Singular Integral Equations

432 Pages·2000·10.412 MB·English

by  Ricardo Estrada, Ram P. Kanwal (auth.)

Dedicated to Ana
- Ricardo Estrada
Dedicated in loving memory of my parents
- Ram P. Kanwal
Ricardo Estrada
Ram P. Kanwal
Singular Integral Equations
Springer Science+Business Media, LLC
Ricardo Estrada Ram P. Kanwal
Escuela de Matematica Department of Mathematics
Universidad de Costa Rica Penn State University
2060 San Jose University Park, PA 16802
Costa Rica U.S.A.
Ubrary of Congress Cataloging-in-Publication Data
Estrada, Ricardo, 1956-
Singular integral equations / Rieardo Estrada, Ram P. Kanwal
p.em.
Includes bibliographical referenees and index.
ISBN 978-1-4612-7123-9 ISBN 978-1-4612-1382-6 (eBook)
DOI 10.1007/978-1-4612-1382-6
1. Integral equations. 2. Kanwal, Ram P. II. Title.
QA431.E73 2000
515'.45-de21 99-050345
CIP
AMS Subjeet Classifieations: 30E25, 4SExx, 4SEOS, 4SE10, 62Exx
Printed on acid-free paper.
©2000 Springer Seience+Business Media New York
Originally published by Birkhl!user Boston in 2000
Softcover reprint ofthe hardcover Ist edition 2000
AII rights reserved. This work may not be translated or copied in whole or in par! without the written
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ISBN 978-1-4612-7123-9 SPIN 10685292
Reformatted from authors' disk by TEXniques, Inc., Cambridge, MA.
9 8 765 4 3 2 1
Contents
Preface ix
1 Reference Material 1
1.1 Introduction......... 1
1.2 Singular Integral Equations . 2
1.3 Improper Integrals ..... 3
1.3.1 The Gamma function. 7
1.3.2 The Beta function. . . 11
1.3.3 Another important improper integral . 12
1.3.4 A few integral identities . . 13
1.4 The Lebesgue Integral ...... . 15
1.5 Cauchy Principal Value for Integrals 18
1.6 The Hadamard Finite Part ..... 26
1.7 Spaces of Functions and Distributions 30
1.8 Integral Transform Methods 33
1.8.1 Fourier transform . 35
1.8.2 Laplace transform 38
1.9 Bibliographical Notes . . . 41
2 Abel's and Related Integral Equations 43
2.1 Introduction ............ . . . . . . . . . . . .. 43
Vl Contents
2.2 Abel's Equation. . . . . . . . . . . . . . . . . . . . 43
2.3 Related Integral Equations . . . . . . . . . . . . . . 46
2.4 The equation J; (5 - t)fJ g(t)dt = f (5) , me f3 > -1 48
2.5 Path of Integration in the Complex Plane ... 51
2.6 The Equation r g(z)dz + k r g(z)dz = f(C) 52
JCor (z-n" JC~b (~-z)" .,
2.7 Equations On a C osed Curve . . . . . . . . . . 57
2.8 Examples . . . . . . . . . 60
2.9 Bibliographical Notes. 64
2.10 Problems ......... 64
3 Cauchy Type Integral Equations 71
3.1 Introduction................... 71
3.2 Cauchy Type Equation of the First Kind . . . . 72
3.3 An Alternative Approach . . . . . . . . . . . . 75
3.4 Cauchy Type Equations of the Second Kind . . 78
3.5 Cauchy Type Equations On a Closed Contour 82
3.6 Analytic Representation of Functions ..... 84
3.7 Sectionally Analytic Functions (z - a)n-v(z - b)m+v 91
3.8 Cauchy's Integral Equation on an Open Contour. . . 93
3.9 Disjoint Contours . . . . . . . . 100
3.10 Contours That Extend to Infinity 103
3.11 The Hilbert Kernel .. 108
3.12 The Hilbert Equation 113
3.13 Bibliographical Notes. 116
3.14 Problems ....... 116
4 Carleman Type Integral Equations 125
4.1 Introduction.................. 125
4.2 Carleman Type Equation over a Real Interval 127
4.3 The Riemann-Hilbert Problem . . . . . . . . 134
4.4 Carleman Type Equations on a Closed Contour 140
4.5 Non-Normal Problems .. ..... 145
4.6 A Factorization Procedure ........... 150
4.7 An Operational Approach. . . . . . . . 152
4.8 Solution of a Related Integral Equation. . . . . . . . 161
4.9 Bibliographical Notes . . . . . . . . . . 168
4.10 Problems ....................... 168
5 Distributional Solutions of Singular Integral Equations 175
5.1 Introduction...................... 175
Contents vii
5.2 Spaces of Generalized Functions . . . . . . . 176
5.3 Generalized Solution of the Abel Equation . 180
5.4 Integral Equations Related to Abel's Equation 187
5.5 The Fractional Integration Operators <l>a . . . • 191
5.6 The Cauchy Integral Equation over a Finite Interval 196
5.7 Analytic Representation of Distributions of E'[ a, b] 205
5.8 Boundary Problems in A[a, b] . . 209
5.9 Disjoint Intervals . . . . . . . . . . . . . . . . . 217
5.9.1 Theprob1em[RjFL =hj . . . . . . . . 221
+
5.9.2 The equation A11l'1 (F) A21l'2(F) = G . 226
5.10 Equations Involving Periodic Distributions. 231
5.11 Bibliographical Notes. 242
5.12 Problems ................. 242
6 Distributional Equations on the Whole Line 251
6.1 Introduction................ 251
6.2 Preliminaries .. . . . . . . . . . . . . 252
6.3 The Hilbert Transform of Distributions . 254
6.4 Analytic Representation. . . . . . . . . 262
6.5 Asymptotic Estimates . . . . . . . . . . 264
6.6 Distributional Solutions of Integral Equations 270
