Introduction to Partial Differential Equations with Applications

417 Pages·1987·5.754 MB·English

by  E. C. Zachmanoglou, Dale W. Thoe

Introduction to
Partial Differential
Equations with
Applications
N
N
Eli Zachrnanoglou
and Dale W. Thoe

Introduction to
Partial Differential Equations
with Applications
E. C. Zachmanoglou
Professor of Mathematics
Purdue University
Dale W. Thoe
Professor of Mathematics
Purdue University
Dover Publications, Inc., New York
Copyright © 1976, 1986 by E. C. Zachmanoglou and Dale W Thoe
All rights reserved under Pan American and International Copyright Conven-
tions.
Published in Canada by General Publishing Company, Ltd ,30Lesmill Road,
Don Mills, Toronto, Ontario.
Published in the United Kingdom by Constable and Company, Ltd
This Dover edition, first published in 1986, is an unabridged, corrected
republication of the work first published by The Williams & Wilkins Company,
Baltimore, 1976
Manufactured in the United States of America
Dover Publications, Inc., 31 East 2nd Street, Mineola, N Y 11501
Library of Congress Cataloging-in-Publication Data
Zachmanoglou, E. C.
Introduction to partial differential equations with applications
Reprint. Originally published. Baltimore: Williams & Wilkins, c1976
Bibliography. p.
Includes index.
1. Differential equations, Partial. I. Thoe, Dale W II Title.
[QA377.Z32 1986] 515 353 86-13604
ISBN 0-486-65251-3

