A Collection of Problems on Mathematical Physics

772 Pages·1964·11.473 MB·English

by  B. M. Budak, A. A. Samarskii, A. N. Tikhonov, I. N. Sneddon, M. Stark and S. Ulam (Auth.)

A C o l l e c t i on of P r o b l e ms on
MATHEMATICAL
P H Y S I CS
B. M. BUDAK, A. A. SAMARSKII
and A. N. TIKHONOV
Translated by
A. R. M. ROBSON
Translation edited by
D. M. BRINK
Clarendon Laboratory, Oxford
PERGAMON PRESS
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1964
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Copyright (g) 1964
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Library of Congress Catalog Card Number 63-17170
This is a translation of the original Russian
C6opHHK aaflaq no MaxeMaTHiiecKOH öH3Hęe
{Sbornik zadach po matematicheskoi fizike)
published by Gostekhizdat, Moscow
Printed in Poland
TRANSLATION EDITORAS NOTE
A NUMBER of the more uninteresting problems which involve the
method of images and the use of special functions have been rejaoved
from the English translation. The collection is still very large and
a student should attempt only a few problems from each section
for himself but will have the solutions of the remaining problems
for reference.
D. M. BRINK
PREFACE
THE PRESENT book is based on the practical work with equations
of mathematical physics done in the Physics Faculty and the
external section of Moscow State University. The problems set
forth were used in the course "Equations of Mathematical Physics"
by A. N. Tikhonov and A. A. Samarskii, and in "A Collection
of Problems on Mathematical Physics" by B. M. Budak. However,
in compihng the present work the range of problems examined
has been considerably enlarged and the number of problems sev­
eral times increased. Much attention has been given to problems
on the derivation of equations and boundary conditions. A con­
siderable number of problems are given with detailed instructions
and solutions. Other problems of similar character are given only
with the answers. The chapters are divided into paragraphs accord­
ing to the method of solution. This has been done in order to give
students the opportunity, by means of independent work, of gain­
ing elementary technical skill in solving problems in the principal
classes of the equations of mathematical physics.
Therefore this book of problems does not claim to include all
methods used in mathematical physics. For example, the opera­
tional method, variational and differential methods and the appli­
cation of integral equations are not considered.
It is hoped, however, that this book will be useful not only to
students but also to engineers and workers in research institutions.
For convenience a set of references is given at the end of
the book. The book "Equations of Mathematical Physics" by
A. N. Tikhonov and A. A. Samarskii is most often referred to,
as the terminology used, and the order in which the material
is set out in this book, most closely corresponds with our own.
In conclusion the authors consider it necessary to point out
that although B. M. Budak and A. N. Tikhonov worked on one
XI
xii PREFACE
group of chapters and A. A. Samarskii and A. N. Tikhonov on
the other group, the joint working out of the general structure
of the book and the joint discussion of the chapters written make
each author responsible in equal measure for its contents.
B. M. BuDAK, A. A. SAMARSKII, A. N. TIKHONOV
CHAPTER I
CLASSIFICATION AND REDUCTION
TO CANONICAL FORM OF SECOND ORDER
PARTIAL DIFFERENTIAL EQUATIONS
IN THIS chapter problems are set on the determination of the type
and on the reduction to canonical form of equations in two and
more independent variables.
In the case of two independent variables equations with con­
stant and variable coefficients are considered. In the case of three
or more independent variables only equations with constant
coefficients are considered, since for three or more independent
variables the equation with variable coefficients cannot, generally
speaking, be reduced to canonical form by the same transforma­
