Elements of Partial Differential Equations

336 Pages·2006·23.29 MB·English

by  Ian N. Sneddon

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ELEMENTS OF
PARTIAL
DIFFERENTIAL
EQUATIONS
SNEDDON
IAN
N.
DOVER
PUBLICATIONS,
INC.
MINEOLA, NEW YORK
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Bibliographical Note
This Dover edition, first published in 2006, is an unabridged
republication of the edition originally published
by McGraw -Hill
Book Company, Inc., New York, 1957.
Library of Congress Cataloging- in- Publication Data
Sneddon, Ian Naismith.
Elements of partial differential equations l Ian N. Sneddon
cm.
P.
Originally published: New York McGraw-Hill,
1957.
:
Includes index.
ISBN 0 -486- 45297 -2 (pbk.)
1. Differential equations, Partial. I. Title.
QA374.S6 2006
515'.353-dc22
2006045445
Manufactured in the United States of America
Dover Publications, Inc., 31 East 2nd Street, Mineola, N.Y. 11501
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PREFACE
The aim of this book is to present the elements of the theory of
partial differential equations in a form suitable for the use of students
and research workers whose main interest in the subject lies in finding
solutions of particular equations rather than in the general theory.
In its present form it has developed from courses given by the author
over the last ten years to audiences of mathematicians, physicists,
and engineers in the University of Glasgow and the University College
of North Staffordshire, and to members of the Research Staff of the
English Electric Company at Stafford. It therefore caters for readers
primarily interested in applied rather than pure mathematics, but it is
hoped that it will be of interest to students of pure mathematics following
a first course in partial differential equations.
A number of worked examples have been included in the text to
aid readers working independently of a teacher. The problems
which are given to test the reader's grasp of the text contain, in some
cases, additional theorems not considered in the body of the text.
Some of them are therefore merely statements of classical results
;
the remainder have been taken from examination papers of the
University of Glasgow and are reproduced here by permission of the
Secretary of the University Court.
The author is indebted to a number of colleagues and students for
critical comments on the lectures upon which this book is based and
for reading portions of the manuscript. In particular, thanks are
due to Ben Noble and Keith Fitch, who worked through the entire
manuscript and checked the problems to Janet Burchnall and
;
Valerie Cook, who prepared the manuscript for the press to John
;
Lowndes, who devoted a great deal of time to reading the proof sheets;
and to Elizabeth Gildart, who prepared the index.
IAN N. SNEDDON
Simson Professor of Mathematics
University of Glasgow
iii
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CONTENTS
f
P r
e a c e
i i i
CHAPTER 1. ORDINARY DIFFERENTIAL EQUATIONS IN MORE THAN Two
VARIABLES
1
Surfaces and Curves in Three Dimensions
1.
. . . . 1
2. Simultaneous Differential Equations of the First Order and the
First Degree in Three Variables 7
= =
3. Methods of Solution of dx jP dyl Q dz/R 10
4. Orthogonal Trajectories of a System of Curves on a Surface 15
. .
5. Pfaffian Differential Forms and Equations 18
.
6. Solution of Pfaffian Differential Equations in Three Variables 26
. .
7. Carathéodory's Theorem 33
8. Application to Thermodynamics 39
. . . . . . . . . . . .
Miscellaneous Problems 42
CHAPTER 2. PARTIAL DIFFERENTIAL EQUATIONS OF THE FIRST ORDER 44
.
1. Partial Differential Equations 44
2. Origins of First -order Partial Differential Equations 45
3. Cauchy's Problem for First -order Equations 47
4. Linear Equations of the First Order 49
5. Integral Surfaces Passing through a Given Curve 56
6. Surfaces Orthogonal to a Given System of Surfaces 57
7. Nonlinear Partial Differential Equations of the First Order 59
. .
8. Cauchy's Method of Characteristics 61
9. Compatible Systems of First -order Equations 67
. . .
10. Charpit's Method 69
11. Special Types of First -order Equations 70
12. Solutions Satisfying Given Conditions 73
13. Jacobi's Method 78
14. Applications of First -order Equations 81
Miscellaneous Problems 85