PREFACE
In writing this introductory book on the old but still rapidly expanding
field of Mathematics known as Partial Differential Equations. our objec-
tive has been to present an elementary treatment of the most important
topics of the theory together with applications to problems from the
physical sciences and engineering. The book should be accessible to
students with a modest mathematical background and should be useful
to those who will actually need to use partial differential equations in
solving physical problems. At the same time we hope that the book will
provide a good basis for those students who will pursue the study of
more advanced topics including what is now known as the modern theory.
Throughout the book, the importance of the proper formulation of
problems associated with partial differential equations is emphasized.
Methods of solution of any particular problem for a given partial differ-
ential equation are discussed only after a large collection of elementary
solutions of the equation has been constructed.
During the last five years, the book has been used in the form of lecture
notes for a semester course at Purdue University. The students are
advanced undergraduate or beginning graduate students in mathematics,
engineering or one of the physical sciences. A course in Advanced
Calculus or a strong course in Calculus with extensive treatment of
functions of several variables, and a very elementary introduction to
Ordinary Differential Equations constitute adequate preparation for the
understanding of the book. In any case, the basic results of advanced
calculus are recalled whenever needed.
The book begins with a short review of calculus and ordinary differ-
ential equations. A new elementary treatment of first order quasi-linear
partial differential equations is then presented. The geometrical back-
ground necessary for the study of these equations is carefully developed.
Several applications are discussed such as applications to problems in
gas dynamics (the development of shocks), traffic flow, telephone net-
works, and biology (birth and death processes and control of disease).
The method of probability generating functions in the study of stochastic
processes is discussed and illustrated by many examples. In recent books
the topic of first order equations is either omitted or treated inadequately.
In older books the treatment of this topic is probably inaccessible to
most students.
A brief discussion of series solutions in connection with one of the
basic results of the theory, known as the Cauchy-Kovalevsky theorem,
is included. The characteristics, classification and canonical forms of
linear partial differential equations are carefully discussed.
For students with little or no background in physics, Chapter VI,
"Equations of Mathematical Physics," should be helpful. In Chapters VII,
VIII and IX where the equations of Laplace, wave and heat are studied, the
physical problems associated with these equations are always used to
motivate and illustrate the theory. The question of determining the well-
posed problems associated with each equation is fundamental throughout
the discussion.
vi Preface
The methods of separation of variables and Fourier series are intro-
duced in the chapter on Laplace's equation and then used again in the
chapters on the wave and heat equations. The method of finite differences
coupled with the use of computers is illustrated with an application to
the Dirichlet problem for Laplace's equation.
The last chapter is devoted to a brief treatment of hyperbolic systems
of equations with emphasis on applications to electrical transmission
lines and to gas dynamics.
The problems at the end of each section fall in three main groups.
The first group consists of problems which ask the student to provide
the details of derivation of some of the items in the text. The problems
in the second group are either straightforward applications of the theory
or ask the student to solve specific problems associated with partial
differential equations. Finally, the problems in the third group introduce
new important topics. For example, the treatment of nonhomogeneous
equations is left primarily to these problems. The student is urged at
least to read these problems.
The references cited in each chapter are listed at the end of that
chapter. A guide to further study, a bibliography for further study and
answers to some of the problems appear at the end of the book.
The book contains roughly twenty-five percent more material than can
be covered in a one-semester course. This provides flexibility for planning
either a more theoretical or a more applied course. For a more theoretical
course, some of the sections on applications should be omitted. For a
more applied version of the course, the instructor should only outline the
results in the following sections: Chapter III, Section 4; Chapter IV,
Sections 1 and 2; Chapter V Sections 5, 6, 7, 8, and 9 (the classification
and characteristics of second order equations should be carefully dis-
cussed, however); Chapter VII, the proof of Theorem 10.1 and Section
11.
We are indebted to many of our colleagues and students for their
comments concerning the manuscript, and extend our thanks for their
help. Finally, to Judy Snider, we express our deep appreciation for her
expert typing of the manuscript.
Purdue University E. C. ZACHMANOGLOU
1975 DALE W. THOE
TABLE OF CONTENTS
PREFACE
CHAPTER I. SOMECONCEPTS FROM CAL-
CULUS AND ORDINARY DIF-
FERENTIAL EQUATIONS
1
1. Sets and functions 1
2. Surfaces and their normals. The implicit function theorem. .. 5
3. Curves and their tangents 11
4. The initial value problem for ordinary differential equations
and systems 16
References for Chapter I 23
CHAPTER II. INTEGRAL CURVES AND SUR-
FACES OF VECTOR FIELDS ... 24
1. Integral curves of vector fields 24
2. Methods of solution of dx/P = dy/Q = dz/R 35
3. The general solution of + + = 0 41
4. Construction of an integral surface of a vector field containing
a given curve 44
5. Applications to plasma physics and to solenoidal vector
fields 51
References for Chapter II 56
CHAPTER III. THEORY AND APPLICATIONS
OF QUASI-LINEAR AND LIN-
EAR EQUATIONS OF FIRST
ORDER
1. First order partial differential equations 57
2. The general integral of + = R 59
3. The initial value problem for quasi-linear first order equations.
Existence and uniqueness of solution 64
4. The initial value problem for quasi-linear first order equations.
Nonexistence and nonuniqueness of solutions 69
5. The initial value problem for conservation laws. The develop-
ment of shocks 72
6. Applications to problems in traffic flow and gas dynamics. .. 76
7. The method of probability generating functions. Applications
to a trunking problem in a telephone network and to the
control of a tropical disease 86
References for Chapter III 95
vU
viii Contents
CHAPTER IV. SERIES SOLUTIONS. THE
CAUCHY-KO VALE VSKY THE.
OREM
96
1. Taylor series. Analytic functions 96
2. The Cauchy-Kovalevsky theorem 100
References for Chapter IV 111
CHAPTER V. LINEAR PARTIAL DIFFEREN
TIAL EQUATIONS. CHARAC-
TERISTICS, CLASSIFICATION
AND CANONICAL FORMS .... 112
1. Linear partial differential operators and their characteristic
curves and surfaces 112
2. Methods for finding characteristic curves and surfaces.
Examples 117
3. The importance of characteristics. A very simple example. .. 124
4. The initial value problem for linear first order equations in
two independent variables 126
5. The general Cauchy problem. The Cauchy-Kovalevsky the-
orem and Holmgren's uniqueness theorem 132
6. Canonical form of first order equations 133
7. Classification and canonical forms of second order equations
in two independent variables 137
8. Second order equations in two or more independent varia-
bles 143
9. The principle of superposition 149
Reference for Chapter V 152
CHAPTER VI. EQUATIONS OF MATHEMAT-
ICAL PHYSICS
153
1. The divergence theorem and the Green's identities 153
2. The equation of heat conduction 156
3. Laplace's equation 161
4. The wave equation 162
5. Well-posed problems 166
Reference for Chapter VI 170
CHAPTER VII. LAPLACE'S EQUATION
171
1. Harmonic functions 172
2. Some elementary harmonic functions. The method of separa-
tion of variables 173
Contents ix
3. Changes of variables yielding new harmonic functions. Inver-
sion with respect to circles and spheres 179
4. Boundary value problems associated with Laplace's equa-
tion 186
5. A representation theorem. The mean value property and the
maximum principle for harmonic functions 191
6. The well-posedness of the Dirichlet problem 197
7. Solution of the Dirichlet problem for the unit disc. Fourier
series and Poisson's integral 199
8. Introduction to Fourier series 206
9. Solution of the Dirichlet problem using Green's functions. .. 223
10. The Green's function and the solution to the Dirichlet problem
for a ball in R3 226
11. Further properties of harmonic functions 232
12. The Dirichlet problem in unbounded domains 235
13. Determination of the Green's function by the method of
electrostatic images 240
14. Analytic functions of a complex variable and Laplace's equa-
tion in two dimensions 245
15. The method of finite differences 249
16. The Neumann problem 257
References for Chapter VII 260
CHAPTER VIII. THE WAVE EQUATION
261
1. Some solutions of the wave equation. Plane and spherical
waves 262
2. The initial value problem 271
3. The domain of dependence inequality. The energy method. 274
.
4. Uniqueness in the initial value problem. Domain of depend-
ence and range of influence. Conservation of energy 280
5. Solution of the initial value problem. Kirchhoff's formula.
The method of descent 284
6. Discussion of the solution of the initial value problem.
Huygens' principle. Diffusion of waves 291
7. Wave propagation in regions with boundaries. Uniqueness of
solution of the initial-boundary value problem. Reflection
of waves 299
8. The vibrating string 308
9. Vibrations of a rectangular membrane 317
10. Vibrations in finite regions. The general method of separation
of variables and eigenfunction expansions. Vibrations of a
circular membrane 322
References for Chapter VIII 330
CHAPTER IX. THE HEAT EQUATION
331
1. Heat conduction in a finite rod. The maximum-minimum
principle and its consequences 331
2. Solution of the initial-boundary value problem for the one-
dimensional heat equation 336