tion, in the entire region, in which the equation belongs to a given
type. In § 1 problems are given for an equation in two independent
variables, and in § 2 for three or more independent variables.
§ 1. The Equation for a Function of Two Independent
Variables
1. The Equation with Variable Coefficients
1. Find the regions where the equation
is hyperboUc, elliptic and parabolic and investigate their depend­
ence on /, where / is a numerical parameter.
In problems Nos. 2-20 reduce the equation to canonical form
in each of the regions.
[1]
2 COLLECTION OF PROBLEMS ON MATHEMATICAL PHYSICS f4
2. ii^^-\-xUj,y = 0.
3. u^^+yuyy = 0.
4. u^^+yu,,+luy = 0.
5. yM„+XM,, = 0.
6. xu^^+yuyy = 0.
8. t/;cxSigny+2W;cy+Wyy = 0.
9. ii^^+2w^^+(l — sign>')M^3, = 0.
10. u^^signy+2u^y+Uyy signX = 0.
11. = 0.
=
12. 0.
13. x2/i,,+yX, = 0.
14. yhi^^+x\y = 0.
15. >;2t^^^+2x>^w^^+x2t/^^ = 0.
16. x2i^^^+2x>;i/^^+/t/^^ = 0.
17. 4A..-e2X^-4>;X = 0.
18. x2M^^+2x>'i/^y—3y^Uy3,—2xu,+4>^Wy+16x^w = 0.
19. (l+x^u,,+(l+y^uyy + xu,+yuy = 0.
20. w^^ sin^x—23;w^y sin x+y^Uyy = 0.
2. The Equation with Constant Coefficients
By means of a substitution u(x,y) = &'^^'^^^ν(χ, y) and reduction
to canonical form simplify the following equations with constant
coefficients.
21. aUxx+4aUxy + aUyy + bUx+cUy+u = 0.
291 Ι· PARTIAL DIFFERENTIAL EQUATIONS 3
22. 2au^^+2aUxy+aUyy+2bu^+2cUy + ii ^ 0.
23. aUxx+2aUxy+aUyy+bu^-\-cUy + u = 0.
§ 2. The Equation with Constant Coefficients for a Function
of η Independent Variables
η η
i, fc = 1 i = 1
Reduce to canonical form equations 24-28.
24. u^^ + 2ii^y+2uyy + 4uy^ + 5u,^ + u^+2uy = 0.
25. u^^—4u^y+2u^, + 4uyy+u,, = 0.
26. U^^+U,, + Uyy + U,,—2U,^+U^, + U,y—2Uy, = U.
27. W^y + Wxz—Wi;c —W);z + tíry + Wíz = 0.
ί=1
i<k
(b) Σ"^.-^* = ο·
i<k
29. Eliminate terms with lowest derivatives in the equation
η η
i=l i = í
CHAPTER I
CLASSIFICATION AND REDUCTION
TO CANONICAL FORM OF SECOND ORDER
PARTIAL DIFFERENTIAL EQUATIONS
§ 1. The Equation for a Function of Two Independent Variables
anUxx-\r2ai2Uxy-\-a22Uyy-\-biUx-\-b2Uy-\-cu = f(x, y)
1. The Equation with Variable Coefficients
1. The discriminant of the equation (l+x)uxx+2xyuxy—y^Uyy = 0 is equal
to α?2-βιι«22 = y^[x^-{-x-{-l] = yKx-Xi)(x-X2)y where
l-|/l-4/ _ l + |/l-4/
^1 = 2 * X2 — ~ 2 ·
Let / < 1/4, then Xi and X2 are real, and for χ < and also for x> X2 the
equation is hyperbolic, and for Xi < χ < X2 it is elliptic; the straight lines
X = Xi and χ = x^ are boundaries of these regions. For / = 1/4 the region
of ellipticity vanishes, since Xi = = —1/2; the straight line χ = —111 forms
the boundary. For /> 1/4 the equation is hyperbolic everywhere.
2. The equation Uxx+xuyy = 0 for χ < 0 belongs to the hyperbolic type
and by the substitution | = l-y-hiY'—xf, η = fj-C/—xf reduces to the
canonical form
For ;c> 0 the equation Uxx+xuyy = 0 belongs to the elliptic type and by
the substitution ξ' = I y, η' = — \/x^ reduces to the canonical form
The characteristics of the equation are the curves (Fig. 14)
y-c=±-j{\/~x)\
where the branches, directed downwards, are given by the equations ξ = const.,
and the branches, directed upwards, are given by the equations η = const.
[163]
164 HINTS, ANSWERS AND SOLUTIONS [3
3. The equation Uxx+yuyy = 0 for :F < 0 is hyperbolic and by the sub­
stitution ξ = x-\-2\^^, η = X—2|/ —y reduces to the canonical form
FIG. 14
For > 0 the equation is elliptic and by substituting I' = x, r{ = 2γγ
reduces to the canonical form
The characteristics of the equation are the parabolae (Fig. 15)
The branches, to the left of the x-axis, are given by the equation ξ = const,
and to the right by τ; = const.
/ X X X ) ^ X X X X X \^
FIG. 15
4. The equation = 0 is of a similar type to the equation
Uxx+yifyy = 0, considered in the preceding problem. By the same substitutions
as in the equation Uxx+yuyy = 0, it reduces to the canonical form d^ujd^dn = 0
in the region where it is hyperbolic {y < 0) and to the canonical form d^u¡dí^+
•\-d^uldn^ = 0 in the region where it is elliptic {y > 0). The characteristics of
the equations Uxx+yuyy+^Uy = 0 and Uxx-{-yUyy = 0 coincide.