A Course of Higher Mathematics. Volume IV

817 Pages·1964·34.992 MB·English

by  V. I. Smirnov and A. J. Lohwater (Auth.)

ADIWES INTERNATIONAL SERIES
IN MATHEMATICS
A. J. LOHWATER
Consulting Editor
A COURSE OF
Higher Mathematics
VOLUME IV
V. I. SMIRNOV
Translated by
D. E. BROWN
Translation edited by
I. N. SNEDDON
Simson Professor in Mathematics
University of Glasgow
PERGAMON PRESS
OXFORDLONDONEDINBURGHNEW YORK
PARIS · FRANKFURT
ADDISON-WESLEY PUBLISHING COMPANY, INC.
READING, MASSACHUSETTS · PALO ALTO · LONDON
1964
Copyright © 1964
PERGAMON PRESS LTD.
U. S. A. Edition distributed by
ADDISON-WESLEY PUBLISHING COMPANY, INC.
Reading, Massachusetts · Palo Alto · London
PERGAMON PRESS
International Series of Monographs in
PURE AND APPLIED MATHEMATICS
Volume 61
Library of Congress Catalog Card No. 63-10134
This translation has been made from
the Russian Edition of
V. I. Smirnov's book
Kypc ebicmeû MamemamUKU (Kurs vysshei matematiki),
published in 1959 by Fizmatgiz, Moscow
MADE IN GREAT BRITAIN
INTRODUCTION
AN ACCOUNT of the over-all plan of Prof. Smirnov's five-volume
course on higher mathematics has been given in the Introduction to
Vol. I of the present English edition.
This fourth volume of the set is devoted to subjects which lie at
the very heart of classical mathematical physics and which supply
the motivation of much recent work of great interest in functional
analysis and in the theory of partial differential equations. The more
elementary parts of the theory of the differential equations of mathe-
matical physics have already been treated at the end of Vol. II.
The present volume begins with full accounts of the theory of inte-
gral equations and of the calculus of variations which together play
an important role in the discussion of the boundary value prob-
lems of mathematical physics. This is followed by a long chapter on
the fundamental theory of partial differential equations and of sys-
tems of equations in which characteristics play a central role. Finally
the boundary value problems of mathematical physics are treated in
a complete and lucid way.
Although this volume is primarily intended for the use of mathema-
ticians whose main interest is in the application of mathematics
to the analysis and elucidation of physical problems, it contains
many topics an acquaintance with which can only serve to deepen
the understanding of anyone embarking on the study of functional
analysis and other branches of analysis and it is to be hoped that
it will find its way into their hands.
I. N. SNEDDON
xi
PREFACE TO THE SECOND EDITION
EVERY chapter except the one on the calculus of variations has been
entirely revised in the present edition of Volume IV. Part of the
material has been transferred to the new edition of Volume II, and
a great deal of new material has been introduced.
I must sincerely thank S. M. Lozinskii for reading the chapter on
integral equations in manuscript and making numerous valuable
suggestions, which I utilized in the final revision. I often discussed the
last two chapters with O. A. Ladyzhenskaya and Kh. L. Smolitskii,
who gave me valuable assistance. Certain sections of these chapters
were written by these authors at my request, as indicated in the text.
O.A. Ladyzhenskaya read Chapter III in manuscript and Kh. L.
Smolitskii Chapter IV. They made a number of helpful suggestions as
regards the final proofs. I thank them most sincerely.
V. SMIRNOV
PREFACE TO THE THIRD EDITION
THE present edition of Volume IV does not differ essentially from
the previous one. The most substantial change is in the treatment
of integral equations with kernels depending on a difference, over a
semi-infinite interval. The new treatment based on the theory of
boundary value problems for analytic functions, is due to Yu. I.
Cherskii. Apart from this, minor modifications have been introduced
into the text in various places.
I wish to express my sincere thanks to Yu. I. Cherskii.
