-6. Reduction of a Volterra Equation to a WienerHopf Equation

Page xv


8.5. Method of Fractional Differentiation8.5-1. The Definition of Fractional Integrals8.5-2. The Definition of Fractional Derivatives8.5-3. Main Properties8.5-4. The Solution of the Generalized Abel Equation

8.6. Equations With Weakly Singular Kernel8.6-1. A Method of Transformation of the Kernel8.6-2. Kernel With Logarithmic Singularity

8.7. Method of Quadratures8.7-1. Quadrature Formulas8.7-2. The General Scheme of the Method8.7-3. An Algorithm Based on the Trapezoidal Rule8.7-4. An Algorithm for an Equation With Degenerate Kernel

8.8. Equations With Infinite Integration Limit8.8-1. An Equation of the First Kind With Variable Lower Limit of Integration 8.8-2. Reduction to a WienerHopf Equation of the First Kind

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


9.1. Volterra Integral Equations of the Second Kind9.1-1. Preliminary Remarks. Equations for the Resolvent9.1-2. A Relationship Between Solutions of Some Integral Equations

9.2. Equations With Degenerate Kernel: K(x, t) = g1(x)h1(t) + + gn(x)hn(t)9.2-1. Equations With Kernel of the Form K(x, t) = (x) + (x)(x t)9.2-2. Equations With Kernel of the Form K(x, t) = (t) + (t)(t x)9.2-3. Equations With Kernel of the Form K(x, t) =

nm=1 m(x)(x t)

m1

9.2-4. Equations With Kernel of the Form K(x, t) =n

m=1 m(t)(t x)m1

9.2-5. Equations With Degenerate Kernel of the General Form

9.3. Equations With Difference Kernel: K(x, t) = K(x t)9.3-1. A Solution Method Based on the Laplace Transform9.3-2. A Method Based on the Solution of an Auxiliary Equation9.3-3. Reduction to Ordinary Differential Equations9.3-4. Reduction to a WienerHopf Equation of the Second Kind9.3-5. Method of Fractional Integration for the Generalized Abel Equation9.3-6. Systems of Volterra Integral Equations

9.4. Operator Methods for Solving Linear Integral Equations9.4-1. Application of a Solution of a Truncated Equation of the First Kind9.4-2. Application of the Auxiliary Equation of the Second Kind9.4-3. A Method for Solving Quadratic Operator Equations9.4-4. Solution of Operator Equations of Polynomial Form9.4-5. A Generalization

9.5. Construction of Solutions of Integral Equations With Special Right-Hand Side9.5-1. The General Scheme9.5-2. A Generating Function of Exponential Form9.5-3. Power-Law Generating Function9.5-4. Generating Function Containing Sines and Cosines

9.6. The Method of Model Solutions9.6-1. Preliminary Remarks9.6-2. Description of the Method9.6-3. The Model Solution in the Case of an Exponential Right-Hand Side

Page xvi


9.6-4. The Model Solution in the Case of a Power-Law Right-Hand Side9.6-5. The Model Solution in the Case of a Sine-Shaped Right-Hand Side9.6-6. The Model Solution in the Case of a Cosine-Shaped Right-Hand Side9.6-7. Some Generalizations

9.7. Method of Differentiation for Integral Equations9.7-1. Equations With Kernel Containing a Sum of Exponential Functions9.7-2. Equations With Kernel Containing a Sum of Hyperbolic Functions9.7-3. Equations With Kernel Containing a Sum of Trigonometric Functions9.7-4. Equations Whose Kernels Contain Combinations of Various Functions

9.8. Reduction of Volterra Equations of the 2nd Kind to Volterra Equations of the 1st Kind9.8-1. The First Method9.8-2. The Second Method

9.9. The Successive Approximation Method9.9-1. The General Scheme9.9-2. A Formula for the Resolvent

9.10. Method of Quadratures9.10-1. The General Scheme of the Method9.10-2. Application of the Trapezoidal Rule9.10-3. The Case of a Degenerate Kernel

9.11. Equations With Infinite Integration Limit9.11-1. An Equation of the Second Kind With Variable Lower Integration Limit9.11-2. Reduction to a WienerHopf Equation of the Second Kind

10. Methods for Solving Linear Equations of the Form b


10.1. Some Definition and Remarks10.1-1. Fredholm Integral Equations of the First Kind10.1-2. Integral Equations of the First Kind With Weak Singularity10.1-3. Integral Equations of Convolution Type10.1-4. Dual Integral Equations of the First Kind

10.2. Kreins Method10.2-1. The Main Equation and the Auxiliary Equation10.2-2. Solution of the Main Equation

10.3. The Method of Integral Transforms10.3-1. Equation With Difference Kernel on the Entire Axis10.3-2. Equations With Kernel K(x, t) = K(x/t) on the Semiaxis10.3-3. Equation With Kernel K(x, t) = K(xt) and Some Generalizations

10.4. The Riemann Problem for the Real Axis10.4-1. Relationships Between the Fourier Integral and the Cauchy Type Integral10.4-2. One-Sided Fourier Integrals10.4-3. The Analytic Continuation Theorem and the Generalized Liouville Theorem10.4-4. The Riemann Boundary Value Problem10.4-5. Problems With Rational Coefficients10.4-6. Exceptional Cases. The Homogeneous Problem10.4-7. Exceptional Cases. The Nonhomogeneous Problem

10.5. The Carleman Method for Equations of the Convolution Type of the First Kind10.5-1. The WienerHopf Equation of the First Kind10.5-2. Integral Equations of the First Kind With Two Kernels

Page xvii


10.6. Dual Integral Equations of the First Kind10.6-1. The Carleman Method for Equations With Difference Kernels10.6-2. Exact Solutions of Some Dual Equations of the First Kind10.6-3. Reduction of Dual Equations to a Fredholm Equation

10.7. Asymptotic Methods for Solving Equations With Logarithmic Singularity10.7-1. Preliminary Remarks10.7-2. The Solution for Large 10.7-3. The Solution for Small 10.7-4. Integral Equation of Elasticity

10.8. Regularization Methods10.8-1. The Lavrentiev Regularization Method10.8-2. The Tikhonov Regularization Method

11. Methods for Solving Linear Equations of the Form y(x) b


11.1. Some Definition and Remarks11.1-1. Fredholm Equations and Equations With Weak Singularity of the 2nd Kind11.1-2. The Structure of the Solution11.1-3. Integral Equations of Convolution Type of the Second Kind11.1-4. Dual Integral Equations of the Second Kind

11.2. Fredholm Equations of the Second Kind With Degenerate Kernel11.2-1. The Simplest Degenerate Kernel11.2-2. Degenerate Kernel in the General Case

11.3. Solution as a Power Series in the Parameter. Method of Successive Approximations11.3-1. Iterated Kernels11.3-2. Method of Successive Approximations11.3-3. Construction of the Resolvent11.3-4. Orthogonal Kernels

