Partial differential equations;: An introduction

357 Pages·1972·8.464 MB·English

by  Eutiquio C Young

PARTIAL DIFFERENTIAL
EQUATIONS
An Introduction
1-:utiquio C. Young
The Florida Seate Unhersi~r
A I lyn an<l Bacon, Inc. Boston
t Copyright 1972 by Allyn and Bacon, Inc.
470 Atlantic AYenuc, Boston.
All rights reserved.
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Librar) of Congress Catalog Cnd Number: 70-166551
Printed in the United States of America.
T O M Y P A R E N T S A N D M Y F A ~I I L Y

Contents
Preface 1x
Chapter 1 Introduction
1. Some Properties of Functions of One Variable 2. Partial
Derivatives 3. Differentiation of Composite Functions; the
Chain Rule 4. Differentiation of Integrals Depending on a
Parameter 5. Uniform Convergence of Series 6. Im
proper Integrals Depending on a Parameter 7. Directional
Derivatives 8. Green's Theorem and Related Formulas
Chapter 2 Linear Partial Differential Equations 36
J. Basic Concepts and Definitions 2. General Solutions
and Auxiliary Conditions 3. Linear Operators and Prin
ciple of Superposition 4. Linear First-Order Equations
5. Solutions of Second-Order Equations with Constant Co
efficients 6. Classification of Second-Order Equations
7. The Canonical Forms
Chapter 3 The Wave Equation 71
1. The Vibrating String 2. The Initial Value Problem
3. Interpretation of the Solution 4. Domain of Dependence
and Characteristic Lines 5. The Nonhomogeneous Wave
Equation 6. Uniqueness of Solution 7. Initial-Boundary
Value Problems 8. Nonhomogeneous Problems and Re
flection of Waves 9. Method of Separation of Variables
vii
\iii Contents
Chapter 4 Green's Function and Sturm-Liouville Problems 121
1. Homogeneous Boundary Value Problems 2. Nonhomo
geneous Problems; Green's Function 3. Modified Green"s
Function 4. Sturrn-Liouville Problems 5. Orthogonality
of Eigenfunctions 6. Eigenfunction Expansions; Mean
Convergence 7. Nonhomogeneous Sturm-Liouville Prob
lems; Bilinear Expansion
Chapter 5 Fourier Series and Fourier Integral 163
I. Orthogonal Trigonometric Functions 2. Fourier Series
3. Fourier Cosine and Sine Series 4. Bessel's Inequality;
Riemann-Lebesgue Theorem 5. Convergence of Fourier
Series 6. Uniform Convergence of Fourier Series 7.
Fourier Integral 8. Fourier Transform
Chapter 6 The Heat Equation 212
1. Derivation of the Heat Equation 2. Initial and Boundary
Conditions 3. The Maximum Principle and Uniqueness
Theorem 4. Initial-Boundary Value Problems 5. Non
hornogeneous Initial-Boundary Value Problems 6. The
Initial Value Problem 7. Initial-Boundary Value Problems
in Infinite Domain
Chapter 7 Laplace's Equation 250
1. Boundary Value Problems 2. Green's Theorem and
Uniqueness of Solutions 3. Maximum Principle for Har
monic Functions 4. Dirichlet Problem in a Rectangle
5. Dirichlet Problem in a Disk 6. Poisson's Integral
Formula 7. Neumann Problem in a Disk 8. Problems
in Infinite Domains 9. Fundamental Solution and Green"s
Functions JO. Examples of Green's Functions 11. Neu
mann's Function and Examples
311
References
Solutions of the Exercises 313
343
Index
Preface
This book is the outgrowth of an introductory course in partial differential
equations which the author has given for a number of years. The students are
mostly undergraduates who are majors of mathematics, engineering, or the
physical sciences.
The purpose of the book is to acquaint the students with some of the tech
niques of applied mathematics and to provide them with basic material necessary
for further study in partial differential equations. As the needs and interests
or the students are varied, the author attempts to strike a balance in emphasis
between theory and application.
The book treats principally linear partial differential equations of the first
and second order involving two independent variables. Problems involving
:,econd-order differential equations are discussed with reference to the three
prominent classical equations of mathematical physics, namely, the wave
equation, the heat equation, and Laplace's equation. These equations serve
as prototypes for the three main types of linear partial differential equations of
second order. The kinds of problems that are treated for each of these equations
in two variables as well as the properties of the solutions are generally typical
of what can be expected with more general differential equations of the same
type in three or more independent variables.
Several techniques of applied mathematics-such as the method of eigen
function expansion, the Fourier transform, and the use of Green's function
are developed and their basic underlying theory are discussed. Along with
many examples, a sufficient number of exercises of varying degree of difficulty
appear in nearly every section. These exercises form an integral part of the text.
They are designed not only to test comprehension of the subject matter pre-
ix
x Preface
sented, but also to introduce other general ideas and procedures. Answers to
almost all exercises are given at the back of the book. A list of selected references
for further reading on the subject is also given.
As a prerequisite for a course based on this book, the student must have a
working knowledge of the topics usually covered in a standard calculus course
and must be familiar with the contents of a basic course in ordinary differential
equations. Many of the topics in the calculus which are extensively used in
later discussion are discussed briefly in the beginning chapter. For those
students who have an adequate background, this material can serve as a review.
Although the book is intended for use in a two-quarter course, it can be
adopted for a one-quarter or a one-semester course, depending on the back
ground of students and the interest of the instructor. For example, topics
selected from Chapters 2, 3, 6, and 7 can be the bases of a one-quarter course
for students who are already familiar with the contents of Chapters 4 and 5.
A word about the numbering of equations and theorems: Unless a different
chapter is explicitly stated, the first number indicating an equation or a theorem
always refers to the section of the particular chapter under study. The exercises
are, however, numbered according to the chapters. Thus, the heading Exercises
5.1 refers to the first set of exercises in Chapter 5.
The author wishes to thank Professors Thomas G. Hallam and Howard E.
Taylor for testing some of the material in its early version and for making
helpful comments, and to Professor C. Y. Chan for reading the manuscript
and for making many invaluable suggestions. The author also acknowledges
with gratitude the secretarial help extended by the Department of Mathematics.
Last but not least, it is a pleasure to thank Mrs. Margaret Parramore for her
skillful typing of the manuscript.
Tallahassee, Florida EUTIQUIO c. YOUNG
Partial Differential Equations
An Introduction