CHAPTER 3. PARTIAL DIFFERENTIAL EQUATIONS OF THE SECOND ORDER 88
The Origin of Second -order Equations 88
1.
2. Second -order Equations in Physics 90
3. Higher -order Equations in Physics 94
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CONTENTS
vi
4. Linear Partial Differential Equations with Constant Coefficients 96
.
5. Equations with Variable Coefficients 105
6. Characteristic Curves of Second -order Equations 110
7. Characteristics of Equations in Three Variables 115
8. The Solution of Linear Hyperbolic Equations 119
9. Separation of Variables 123
10. The Method of Integral Transforms 126
11. Nonlinear Equations of the Second Order 131
Miscellaneous Problems 136
CHAPTER 4. LAPLACE'S EQUATION 141
1. The Occurrence of Laplace's Equation in Physics 141
. .
2. Elementary Solutions of Laplace's Equation 145
. . . .
3. Families of Equipotential Surfaces 148
4. Boundary Value Problems 151
5. Separation of Variables 156
6. Problems with Axial Symmetry 161
7. Kelvin's Inversion Theorem 164
8. The Theory of Green's Function for Laplace's Equation 167
. . . .
9. The Relation of Dirichlet's Problem to the Calculus of Variations 174
10. "Mixed" Boundary Value Problems 175
.
11. The Two -dimensional Laplace Equation 180
12. Relation of the Logarithmic Potential to the Theory of Functions 184
13. Green's Function for the Two- dimensional Equation 193
Miscellaneous Problems 197
CHAPTER 5. THE WAVE EQUATION 209
The Occurrence of the Wave Equation in Physics 209
1.
2. Elementary Solutions of the One -dimensional Wave Equation 215
.
3. The Riemann- Volterra Solution of the One -dimensional Wave
Equation 221
4. Vibrating Membranes Application of the Calculus of Variations 226
:
5. Three -dimensional Problems 232
6. General Solutions of the Wave Equation 239
7. Green's Function for the Wave Equation 244
.
8. The Nonhomogeneous Wave Equation 249
9. Riesz's Integrals 254
10. The Propagation of Sound Waves of Finite Amplitude 257
. . .
Miscellaneous Problems 262
CHAPTER 6. THE DIFFUSION EQUATION 274
.
The Occurrence of the Diffusion Equation in Physics 274
1.
2. The Resolution of Boundary Value Problems for the Diffusion
Equation 278
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CONTENTS
vii
3. Elementary. Solutions of the Diffusion Equation 282
4. Separation of Variables 286
5. The Use of Integral Transforms 290
6. The Use of Green's Functions 294
7. The Diffusion Equation with Sources 299
Miscellaneous Problems 303
APPENDIX. SYSTEMS OF SURFACES 309
1. One -parameter Systems 309
2. Two -parameter Systems 311
3. The Edge of Regression 312
4. Ruled Surfaces 314
Solutions to the Odd- numbered Problems 315
.
Index 323
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Chapter
I
ORDINARY DIFFERENTIAL EQUATIONS IN
MORE THAN TWO VARIABLES
In this chapter we shall discuss the properties of ordinary differential
equations in more than two variables. Parts of the theory of these
equations play important roles in the theory of partial differential
equations, and it is essential that they should be understood thoroughly
before the study of partial differential equations is begun. Collected
in the first section are the basic concepts from solid geometry which
are met with most frequently in the study of differential equations.
I. Surfaces and Curves in Three Dimensions
By considering special examples it is readily seen that if the rectangular
Cartesian coordinates (x,y,z) of a point in three dimensional space
-
are connected by a single relation of the type
f(x,y,z) = (l)
O
the point lies on a surface. For that reason we call the relation (1) the
equation of a surface S.
To demonstrate this generally we suppose a point (x,y,z) satisfying
equation (1). Then any increments (x,ôy,ôz) in (x,y,z) are related by
the equation
á
s
+ + =
o
(5y áZ (5z
ax
y
so that two of them can be chosen arbitrarily. In other words, in the
neighborhood of P(x,y,z) there are points P'(x + y + n, z + o satis-
,
fying (1) and for which any two of are chosen arbitrarily and the
, 77,
third is given by ,á
+á
+ßáy =o
The projection of the initial direction PP' on the plane xOy may there-
fore be chosen arbitrarily. In other words, equation (1) is, in general,
a relation satisfied by points which lie on a surface.
1
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