11.4. Method of Fredholm Determinants11.4-1. A Formula for the Resolvent11.4-2. Recurrent Relations

11.5. Fredholm Theorems and the Fredholm Alternative11.5-1. Fredholm Theorems11.5-2. The Fredholm Alternative

11.6. Fredholm Integral Equations of the Second Kind With Symmetric Kernel11.6-1. Characteristic Values and Eigenfunctions11.6-2. Bilinear Series11.6-3. The HilbertSchmidt Theorem11.6-4. Bilinear Series of Iterated Kernels11.6-5. Solution of the Nonhomogeneous Equation11.6-6. The Fredholm Alternative for Symmetric Equations11.6-7. The Resolvent of a Symmetric Kernel11.6-8. Extremal Properties of Characteristic Values and Eigenfunctions11.6-9. Integral Equations Reducible to Symmetric Equations11.6-10. Skew-Symmetric Integral Equations

11.7. An Operator Method for Solving Integral Equations of the Second Kind11.7-1. The Simplest Scheme11.7-2. Solution of Equations of the Second Kind on the Semiaxis

Page xviii


11.8. Methods of Integral Transforms and Model Solutions11.8-1. Equation With Difference Kernel on the Entire Axis11.8-2. An Equation With the Kernel K(x, t) = t1 Q(x/t) on the Semiaxis11.8-3. Equation With the Kernel K(x, t) = tQ(xt) on the Semiaxis11.8-4. The Method of Model Solutions for Equations on the Entire Axis

11.9. The Carleman Method for Integral Equations of Convolution Type of the Second Kind11.9-1. The WienerHopf Equation of the Second Kind11.9-2. An Integral Equation of the Second Kind With Two Kernels11.9-3. Equations of Convolution Type With Variable Integration Limit11.9-4. Dual Equation of Convolution Type of the Second Kind

11.10. The WienerHopf Method11.10-1. Some Remarks11.10-2. The Homogeneous WienerHopf Equation of the Second Kind11.10-3. The General Scheme of the Method. The Factorization Problem11.10-4. The Nonhomogeneous WienerHopf Equation of the Second Kind11.10-5. The Exceptional Case of a WienerHopf Equation of the Second Kind

11.11. Kreins Method for WienerHopf Equations11.11-1. Some Remarks. The Factorization Problem11.11-2. The Solution of the WienerHopf Equations of the Second Kind11.11-3. The HopfFock Formula

11.12. Methods for Solving Equations With Difference Kernels on a Finite Interval11.12-1. Kreins Method11.12-2. Kernels With Rational Fourier Transforms11.12-3. Reduction to Ordinary Differential Equations

11.13. The Method of Approximating a Kernel by a Degenerate One11.13-1. Approximation of the Kernel11.13-2. The Approximate Solution

11.14. The Bateman Method11.14-1. The General Scheme of the Method11.14-2. Some Special Cases

11.15. The Collocation Method11.15-1. General Remarks11.15-2. The Approximate Solution11.15-3. The Eigenfunctions of the Equation

11.16. The Method of Least Squares11.16-1. Description of the Method11.16-2. The Construction of Eigenfunctions

11.17. The BubnovGalerkin Method11.17-1. Description of the Method11.17-2. Characteristic Values

11.18. The Quadrature Method11.18-1. The General Scheme for Fredholm Equations of the Second Kind11.18-2. Construction of the Eigenfunctions11.18-3. Specific Features of the Application of Quadrature Formulas

11.19. Systems of Fredholm Integral Equations of the Second Kind11.19-1. Some Remarks11.19-2. The Method of Reducing a System of Equations to a Single Equation

Page xix


11.20. Regularization Method for Equations With Infinite Limits of Integration11.20-1. Basic Equation and Fredholm Theorems11.20-2. Regularizing Operators11.20-3. The Regularization Method

12. Methods for Solving Singular Integral Equations of the First Kind

12.1. Some Definitions and Remarks12.1-1. Integral Equations of the First Kind With Cauchy Kernel12.1-2. Integral Equations of the First Kind With Hilbert Kernel

12.2. The Cauchy Type Integral12.2-1. Definition of the Cauchy Type Integral12.2-2. The Holder Condition12.2-3. The Principal Value of a Singular Integral12.2-4. Multivalued Functions12.2-5. The Principal Value of a Singular Curvilinear Integral12.2-6. The PoincareBertrand Formula

12.3. The Riemann Boundary Value Problem12.3-1. The Principle of Argument. The Generalized Liouville Theorem12.3-2. The Hermite Interpolation Polynomial12.3-3. Notion of the Index12.3-4. Statement of the Riemann Problem12.3-5. The Solution of the Homogeneous Problem12.3-6. The Solution of the Nonhomogeneous Problem12.3-7. The Riemann Problem With Rational Coefficients12.3-8. The Riemann Problem for a Half-Plane12.3-9. Exceptional Cases of the Riemann Problem12.3-10. The Riemann Problem for a Multiply Connected Domain12.3-11. The Cases of Discontinuous Coefficients and Nonclosed Contours12.3-12. The Hilbert Boundary Value Problem

12.4. Singular Integral Equations of the First Kind12.4-1. The Simplest Equation With Cauchy Kernel12.4-2. An Equation With Cauchy Kernel on the Real Axis12.4-3. An Equation of the First Kind on a Finite Interval12.4-4. The General Equation of the First Kind With Cauchy Kernel12.4-5. Equations of the First Kind With Hilbert Kernel

12.5. MulthoppKalandiya Method12.5-1. A Solution That is Unbounded at the Endpoints of the Interval12.5-2. A Solution Bounded at One Endpoint of the Interval12.5-3. Solution Bounded at Both Endpoints of the Interval

13. Methods for Solving Complete Singular Integral Equations

13.1. Some Definitions and Remarks13.1-1. Integral Equations With Cauchy Kernel13.1-2. Integral Equations With Hilbert Kernel13.1-3. Fredholm Equations of the Second Kind on a Contour

13.2. The Carleman Method for Characteristic Equations13.2-1. A Characteristic Equation With Cauchy Kernel13.2-2. The Transposed Equation of a Characteristic Equation13.2-3. The Characteristic Equation on the Real Axis13.2-4. The Exceptional Case of a Characteristic Equation

Page xx


13.2-5. The Characteristic Equation With Hilbert Kernel13.2-6. The Tricomi Equation

13.3. Complete Singular Integral Equations Solvable in a Closed Form13.3-1. Closed-Form Solutions in the Case of Constant Coefficients13.3-2. Closed-Form Solutions in the General Case

13.4. The Regularization Method for Complete Singular Integral Equations13.4-1. Certain Properties of Singular Operators13.4-2. The Regularizer13.4-3. The Methods of Left and Right Regularization13.4-4. The Problem of Equivalent Regularization13.4-5. Fredholm Theorems13.4-6. The CarlemanVekua Approach to the Regularization13.4-7. Regularization in Exceptional Cases13.4-8. The Complete Equation With Hilbert Kernel

14. Methods for Solving Nonlinear Integral Equations

14.1. Some Definitions and Remarks14.1-1. Nonlinear Volterra Integral Equations14.1-2. Nonlinear Equations With Constant Integration Limits

