Integral equations of first kind

269 Pages·1995·9.913 MB·English

by  Bitsadze A.V.

Integral Equations of
First Kind
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A V Bitsadze
Steklov Institute for Mathematics
Moscow, Russia
Series on
Soviet and
East European
Mathematics
Vol.7
Integral Equations of
First Kind
V fe World Scientific
wb Singapore • New Jersey • London • Hong Kong
Published by
World Scientific Publishing Co. Pte. Ltd.
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British Library Cataloguing-in-Publication Data
A catalogue record for this book is available from the British Library.
INTEGRAL EQUATIONS OF THE FIRST KIND
Copyright © 1995 by World Scientific Publishing Co. Pte. Ltd.
All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means,
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ISBN: 981-02-2263-7
This book is printed on acid-free paper.
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V
PREFACE
The book is devoted to the study of classes of linear integral equations of the first
kind most often met in applications. Since the general theory of integral equations
of the first kind has not been formed yet we shall confine ourselves here to
considering the equations whose solutions are either constructed in quadratures or
can be reduced to the well-investigated classes of integral equations of the second
kind.
One-dimensional integral equations are treated on the basis of the theory of one-
dimensional Cauchy-type integrals. Consideration various multi-dimensional analogs
of such integrals makes it possible to study some multi-dimensional integral
equations of the first kind as well.
Simple models of equations have been used in this book keeping in mind the fact
that if necessary the reader himself, while investigating the problems he is interested
in, might apply the methods developed here whenever it is possible in principle.
The author wishes to express his thanks to Professors E. I. Moiseev and A. P.
Soldatov, who after reading the manuscript made a series of valuable suggestions.
A. V. Bitsadze
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VII
CONTENTS
Preface v
Chapter 1. Brief Review of the General Theory of the Linear
Equations in Metric Spaces 1
Chapter 2. General Remarks on Linear Integral Equations of the
First Kind 27
Chapter 3. The Picard Theorem of Solvability of a Class of Integral
Equations of the First Kind 51
Chapter 4. Integral Equations of the First Kind with Kernels
Generated by the Schwarz Kernel 81
Chapter 5. Integral Equations of the First Kind with the Kernels
Generated by the Poisson and Neumann Kernels 103
Chapter 6. Some Other Classes of Integral Equation of the First Kind 125
Chapter 7. The Abel Integral Equation and Some of Its Generalizations 181
Chapter 8. A Two-Dimensional Analogue of the Cauchy Type Integral
and Some of Its Applications 215
References 259
1
CHAPTER 1
BRIEF REVIEW OF THE GENERAL THEORY OF THE LINEAR EQUATIONS
IN METRIC SPACES
Let E and E be linear metric spaces and let T be a linear
x y
mapping of E into E
x y
Tx=y. (1.1)
Elements x e E and y e E related to each other by equality
(1.1) are said to be an inverse image and image under
mEa pbpiyn gL (=1 .T1E) .r espectively. Let us denote the image of space
x y y y
If the mapping of space E onto L is one-to-one and,
together with T, the inverse maxp ping x y= ~Ty is also linear,
then E and L are called linearly homeomorphic. When image
y is given beforehand and its inverse image is to be found,
equality (1.1) is called a linear equation. The study of
solvability of this equation and constructing its solutions
(exact, if possible, or approximate) constitute one of the
main subjects of mathematics.
To study equation (1.1), special methods had to be
developed. These methods together with results obtained with
their aid form a unified branch of mathematical knowledge
referred to as the theory of linear equations in metric
spaces. Vital interest drawn to this theory is due to its
general scientific and applied importance.
1. Linear Equation in Finite-Dimensional Spaces
In this section, a finite-dimensional space is understood as
2
the ^-dimensional Euclidean space E of points x with
Cartesian orthogonal coordinates x+ , ..., x . A point x e E
is interpreted as a n-dimensional vector with components
x ...,
v V
The linear equation (1.1) is then represented in the
form
n
z Tikxk = yf l = *> •••• »• ( x-2>
it=i
where T = W. ,\\ is a quadratic n * n matrix whose elements
7*., are real numbers.
A scalar product of two vectors x and y is understood
as the number
n
X xy = xy.
t t
i=\
A norm II* II of vector x is the non-negative number
IIJCII = (JCJC)1/2.
On defining the distance between x and y as II x - y II, space
£ becomes a n-dimensional vector metric space.
Matrix T is called degenerate or non-degenerate
depending on whether detT = 0 or detT * 0, respectively.
A simple but, at the same time, rather important
statement is valid: if matrix T is non-degenerate, equation
(1.2) has a unique solution
x = T~ly (1.3)
for any its right-hand side y e E Here T~ denotes a
matrix inverse to matrix T. Formula (1.3) is known as the
Kramer formula.
The equation
3
Tx = 0 (1.4)
0
is called the homogeneous equation corresponding to equation
(1.2). The equation
T*x* = 0, (1.5)
Q
where
r = , \ i, T\ = T
T k k ki
is referred to as the equation adjoint or conjugate to
equation (1.4).
Matrices T and T* are simultaneously either degenerate
or non-degenerate. When matrix T is non-degenerate, both
equation (1.4) and (1.5) possess only the trivial solutions
*0 = °
and
x* = 0.
Q
In the theory of equation (1.2), the following
statements are of primary importance:
1) Homogeneous equations (1.4) and (1.5) possess equal
finite number I = n - r of linearly independent solutions
xk ', ..., xk ' and x^ ] ..., x^S ' where r is the rank of
matrix T.
2) In order that the non-homogeneous equation (1.2) be
solvable, it is necessary and sufficient that vector y in
its right-hand side be orthogonal to all the linearly
independent solutions of equation (1.5), i.e.,