V. SMTRNOV
xiii
PREFACE TO THE SECOND EDITION
EVERY chapter except the one on the calculus of variations has been
entirely revised in the present edition of Volume IV. Part of the
material has been transferred to the new edition of Volume II, and
a great deal of new material has been introduced.
I must sincerely thank S. M. Lozinskii for reading the chapter on
integral equations in manuscript and making numerous valuable
suggestions, which I utilized in the final revision. I often discussed the
last two chapters with O. A. Ladyzhenskaya and Kh. L. Smolitskii,
who gave me valuable assistance. Certain sections of these chapters
were written by these authors at my request, as indicated in the text.
O.A. Ladyzhenskaya read Chapter III in manuscript and Kh. L.
Smolitskii Chapter IV. They made a number of helpful suggestions as
regards the final proofs. I thank them most sincerely.
V. SMIRNOV
PREFACE TO THE THIRD EDITION
THE present edition of Volume IV does not differ essentially from
the previous one. The most substantial change is in the treatment
of integral equations with kernels depending on a difference, over a
semi-infinite interval. The new treatment based on the theory of
boundary value problems for analytic functions, is due to Yu. I.
Cherskii. Apart from this, minor modifications have been introduced
into the text in various places.
I wish to express my sincere thanks to Yu. I. Cherskii.
V. SMTRNOV
xiii
CHAPTER I
INTEGRAL EQUATIONS
1. Examples of the formation of integral equations. Any equation
containing the required function under the integral sign is an integral
equation. Suppose we want to find the solution of the differential
equation y' = f(x, y), satisfying the initial condition y(x ) = y .
0 0
We saw previously [II, 51] that the problem amounts to solving the
integral equation:
X
y{x) = I /(*> y)dx + Vo ·
Xp
In the same way, the problem of integrating the second order dif-
ferential equation y" = f(x, y) with the initial conditions y(x ) — y ;
Q 0
y'(x ) = y' leads to the integral equation:
0 0
X X
y(x) = J" àx J f[z, y(z)] dz + y + y' {x - x ).
0 0 0
X9 X0
We can rewrite this as follows by transforming the double integral
into a single integral [II, 15]:
X
y(x) = J (x - z) f[z, y(z)] dz + y + y' {x - x ).
0 0 0
The general solution of 'y" = f(x, y) is obtained from the integral
equation
X
y{x)= J (x — z)f[zy(z)]dz + c + c x, (1)
f i 2
o
where c and c are arbitrary constants and the lower limit of integra-
x 2
tion has been taken equal to zero. We now consider a boundary value
problem for our second order differential equation; we seek the
solution satisfying the boundary conditions y(0) = a; y(l) = 6.
On substituting first x = 0 then x — I in equation (1), we obtain
two equations for the arbitrary constants; these give us
c = a; c = —=^- - j- J (Z - 2) /[>, y(z)] dz .
x 2
o
1
2 INTEGRAL EQUATIONS [1
By substituting the values obtained in (1), we reduce our boundary
value problem to the integral equation:
X l
y(x) = F(x) +]"(*- 2) f[z, y(z)] dz - -Ξ- J(Î - z)/[*, */(*)] dz, (2)
6 o
where
We can rewrite (2) as follows:
x ι
y(x) = F(x) - J"^^-/[z, 2/(2)] dz - Jü£z^-/[ , „(g)] dz. (3)
z
0 X
We introduce the function of two variables:
I -*—j—- for z < x
K(x,z) =
n ^ (4)
^Lfovx<z.
Equation (3) can be written as follows with the aid of this function:
i
y(x) = F(x) - j" K(x, z) f[z, y(z)] dz. (5)
o
We apply the results obtained to the linear equation
y' + p{*)y = <»(*)· (6)
We can say that the problem of finding the solution of this equation
with the boundary conditions
2/(0)= a; y(l) = b (7)
is equivalent to finding the function y(x) satisfying the linear integral
equation
ι
y(x) = F (x) + j K(x, z) p(z) y(z) dz, (8)
±
where
F(x) = F(x) — J K{x z) ω(ζ) dz
x 9
o
is a known function of the independent variable x.