14.2. Nonlinear Volterra Integral Equations14.2-1. The Method of Integral Transforms14.2-2. The Method of Differentiation for Integral Equations14.2-3. The Successive Approximation Method14.2-4. The NewtonKantorovich Method14.2-5. The Collocation Method14.2-6. The Quadrature Method

14.3. Equations With Constant Integration Limits14.3-1. Nonlinear Equations With Degenerate Kernels14.3-2. The Method of Integral Transforms14.3-3. The Method of Differentiating for Integral Equations14.3-4. The Successive Approximation Method14.3-5. The NewtonKantorovich Method14.3-6. The Quadrature Method14.3-7. The Tikhonov Regularization Method

SupplementsSupplement 1. Elementary Functions and Their Properties

1.1. Trigonometric Functions

1.2. Hyperbolic Functions

1.3. Inverse Trigonometric Functions

1.4. Inverse Hyperbolic Functions

Supplement 2. Tables of Indefinite Integrals

2.1. Integrals Containing Rational Functions

2.2. Integrals Containing Irrational Functions

2.3. Integrals Containing Exponential Functions

2.4. Integrals Containing Hyperbolic Functions

2.5. Integrals Containing Logarithmic Functions

Page xxi


2.6. Integrals Containing Trigonometric Functions

2.7. Integrals Containing Inverse Trigonometric Functions

Supplement 3. Tables of Definite Integrals3.1. Integrals Containing Power-Law Functions

3.2. Integrals Containing Exponential Functions

3.3. Integrals Containing Hyperbolic Functions

3.4. Integrals Containing Logarithmic Functions

3.5. Integrals Containing Trigonometric Functions

Supplement 4. Tables of Laplace Transforms4.1. General Formulas

4.2. Expressions With Power-Law Functions

4.3. Expressions With Exponential Functions

4.4. Expressions With Hyperbolic Functions

4.5. Expressions With Logarithmic Functions

4.6. Expressions With Trigonometric Functions

4.7. Expressions With Special Functions

Supplement 5. Tables of Inverse Laplace Transforms5.1. General Formulas

5.2. Expressions With Rational Functions

5.3. Expressions With Square Roots

5.4. Expressions With Arbitrary Powers






Supplement 6. Tables of Fourier Cosine Transforms6.1. General Formulas







Supplement 7. Tables of Fourier Sine Transforms7.1. General Formulas







Page xxii


Supplement 8. Tables of Mellin Transforms

8.1. General Formulas






Supplement 9. Tables of Inverse Mellin Transforms


9.2. Expressions With Exponential and Logarithmic Functions



Supplement 10. Special Functions and Their Properties

10.1. Some Symbols and Coefficients

10.2. Error Functions and Integral Exponent

10.3. Integral Sine and Integral Cosine. Fresnel Integrals

10.4. Gamma Function. Beta Function

10.5. Incomplete Gamma Function

10.6. Bessel Functions

10.7. Modified Bessel Functions

10.8. Degenerate Hypergeometric Functions

10.9. Hypergeometric Functions

10.10. Legendre Functions

10.11. Orthogonal Polynomials

References

Page xxiii


Part I

Exact Solutions ofIntegral Equations

Page 1


Chapter 1

Linear Equations of the First KindWith Variable Limit of Integration

Notation: f = f (x), g = g(x), h = h(x),K =K(x), andM =M (x) are arbitrary functions (thesemay be composite functions of the argument depending on two variables x and t); A, B, C, D, E,a, b, c, , , , , and are free parameters; and m and n are nonnegative integers.

Preliminary remarks. For equations of the form xa

K(x, t)y(t) dt = f (x), a x b,

where the functions K(x, t) and f (x) are continuous, the right-hand side must satisfy the followingconditions:

1. If K(a, a) 0, then we must have f (a) = 0 (for example, the right-hand sides of equations 1.1.1and 1.2.1 must satisfy this condition).

2. If K(a, a) = K x(a, a) = = K (n1)x (a, a) = 0, 0 n, the solution has the form

y(x) =Am!

n! (m n 1)!xmn1.

24. x

a

(xn tn)y(t) dt = f (x), f (a) = f x(a) = 0, n = 1, 2, . . .

Solution: y(x) =1n

d

dx

[f x(x)xn1

].

25. x

a

(tnxn+1 xntn+1

)y(t) dt = f (x), n = 2, 3, . . .

This is a special case of equation 1.9.11 with g(x) = xn+1 and h(x) = xn.

Solution: y(x) =1xn

d2

dx2

[f (x)xn

].

Page 6


1.1-5. Kernels Containing Rational Functions

26. x

0

y(t) dt

x + t= f (x).

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

n=0Anx

n, the solution has the form

y(x) =N

n=0

AnBn

xn, Bn = (1)n

[ln 2 +

nk=1

(1)k

k

].

2. For f (x) = xN

n=0Anx

n, where is an arbitrary number ( > 1), the solution has the

form

y(x) = xN

n=0

AnBn

xn, Bn = 1

0

t+n dt

1 + t.

3. For f (x) = lnx( N

n=0Anx

n)

, the solution has the form

y(x) = lnxN

n=0

AnBn

xn +N

n=0

AnInB2n

xn,

Bn = (1)n

[ln 2 +

nk=1

(1)k

k

], In = (1)

n

[2

12+

nk=1

(1)k

k2

].

4. For f (x) =N

n=0An

(lnx)n, the solution of the equation has the form

y(x) =N

n=0

AnYn(x),

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

Yn(x) =

{dn

dn

[x

I()

]}=0

, I() = 1

0

z dz

1 + z.

5. For f (x) =N

n=1An cos(n lnx) +

Nn=1Bn sin(n lnx), the solution of the equation has the

form

y(x) =N

n=1

Cn cos(n lnx) +N

n=1

Dn sin(n lnx),

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 e

2 , y(t) = ew( ), f (x) = ezg(z)

leads to an integral equation with difference kernel of the form 1.9.26: z

w( ) dcosh(z )

= g(z).

Page 7


27. x

0

y(t) dt

ax + bt= f (x), a > 0, a + b > 0.


n=0Anx


y(x) =N

n=0

AnBn

xn, Bn = 1

0

tn dt

a + bt.

2. For f (x) = xN

n=0Anx


form

y(x) = xN

n=0

AnBn

xn, Bn = 1

0

t+n dt

a + bt.


n=0Anx

n)


y(x) = lnxN

n=0

AnBn

xn N

n=0

AnCnB2n

xn, Bn = 1

0

tn dt

a + bt, Cn =

10

tn ln ta + bt

dt.

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. x

0

y(t) dt

ax2 + bt2= f (x), a > 0, a + b > 0.


n=0Anx


y(x) =N

n=0

AnBn

xn+1, Bn = 1

0

tn+1 dt

a + bt2.

Example. For a = b = 1 and f (x) = Ax2 + Bx + C, the solution of the integral equation is:

y(x) =2A

1 ln 2x3 +

4B

4 x2 +

2C

ln 2x.

2. For f (x) = xN

n=0Anx


form

y(x) = xN

n=0

AnBn

xn+1, Bn = 1

0

t+n+1 dt

a + bt2.


n=0Anx

n)


y(x) = lnxN

n=0

AnBn

xn+1 N

n=0

AnCnB2n

xn+1, Bn = 1

0

tn+1 dt

a + bt2, Cn =

10

tn+1 ln ta + bt2

dt.