6.7 Non-Normal Equations 282
6.8 Bibliographical Notes. 286
6.9 Problems ....... 287
7 Integral Equations with Logarithmic Kernels 295
7.1 Introduction................. 295
7.2 Expansion of the Kernel In Ix - y I . . . . . 296
1:
7.3 The Equation In Ix - yl g(y) dy = I(x) 298
7.4 Two Related Operators . . . . . . . . . . . 300
7.5 Generalized Solutions of Equations with
Logarithmic Kfe:r nels ................ 302
7.6 The Operator (P(x - y) In Ix - yl + Q(x, y)) g(y) dy 308
7.7 Disjoint Intervals of Integration. . . . . . 311
7.8 An Equation Over a Semi-Infinite Interval ......... 313
7.9 The Equation of the Second Kind Over a
Semi-Infinite Interval . . . . . . . . . 314
7.10 Asymptotic Behavior of Eigenvalues. 318
7.11 Bibliographical Notes. 322
7.12 Problems .............. . 323
viii Contents
8 Wiener-Hopf Integral Equations 339
8.1 Introduction................. 339
8.2 The Holomorphic Fourier Transform 340
8.3 The Mathematical Technique . . . . . . . . 345
8.4 The Distributional Wiener-Hopf Operators 355
8.5 Illustrations.......... 361
8.6 Bibliographical Notes . 369
8.7 Problems ........... 369
9 Dual and Triple Integral Equations 375
9.1 Introduction................... 375
9.2 The Hankel Transform ............ 377
9.3 Dual Equations with Trigonometric Kernels 379
9.4 Beltrami's Dual Integral Equations . 382
9.5 Some Triple Integral Equations. 384
9.6 Erdelyi-Kober Operators . . . . 386
9.7 Dual Integral Equations of the
Titchmarsh Type ............. 390
9.8 Distributional Solutions of Dual Integral Equations 391
9.8.1 Fractional integration in H~,v ..... 395
9.8.2 Solution of the distributional problem 399
9.8.3 Uniqueness.. 402
9.9 Bibliographical Notes . 403
9.10 Problems ....... 403
References 413
Index 423
Preface
Many physical problems that are usually solved by differential equation
techniques can be solved more effectively by integral equation methods.
This work focuses exclusively on singular integral equations and on the
distributional solutions of these equations. A large number of beautiful
mathematical concepts are required to find such solutions, which in tum,
can be applied to a wide variety of scientific fields - potential theory, me
chanics, fluid dynamics, scattering of acoustic, electromagnetic and earth
quake waves, statistics, and population dynamics, to cite just several.
An integral equation is said to be singular if the kernel is singular within
the range of integration, or if one or both limits of integration are infinite.
The singular integral equations that we have studied extensively in this
book are of the following type. In these equations f (x) is a given function
and g(y) is the unknown function.
1. The Abel equation
l
x
/ (x) = g (y) d y, 0 < a < 1.
Ct
( Y )
a X -
2. The Cauchy type integral equation
l
b g (y)
g(x)=/(x)+).. --dy,
a y-x
where).. is a parameter.
x Preface
3. The extension
l
b g (y)
a (x) g (x) = J (x) +).. --dy ,
a y-x
of the Cauchy equation. This is called the Carle man equation.
4. The integral equation of logarithmic kernel,
lb
In Ix - yl g(y) dy = J(x)
5. The Wiener-Hopf integral equation
1
00
g(x)+).. K(x-y)g(y)dy=J(x)
The distinguishing feature of this equation is that the kernel is a
difference kernel and that the interval of integration is [0, (0).
We examine several variants and extensions of these equations, for ex
ample, on contours of the complex plane. Similarly, we present the solu
tions of double and triple integral equations. But unlike regular equations,
no general theory is available for singular integral equations, so that all
of the above-mentioned singular equations are studied on an ad-hoc basis.
We have therefore introduced generalized functions to provide a common
thread in our analysis of these equations.
The plan of the book is as follows.
In the first chapter we have included some reference material that will be
needed throughout the book. We present the basic principles of improper
integrals and singular integrals, as well as the fundamentals of Lebesgue
integration. We also consider two very important methods to assign a
value to a divergent integral, namely, the Cauchy principal value and the
Hadamard finite part. We also present the concepts of boundary values
of analytic functions and the various formulas related to this subject. We
introduce several spaces of functions and distributions to be used in our
studies. Finally, we give some ideas on transform analysis, considering in
particular, the Fourier and Laplace transforms.
The Abel integral equation is one of the simplest integral equations.
We present its solution in Chapter 2. We also study related equations.
The analysis is then extended to the case when the path of integration is
a contour in the complex plane. In addition to being very applicable, the