Notice that the upper limit of integration is variable in equa-
tion (1), whereas in (8) both limits are constant. Notice also that,
both in (1) and in (8), the required function appears outside as well
1] EXAMPLES OP THE FORMATION OF INTEGRAL EQUATIONS 3
as under the integral sign. As we saw previously [II, 50], this is of great
importance in enabling us to apply the method of successive approxi-
mations to the solution of the equation.
We multiply the coefficient p(x) in (6) by a parameter λ and consider
the homogeneous equation
y" + Xp{x)y = 0 (9)
with the homogeneous boundary conditions
2/(0) = 0; y(Z) = 0. (10)
This homogeneous boundary value problem leads us to a homogeneous
integral equation containing the parameter λ:
ι
y(x)=?^K(x,z)p(z)y(z)dz. (H)
o
One of the main problems that will confront us later is this: for
what values of the parameter λ has our problem solutions that
do not vanish identically? We have already met this question when
applying Fourier's method to boundary value problems of mathemati-
cal physics. We notice further some characteristic properties of the
function K(x, z), which is known as the kernel of the integral equation.
The kernel is continuous in the square Jc , defined by the inequalities
0
0 < x < I and 0 < z < I. On the diagonal of the square where x = z
the first derivative of the kernel possesses a discontinuity:
&x(x> z) l*-z+o - Κχ{*> z) Uz-o = - 1.
Furthermore, outside the diagonal the kernel, considered as a function
of x, is the solution of the homogeneous equation y" = 0 satisfying
the boundary conditions (10). We observe finally the symmetrical
nature of the kernel, expressed by the equation:
K{z, x) = K(x, z). (12)
All these properties of the kernel follow immediately from (4).
The kernel K(x, z) has a simple physical meaning. We recall that,
when a concentrated force acts at a point x = z of a string fixed at
both ends, the condition must hold at the point of application [II, 163]:
^[(«Οχ-ζ+ο - K)x=*-o] = —P,
where P is the magnitude of the force. The function
η(χ) = -γ-Κ(χ,ζ)
4 INTEGRAL EQUATIONS [1
may easily be seen to give the statical shape of the bent string due
to this concentrated force. It may be mentioned here that the equation
of vibration of the string in the statical case amounts simply to the
equation u = 0. All these ideas concerning the reduction of a
xx
boundary value problem to an integral equation, discussed here for
an elementary case, will be analysed in detail in Chapter IV.
We shall further mention a characteristic method of reducing the
boundary value problems of mathematical physics to integral equations.
We defined earlier the potential of a spherical layer by the expression:
S
where q(M') is a function given on the surface of the sphere S, ds
is an elementary area of the sphere, and d is the distance from a
variable point M of space to a variable point M' of the sphere. Let
n be the normal direction at a point M of the sphere. Let (du(M )ldn)i
0 0
and (du(M )[3n) denote respectively the limits of the derivative
0 e
du(M)jdn as the variable point M of space approaches the point M
0
from inside and outside the sphere. We previously [III , 138] deduced
2
the following expressions:
^ ')-?ψ-ά8 + 2π (Μ ),
{ dn )i ~ ρ{Μ ρ 0
(13)
$\ (3Ι')^ψ-ά8-2πρ(Μ ),
dn )e ρ 0
S
where d is the distance from the point M to the variable point M'
0
of the sphere and ω is the angle formed by the radius vector M'M
Q
with the direction n.
We shall see in the next chapter that these expressions are not
only valid for a sphere. We now pose the interior Neumann problem
for the sphere, i.e. we take it that the function is required which is
harmonic inside the sphere and whose normal derivative has assigned
boundary values on the surface of the sphere:
du
= fWo). (I*)
dn s
We shall seek u as the potential of a spherical layer. This potential is
a harmonic function inside the sphere and we only have to choose
the density ρ(Μ') of the potential such that the boundary condition