Page 8


29. x

0

y(t) dt

axm + btm= f (x), a > 0, a + b > 0, m = 1, 2, . . .


n=0Anx


y(x) =N

n=0

AnBn

xm+n1, Bn = 1

0

tm+n1 dt

a + btm.

2. For f (x) = xN

n=0Anx


form

y(x) = xN

n=0

AnBn

xm+n1, Bn = 1

0

t+m+n1 dt

a + btm.


n=0Anx

n)


y(x) = lnxN

n=0

AnBn

xm+n1 N

n=0

AnCnB2n

xm+n1,

Bn = 1

0

tm+n1 dt

a + btm, Cn =

10

tm+n1 ln ta + btm

dt.

1.1-6. Kernels Containing Square Roots

30. x

a

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

Differentiating with respect to x, we arrive at Abels equation 1.1.36:

xa

y(t) dtx t

= 2f x(x).

Solution:

y(x) =2

d2

dx2

xa

f (t) dtx t

.

31. x

a

(x

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

This is a special case of equation 1.1.44 with = 12 .

Solution: y(x) = 2d

dx

[x f x(x)

].

32. x

a

(A

x + B

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


Page 9


33. x

a

(1 + b

x t

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

Differentiating with respect to x, we arrive at Abels equation of the second kind 2.1.46:

y(x) +b

2

xa

y(t) dtx t

= f x(x).

34. x

a

(tx x

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

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

35. x

a

(At

x + Bx

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

This is a special case of equation 1.9.12 with g(x) =x and h(t) = t.

36. x

a

y(t) dtx t

= f (x).

Abels equation.Solution:

y(x) =1

d

dx

xa

f (t) dtx t

=f (a)

x a

+1

xa

f t(t) dtx t

.

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

37. x

a

(b +

1x t

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

Let us rewrite the equation in the form

xa

y(t) dtx t

= f (x) b x

a

y(t) dt.

Assuming the right-hand side to be known, we solve this equation as Abels equation 1.1.36.After some manipulations, we arrive at Abels equation of the second kind 2.1.46:

y(x) +b

xa

y(t) dtx t

= F (x), where F (x) =1

d

dx

xa

f (t) dtx t

.

38. x

a

(1

x

1

t

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


Solution: y(x) = 2[x3/2f x(x)

]x

, a > 0.

39. x

a

(A

x

+Bt

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


Page 10


40. x

a

y(t) dtx2 t2

= f (x).

Solution: y =2

d

dx

xa

tf (t) dtx2 t2

.

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

41. x

0

y(t) dtax2 + bt2

= f (x), a > 0, a + b > 0.


n=0Anx


y(x) =N

n=0

AnBn

xn, Bn = 1

0

tn dta + bt2

.

2. For f (x) = xN

n=0Anx


form

y(x) = xN

n=0

AnBn

xn, Bn = 1

0

t+n dta + bt2

.


n=0Anx

n)


y(x) = lnxN

n=0

AnBn

xn N

n=0

AnCnB2n

xn, Bn = 1

0

tn dta + bt2

, Cn = 1

0

tn ln ta + bt2

dt.

4. For f (x) =N

n=0An

(lnx)n, the solution of the equation has the form

y(x) =N

n=0

AnYn(x),

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

Yn(x) =

{dn

dn

[x

I()

]}=0

, I() = 1

0

z dza + bz2

.

5. For f (x) =N

n=1An cos(n lnx) +

Nn=1Bn sin(n lnx), the solution of the equation has the

form

y(x) =N

n=1

Cn cos(n lnx) +N

n=1

Dn sin(n lnx),

where the constants Cn and Dn are found by the method of undetermined coefficients.

Page 11


1.1-7. Kernels Containing Arbitrary Powers

42. x

a

(x t)y(t) dt = f (x), f (a) = 0, 0 < < 1.

Differentiating with respect to x, we arrive at the generalized Abel equation 1.1.46: xa

y(t) dt(x t)1

=1f x(x).

Solution:

y(x) = kd2

dx2

xa

f (t) dt(x t)

, k =sin()

.

Reference: F. D. Gakhov (1977).

43. x

a

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

For = 0, 1, 2, . . . , see equations 1.1.1, 1.1.2, 1.1.4, 1.1.12, and 1.1.23. For 1 < < 0, seeequation 1.1.42.

Set = n , where n = 1, 2, . . . and 0 < 1, and f (a) = f x(a) = = f (n1)x (a) = 0.On differentiating the equation n times, we arrive at an equation of the form 1.1.46: x

a

y(t) d(x t)

=( n + 1)

( + 1)f (n)x (x),

where () is the gamma function.

Example. Set f (x) = Ax , where 0, and let > 1 and 0, 1, 2, . . . In this case, the solution has

the form y(x) =A ( + 1)

( + 1) ( )x1.

Reference: M. L. Krasnov, A. I. Kisilev, and G. I. Makarenko (1971).

44. x

a

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

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

Solution: y(x) =1

[x1f x(x)

]x

.

45. x

a

(Ax + Bt

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

For B = A, see equation 1.1.44. This is a special case of equation 1.9.4 with g(x) = x.

Solution: y(x) =1

A +Bd

dx

[x

AA+B

xa

tB

A+B f t(t) dt

].

46. x

a

y(t) dt

(x t)= f (x), 0 < < 1.

The generalized Abel equation.Solution:

y(x) =sin()

d

dx

xa

f (t) dt(x t)1

=sin()

[f (a)

(x a)1+

xa

f t(t) dt(x t)1

].

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

Page 12


47. x

a

[b +

1

(x t)

]y(t) dt = f (x), 0 < < 1.

Rewrite the equation in the form

xa

y(t) dt(x t)

= f (x) b x

a

y(t) dt.

Assuming the right-hand side to be known, we solve this equation as the generalized Abelequation 1.1.46. After some manipulations, we arrive at Abels equation of the secondkind 2.1.60:

y(x) +b sin()

xa

y(t) dt(x t)1

= F (x), where F (x) =sin()

d

dx

xa

f (t) dt(x t)1

.

48. x

a

(x

t)y(t) dt = f (x), 0 < < 1.

Solution:

y(x) =kx

(xd

dx

)2 xa

f (t) dtt(x

t) , k = sin() .

49. x

a

y(t) dt(x

t) = f (x), 0 < < 1.

Solution:

y(x) =sin()

2d

dx

xa

f (t) dtt(x

t)1 .

50. x

a

(Ax + Bt

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

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

51. x

a

[1 + A(xt x+)

]y(t) dt = f (x).

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

y(x) =d

dx

{x

(x)

xa

[tf (t)

]t(t) dt

}, (x) = exp

(A

+ x+

).

52. x

a

(Axt + Bxt

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

This is a special case of equation 1.9.15 with g1(x) = Ax , h1(t) = t , g2(x) = Bx , andh2(t) = t.

53. x

a

[Ax(t x) + Bx(t x)

]y(t) dt = f (x).

This is a special case of equation 1.9.45 with g1(x) = Ax, h1(x) = x, g2(x) = Bx , andh2(x) = x .

Page 13


54. x

a

[Axt + Bx+t (A + B)x+t

]y(t) dt = f (x).

This is a special case of equation 1.9.47 with g(x) = x.

55. x

a

t(x t)y(t) dt = f (x), > 1, > 0, > 1.

The transformation = t, z = x, w( ) = t+1y(t) leads to an equation of the form 1.1.42: zA

(z )w( ) d = F (z),

where A = a and F (z) = f (z1/).Solution with 1 < < 0:

y(x) = sin()x

d

dx

[ xa

t1(x t)1f (t) dt

].

56. x

0

y(t) dt

(x + t)= f (x).

This is a special case of equation 1.1.57 with = 1 and a = b = 1.The transformation

x = 12 e2z , t = 12 e

2 , y(t) = e(2)w( ), f (x) = ezg(z)

leads to an equation with difference kernel of the form 1.9.26: z

w( ) dcosh(z )

= g(z).

57. x

0

y(t) dt

(ax + bt)= f (x), a > 0, a + b > 0.

1. The substitution t = xz leads to a special case of equation 3.8.45: 10

y(xz) dz(a + bz)

= x1f (x). (1)

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

m=0Amx

m, the solution has the form

y(x) = x1n

m=0

AmIm

xm, Im = 1

0

zm+1 dz

(a + bz).

The integrals Im are supposed to be convergent.

3. The solution structure for some other right-hand sides of the integral equation may beobtained using (1) and the results presented for the more general equation 3.8.45 (see alsoequations 3.8.263.8.32).

4. For a = b, the equation can be reduced, just as equation 1.1.56, to an integral equationwith difference kernel of the form 1.9.26.

58. x

a

(x +

x t

)2+

(x

x t

)22t

x t

y(t) dt = f (x).

The equation can be rewritten in terms of the Gaussian hypergeometric functions in the form xa

(x t)1F(, , ; 1

x

t

)y(t) dt = f (x), where = 12 .

See 1.8.86 for the solution of this equation.

Page 14


1.2. Equations Whose Kernels Contain ExponentialFunctions

1.2-1. Kernels Containing Exponential Functions

1. x

a

e(xt)y(t) dt = f (x).

Solution: y(x) = f x(x) f (x).

Example. In the special case a = 0 and f (x) = Ax, the solution has the form y(x) = A(1 x).

2. x

a

ex+ty(t) dt = f (x).

Solution: y(x) = e(+)x[f x(x) f (x)

].

Example. In the special case a = 0 and f (x) = A sin(x), the solution has the form y(x) = Ae(+)x [ cos(x) sin(x)].

3. x

a

[e(xt) 1

]y(t) dt = f (x), f (a) = f x(a) = 0.

Solution: y(x) = 1 fxx(x) f

x(x).

4. x

a

[e(xt) + b

]y(t) dt = f (x).

For b = 1, see equation 1.2.3. Differentiating with respect to x yields an equation of theform 2.2.1:

y(x) +

b + 1

xa

e(xt)y(t) dt =f x(x)b + 1

.

Solution:

y(x) =f x(x)b + 1

(b + 1)2

xa

exp

[b

b + 1(x t)

]f t(t) dt.

5. x

a

(ex+t + b

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

For = , see equation 1.2.4. This is a special case of equation 1.9.15 with g1(x) = ex,h1(t) = et, g2(x) = 1, and h2(t) = b.

6. x

a

(ex et

)y(t) dt = f (x), f (a) = f x(a) = 0.

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

Solution: y(x) = ex[

1f xx(x) f

x(x)

].

7. x

a

(ex et + b

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

For b = 0, see equation 1.2.6. This is a special case of equation 1.9.3 with g(x) = ex.Solution:

y(x) =1bf x(x)

b2ex

xa

exp

(et ex

b

)f t(t) dt.

Page 15


8. x

a

(Aex + Bet

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

For B = A, see equation 1.2.6. This is a special case of equation 1.9.4 with g(x) = ex.

Solution: y(x) =1

A +Bd

dx

[exp

(A

A +Bx) x

a

exp(

B

A +Bt)f t(t) dt

].

9. x

a

(Aex + Bet + C

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

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

10. x

a

(Aex + Bet

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

For = , see equation 1.2.8. This is a special case of equation 1.9.6 with g(x) = Aex andh(t) = Bet.

11. x

a

[e(xt) e(xt)

]y(t) dt = f (x), f (a) = f x(a) = 0.

Solution:

y(x) =1

[f xx ( + )f

x + f

], f = f (x).

12. x

a

[Ae(xt) + Be(xt)

]y(t) dt = f (x).

ForB = A, see equation 1.2.11. This is a special case of equation 1.9.15 with g1(x) = Aex,h1(t) = et, g2(x) = Bex, and h2(t) = et.

Solution:

y(x) =ex

A +Bd

dx

{e()x(x)

xa

[f (t)et

]t

dt

(t)

}, (x) = exp

[B( )A +B

x

].

13. x

a

[Ae(xt) + Be(xt) + C

]y(t) dt = f (x).

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

14. x

a

[Ae(xt) + Be(xt) + Ce(xt)

]y(t) dt = f (x).

Differentiating the equation with respect to x yields

(A +B + C)y(x) + x

a

[Ae(xt) +Be(xt) + Ce(xt)

]y(t) dt = f x(x).

Eliminating the term with e(xt) with the aid of the original equation, we arrive at an equationof the form 2.2.10:

(A +B + C)y(x) + x

a

[A( )e(xt) +B( )e(xt)

]y(t) dt = f x(x) f (x).

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

Page 16


15. x

a

[Ae(xt) + Be(xt) + Ce(xt) A B C

]y(t) dt = f (x), f (a) = f x(a) = 0.

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

[Ae(xt) +Be(xt) + Ce(xt)

]y(t) dt = f x(x).

16. x

a

(ex+t ex+t

)y(t) dt = f (x), f (a) = f x(a) = 0.

This is a special case of equation 1.9.11 with g(x) = ex and h(t) = et.Solution:

y(x) =f xx ( + )f

x(x) + f (x)

( ) exp[( + )x].

17. x

a

(Aex+t + Bex+t

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

For B = A, see equation 1.2.16. This is a special case of equation 1.9.12 with g(x) = ex

and h(t) = et.Solution:

y(x) =1

(A +B)exd

dx

{A(x)

xa

B(t)d

dt

[f (t)et

]dt

}, (x) = exp

( A +B

x)

.

18. x

a

(Aex+t + Bex+t

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

This is a special case of equation 1.9.15 with g1(x) = Aex, h1(t) = et, g2(x) = Bex, andh2(t) = et.

19. x

a

(Ae2x + Be2t + Cex + Det + E

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

This is a special case of equation 1.9.6 with g(x) =Ae2x +Cex and h(t) =Be2t +Det +E.

20. x

a

(Aex+t + Be2t + Cex + Det + E

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

This is a special case of equation 1.9.15 with g1(x) = ex, h1(t) = Aet +D, and g2(x) = 1,h2(t) = Be2t +Det + E.

21. x

a

(Ae2x + Bex+t + Cex + Det + E

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

This is a special case of equation 1.9.15 with g1(x) = Bex + D, h1(t) = et, and g2(x) =Ae2x + Cex + E, h2(t) = 1.

22. x

a

[1 + Aex(et ex)y(t) dt = f (x).

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

y(x) =d

dx

{ex(x)

xa

[f (t)et

]t

dt

(t)

}, (x) = exp

[A

+ e(+)x

].

Page 17


23. x

a

[Aex(ex et) + Bex(ex et)

]y(t) dt = f (x).

This is a special case of equation 1.9.45 with g1(x) = Aex, h1(t) = et, g2(x) = Bex, andh1(t) = et.

24. x

a

{A exp(x + t) + B exp[( + )x + ( )t]

(A + B) exp[( + )x + ( )t]}y(t) dt = f (x).

This is a special case of equation 1.9.47 with g1(x) = ex.

25. x

a

(ex et

)ny(t) dt = f (x), n = 1, 2, . . .

Solution:

y(x) =1

nn!ex

( 1ex

d

dx

)n+1f (x).

26. x

a

ex et y(t) dt = f (x), > 0.

Solution:

y(x) =2ex

(ex

d

dx

)2 xa

etf (t) dtex et

.

27. x

a

y(t) dtex et

= f (x), > 0.

Solution:

y(x) =

d

dx

xa

etf (t) dtex et

.

28. x

a

(ex et)y(t) dt = f (x), > 0, 0 < < 1.

Solution:

y(x) = kex(ex

d

dx

)2 xa

etf (t) dt(ex et)

, k =sin()

.

29. x

a

y(t) dt

(ex et)= f (x), > 0, 0 < < 1.

Solution:

y(x) = sin()

d

dx

xa

etf (t) dt(ex et)1

.

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

30. x

a

[A(x t) + Be(xt)

]y(t) dt = f (x).

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

By(x) + x

a

[A +Be(xt)

]y(t) dt = f x(x).

Page 18


31. x

a

(x t)e(xt)y(t) dt = f (x), f (a) = f x(a) = 0.

Solution: y(x) = f xx(x) 2fx(x) +

2f (x).

32. x

a

(Ax + Bt + C)e(xt)y(t) dt = f (x).

The substitution u(x) = exy(x) leads to an equation of the form 1.1.3: xa

(Ax +Bt + C)u(t) dt = exf (x).

33. x

a

(Axet + Btex)y(t) dt = f (x).

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

34. x

a

[Axe(xt) + Bte(xt)

]y(t) dt = f (x).

This is a special case of equation 1.9.15 with g1(x) = Axex, h1(t) = et, g2(x) = Bex, andh2(t) = tet.

35. x

a

(x t)2e(xt)y(t) dt = f (x), f (a) = f x(a) = fxx(a) = 0.

Solution: y(x) = 12[f xxx(x) 3f

xx(x) + 3

2f x(x) 3f (x)

].

36. x

a

(x t)ne(xt)y(t) dt = f (x), n = 1, 2, . . .

It is assumed that f (a) = f x(a) = = f (n)x (a) = 0.Solution: y(x) =

1n!ex

dn+1

dxn+1[exf (x)

].

37. x

a

(Ax + Bet)y(t) dt = f (x).

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

38. x

a

(Aex + Bt)y(t) dt = f (x).

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

39. x

a

(Axet + Btex)y(t) dt = f (x).

This is a special case of equation 1.9.15 with g1(x) = Ax , h1(t) = et, g2(x) = Bex, andh2(t) = t .

40. x

a

e(xt)x t y(t) dt = f (x).

Solution:

y(x) =2ex

d2

dx2

xa

etf (t) dtx t

.

Page 19


41. x

a

e(xt)x t

y(t) dt = f (x).

Solution:

y(x) =1ex

d

dx

xa

etf (t) dtx t

.

42. x

a

(x t)e(xt)y(t) dt = f (x), 0 < < 1.

Solution:

y(x) = kexd2

dx2

xa

etf (t) dt(x t)

, k =sin()

.

43. x

a

e(xt)

(x t)y(t) dt = f (x), 0 < < 1.

Solution:

y(x) =sin()

exd

dx

xa

etf (t)(x t)1

dt.

44. x

a

(x

t)e(xt)y(t) dt = f (x), 0 < < 1.

The substitution u(x) = exy(x) leads to an equation of the form 1.1.48:

xa

(x

t)u(t) dt = exf (x).

45. x

a

e(xt)y(t) dt(x

t) = f (x), 0 < < 1.

The substitution u(x) = exy(x) leads to an equation of the form 1.1.49:

xa

u(t) dt

(x

t)

= exf (x).

46. x

a

e(xt)x2 t2

y(t) dt = f (x).

Solution: y =2ex

d

dx

xa

tetx2 t2

f (t) dt.

47. x

a

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

Solution: y(x) = f x(x) 2xf (x).

48. x

a

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

This is a special case of equation 1.9.2 with g(x) = exp(x2).

Solution: y(x) =1

2d

dx

[f x(x)

x exp(x2)

].

Page 20


49. x

a

[A exp(x2) + B exp(t2) + C

]y(t) dt = f (x).

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

50. x

a

[A exp(x2) + B exp(t2)

]y(t) dt = f (x).

This is a special case of equation 1.9.6 with g(x) = A exp(x2) and h(t) = B exp(t2).

51. x

a

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

Solution:

y(x) =2

exp(x2)d2

dx2

xa

exp(t2)x t

f (t) dt.

52. x

a

exp[(x2 t2)]x t

y(t) dt = f (x).

Solution:

y(x) =1

exp(x2)d

dx

xa

exp(t2)x t

f (t) dt.

53. x

a

(x t) exp[(x2 t2)]y(t) dt = f (x), 0 < < 1.

Solution:

y(x) = k exp(x2)d2

dx2

xa

exp(t2)(x t)

f (t) dt, k =sin()

.

54. x

a

exp[(x t)]y(t) dt = f (x).

Solution: y(x) = f x(x) x1f (x).

1.3. Equations Whose Kernels Contain HyperbolicFunctions

1.3-1. Kernels Containing Hyperbolic Cosine

1. x

a

cosh[(x t)]y(t) dt = f (x).

Solution: y(x) = f x(x) 2 x

a

f (x) dx.

2. x

a

{cosh[(x t)] 1

}y(t) dt = f (x), f (a) = f x(a) = f

xx(x) = 0.

Solution: y(x) =12f xxx(x) f

x(x).

Page 21


3. x

a

{cosh[(x t)] + b

}y(t) dt = f (x).

For b = 0, see equation 1.3.1. For b = 1, see equation 1.3.2. For = 0, see equation 1.1.1.Differentiating the equation with respect to x, we arrive at an equation of the form 2.3.16:

y(x) +

b + 1

xa

sinh[(x t)]y(t) dt =f x(x)b + 1

.

1. Solution with b(b + 1) < 0:

y(x) =f x(x)b + 1

2

k(b + 1)2

xa

sin[k(x t)]f t(t) dt, where k =

bb + 1

.

2. Solution with b(b + 1) > 0:

y(x) =f x(x)b + 1

2

k(b + 1)2

xa

sinh[k(x t)]f t(t) dt, where k =

b

b + 1.

4. x

a

cosh(x + t)y(t) dt = f (x).

For = , see equation 1.3.1.Differentiating the equation with respect to x twice, we obtain

cosh[(+)x]y(x)+ x

a

sinh(x+t)y(t) dt = f x(x), (1)

{cosh[(+)x]y(x)

}x

+ sinh[(+)x]y(x)+2 x

a

cosh(x+t)y(t) dt = f xx(x). (2)

Eliminating the integral term from (2) with the aid of the original equation, we arrive atthe first-order linear ordinary differential equation

wx + tanh[( + )x]w = fxx(x)

2f (x), w = cosh[( + )x]y(x). (3)

Setting x = a in (1) yields the initial conditionw(a) = f x(a). On solving equation (3) with thiscondition, after some manipulations we obtain the solution of the original integral equationin the form

y(x) =1

cosh[( + )x]f x(x)

sinh[( + )x]

cosh2[( + )x]f (x)

+

coshk+1[( + )x]

xa

f (t) coshk2[( + )t] dt, k =

+ .

5. x

a

[cosh(x) cosh(t)]y(t) dt = f (x).

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

Solution: y(x) =1

d

dx

[f x(x)

sinh(x)

].

6. x

a

[A cosh(x) + B cosh(t)]y(t) dt = f (x).

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

Solution: y(x) =1

A +Bd

dx

{[cosh(x)

] AA+B xa

[cosh(t)

] BA+B f t(t) dt}

.

Page 22


7. x

a

[A cosh(x) + B cosh(t) + C

]y(t) dt = f (x).

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

8. x

a

{A1 cosh[1(x t)] + A2 cosh[2(x t)]

}y(t) dt = f (x).

The equation is equivalent to the equation xa

{B1 sinh[1(x t)] +B2 sinh[2(x t)]

}y(t) dt = F (x),

B1 =A11

, B2 =A22

, F (x) = x

a

f (t) dt,

of the form 1.3.41. (Differentiating this equation yields the original equation.)

9. x

a

cosh2[(x t)]y(t) dt = f (x).

Differentiation yields an equation of the form 2.3.16:

y(x) + x

a

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

Solution:

y(x) = f x(x) 22

k

xa

sinh[k(x t)]f t(t) dt, where k =

2.

10. x

a

[cosh2(x) cosh2(t)

]y(t) dt = f (x), f (a) = f x(a) = 0.

Solution: y(x) =1

d

dx

[f x(x)

sinh(2x)

].

11. x

a

[A cosh2(x) + B cosh2(t)

]y(t) dt = f (x).

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

y(x) =1

A +Bd

dx

{[cosh(x)

] 2AA+B xa

[cosh(t)

] 2BA+B f t(t) dt}

.

12. x

a

[A cosh2(x) + B cosh2(t) + C

]y(t) dt = f (x).

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

13. x

a

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

Using the formula

cosh( ) cosh( + ) = 12 [cos(2) + cos(2)], = x, = t,

we transform the original equation to an equation of the form 1.4.6 with A = B = 1: xa

[cosh(2x) + cosh(2t)]y(t) dt = 2f (x).

Solution:

y(x) =d

dx

[1

cosh(2x)

xa

f t(t) dtcosh(2t)

].

Page 23


14. x

a

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

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

15. x

a


Using the formula cosh3 = 14 cosh 3 +34 cosh, we arrive at an equation of the form 1.3.8: x

a

{14 cosh[3(x t)] +

34 cosh[(x t)]

}y(t) dt = f (x).

16. x

a

[cosh3(x) cosh3(t)

]y(t) dt = f (x), f (a) = f x(a) = 0.

Solution: y(x) =1

3d

dx

[f x(x)

sinh(x) cosh2(x)

].

17. x

a

[A cosh3(x) + B cosh3(t)

]y(t) dt = f (x).

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

y(x) =1

A +Bd

dx

{[cosh(x)

] 3AA+B xa

[cosh(t)

] 3BA+B f t(t) dt}

.

18. x

a

[A cosh2(x) cosh(t) + B cosh(x) cosh2(t)

]y(t) dt = f (x).

This is a special case of equation 1.9.15 with g1(x) = A cosh2(x), h1(t) = cosh(t), g2(x) =

B cosh(x), and h2(t) = cosh2(t).

19. x

a


Let us transform the kernel of the integral equation using the formula

cosh4 = 18 cosh 4 +12 cosh 2 +

38 , where = (x t),

and differentiate the resulting equation with respect to x. Then we obtain an equation of theform 2.3.18:

y(x) + x

a

{12 sinh[4(x t)] + sinh[2(x t)]

}y(t) dt = f x(x).

20. x

a

[cosh(x) cosh(t)]ny(t) dt = f (x), n = 1, 2, . . .

The right-hand side of the equation is assumed to satisfy the conditions f (a) = f x(a) = =f (n)x (a) = 0.

Solution: y(x) =sinh(x)nn!

[1

sinh(x)d

dx

]n+1f (x).

Page 24


21. x

a

coshx cosh t y(t) dt = f (x).

Solution:

y(x) =2

sinhx( 1

sinhxd

dx

)2 xa

sinh t f (t) dtcoshx cosh t

.

22. x

a

y(t) dt

coshx cosh t= f (x).

Solution:

y(x) =1

d

dx

xa

sinh t f (t) dtcoshx cosh t

.

23. x

a

(coshx cosh t)y(t) dt = f (x), 0 < < 1.

Solution:

y(x) = k sinhx( 1

sinhxd

dx

)2 xa

sinh t f (t) dt(coshx cosh t)

, k =sin()

.

24. x

a

(cosh x cosh t)y(t) dt = f (x).

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

Solution: y(x) =1

d

dx

[f x(x)

sinhx cosh1 x

].

25. x

a

(A cosh x + B cosh t

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

ForB = A, see equation 1.3.24. This is a special case of equation 1.9.4 with g(x) = cosh x.Solution:

y(x) =1

A +Bd

dx

{[cosh(x)

] AA+B xa

[cosh(t)

] BA+B f t(t) dt}

.

26. x

a

y(t) dt

(coshx cosh t)= f (x), 0 < < 1.

Solution:

y(x) =sin()

d

dx

xa

sinh t f (t) dt(coshx cosh t)1

.

27. x

a

(x t) cosh[(x t)]y(t) dt = f (x), f (a) = f x(a) = 0.

Differentiating the equation twice yields

y(x) + 2 x

a

sinh[(x t)]y(t) dt + 2 x

a

(x t) cosh[(x t)]y(t) dt = f xx(x).

Eliminating the third term on the right-hand side with the aid of the original equation, wearrive at an equation of the form 2.3.16:

y(x) + 2 x

a

sinh[(x t)]y(t) dt = f xx(x) 2f (x).

Page 25


28. x

a

cosh(

x t

)x t

y(t) dt = f (x).

Solution:

y(x) =1

d

dx

xa

cos(x t

)x t

f (t) dt.

29. x

0

cosh(

x2 t2

)x2 t2

y(t) dt = f (x).

Solution:

y(x) =2

d

dx

x0t

cos(x2 t2

)x2 t2

f (t) dt.

30.

x

cosh(

t2 x2

)t2 x2

y(t) dt = f (x).

Solution:

y(x) = 2

d

dx

x

tcos

(t2 x2

)t2 x2

f (t) dt.

31. x

a

[Ax + B cosh(t) + C]y(t) dt = f (x).

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

32. x

a

[A cosh(x) + Bt + C]y(t) dt = f (x).

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

33. x

a

(Ax cosh t + Bt cosh x

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

This is a special case of equation 1.9.15 with g1(x) =Ax, h1(t) = cosh t, g2(x) =B cosh

x,and h2(t) = t .

1.3-2. Kernels Containing Hyperbolic Sine

34. x

a

sinh[(x t)]y(t) dt = f (x), f (a) = f x(a) = 0.

Solution: y(x) =1f xx(x) f (x).

35. x

a

{sinh[(x t)] + b

}y(t) dt = f (x).

Differentiating the equation with respect to x, we arrive at an equation of the form 2.3.3:

y(x) +

b

xa

cosh[(x t)]y(t) dt =1bf x(x).

Solution:

y(x) =1bf x(x) +

xa

R(x t)f t(t) dt,

R(x) =

b2exp

(x

2b

)[

2bksinh(kx) cosh(kx)

], k =

1 + 4b2

2b.

Page 26


36. x

a

sinh(x + t)y(t) dt = f (x).

For = , see equation 1.3.34. Assume that .Differentiating the equation with respect to x twice yields

sinh[( + )x]y(x) + x

a

cosh(x + t)y(t) dt = f x(x), (1)

{sinh[( + )x]y(x)

}x

+ cosh[( + )x]y(x) + 2 x

a

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

Eliminating the integral term from (2) with the aid of the original equation, we arrive at thefirst-order linear ordinary differential equation

wx + coth[( + )x]w = fxx(x)

2f (x), w = sinh[( + )x]y(x). (3)

Setting x = a in (1) yields the initial conditionw(a) = f x(a). On solving equation (3) with thiscondition, after some manipulations we obtain the solution of the original integral equationin the form

y(x) =1

sinh[( + )x]f x(x)

cosh[( + )x]

sinh2[( + )x]f (x)

sinhk+1[( + )x]

xa

f (t) sinhk2[( + )t] dt, k =

+ .

37. x

a

[sinh(x) sinh(t)]y(t) dt = f (x), f (a) = f x(a) = 0.

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

Solution: y(x) =1

d

dx

[f x(x)

cosh(x)

].

38. x

a

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

ForB = A, see equation 1.3.37. This is a special case of equation 1.9.4 with g(x) = sinh(x).

Solution: y(x) =1

A +Bd

dx

{[sinh(x)

] AA+B xa

[sinh(t)

] BA+B f t(t) dt}

.

39. x

a

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

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

40. x

a

{ sinh[(x t)] sinh[(x t)]

}y(t) dt = f (x).

It is assumed that f (a) = f x(a) = fxx(a) = f

xxx(a) = 0.

Solution:

y(x) =f xxxx (

2 + 2)f xx + 22f

3 3, f = f (x).

Page 27


41. x

a

{A1 sinh[1(x t)] + A2 sinh[2(x t)]

}y(t) dt = f (x), f (a) = f x(a) = 0.

1. Introduce the notation

I1 = x

a

sinh[1(x t)]y(t) dt, I2 = x

a

sinh[2(x t)]y(t) dt,

J1 = x

a

cosh[1(x t)]y(t) dt, J2 = x

a

cosh[2(x t)]y(t) dt.

Let us successively differentiate the integral equation four times. As a result, we have (thefirst line is the original equation):

A1I1 +A2I2 = f , f = f (x), (1)

A11J1 +A22J2 = fx, (2)

(A11 +A22)y +A121I1 +A2

22I2 = f

xx, (3)

(A11 +A22)yx +A1

31J1 +A2

32J2 = f

xxx, (4)

(A11 +A22)yxx + (A1

31 +A2

32)y +A1

41I1 +A2

42I2 = f

xxxx. (5)

Eliminating I1 and I2 from (1), (3), and (5), we arrive at the following second-order linearordinary differential equation with constant coefficients:

(A11 +A22)yxx 12(A12 +A21)y = f

xxxx (

21 +

22)f

xx +

21

22f . (6)

The initial conditions can be obtained by substituting x = a into (3) and (4):

(A11 +A22)y(a) = fxx(a), (A11 +A22)y

x(a) = f

xxx(a). (7)

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

2. Denote

= 12A12 +A21A11 +A22

.

2.1. Solution for > 0:

(A11 +A22)y(x) = fxx(x) +Bf (x) + C

xa

sinh[k(x t)]f (t) dt,

k =

, B = 21 22, C =

1

[2 (21 +

22) +

21

22

].

2.2. Solution for < 0:

(A11 +A22)y(x) = fxx(x) +Bf (x) + C

xa

sin[k(x t)]f (t) dt,

k =

, B = 21 22, C =

1

[2 (21 +

22) +

21

22

].

2.3. Solution for = 0:

(A11 +A22)y(x) = fxx(x) (

21 +

22)f (x) +

21

22

xa

(x t)f (t) dt.

2.4. Solution for = :

y(x) =f xxxx (

21 +

22)f

xx +

21

22f

A131 +A2

32

, f = f (x).

In the last case, the relation A11 + A22 = 0 is valid, and the right-hand side of theintegral equation is assumed to satisfy the conditions f (a) = f x(a) = f

xx(a) = f

xxx(a) = 0.

Page 28


42. x

a

{A sinh[(x t)] + B sinh[(x t)] + C sinh[(x t)]

}y(t) dt = f (x).

It assumed that f (a) = f x(a) = 0. Differentiating the integral equation twice yields

(A +B + C)y(x) + x

a

{A2 sinh[(x t)] +B2 sinh[(x t)]

}y(t) dt

+ C2 x

a

sinh[(x t)]y(t) dt = f xx(x).

Eliminating the last integral with the aid of the original equation, we arrive at an equation ofthe form 2.3.18:

(A +B + C)y(x)

+ x

a

{A(2 2) sinh[(x t)] +B(2 2) sinh[(x t)]

}y(t) dt = f xx(x)

2f (x).

In the special case A +B + C = 0, this is an equation of the form 1.3.41.

43. x

a

sinh2[(x t)]y(t) dt = f (x), f (a) = f x(a) = fxx(a) = 0.

Differentiating yields an equation of the form 1.3.34: xa

sinh[2(x t)]y(t) dt =1f x(x).

Solution: y(x) = 122f xxx(x) 2f

x(x).

44. x

a

[sinh2(x) sinh2(t)

]y(t) dt = f (x), f (a) = f x(a) = 0.

Solution: y(x) =1

d

dx

[f x(x)

sinh(2x)

].

45.

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