Elementary Applied Partial Differential Equations: With Fourier Series and Boundary Value Problems

545 Pages·1987·23.61 MB·English

by  Richard Haberman

Second Edition
Elementary Rpplied Partial
Differential Equations
with Fourier Series
and Boundory Value Problems
RICHARD HABERMAN
Department of Mathematics
Southern Methodist University
Prentice Hall, Englewood Cliffs, New Jersey 07632
Library of Congress Cataloging-in-Publication Data
Haberman, Richard, (date)
Elementary applied partial differential equations.
Bibliography: p.
Includes index.
1. Differential equations, Partial. 2. Fourier
series. 3. Boundary value problems. I. Title.
QA377.H27 1987 515.3'53 86- 1705 1
ISBN 0- 13-252875-4
Editorial/production supervision and
interior design: Maria McColligan
Cover design: Karen A. Stephens
Manufacturing buyer: John B. Hall
01 987, 1983 by Prentice-Hall, Inc.
A Paramount Communications Company
Englewood Cliffs, New Jersey 07632
All rights reserved. No part of this book may be reproduced,
in any form or by any means, without permission in writing
from the publisher.
Printed in the United States of America
I S B N 0-13-252875-4
Prentice-Hall International, Inc., London
Prentice-Hall of Australia Pty. Limited, Sydney
Editora Prentice-Hall do Brasil, Ltda., Rio de Janeiro
Prentice-Hall Canada Inc., Toronto
Prentice-Hall Hispanoamericana, S.A., Mexico
Prentice-Hall of India Private Limited, New Delhi
Prentice-Hall of Japan, Inc., Tokyo
Prentice-Hall of Southeast Asia Pte. Ltd., Singapore
To Liz, Ken, and Vicki
Contents
PREFACE xi
CHAPTER 1 HEAT EQUATION 1
1 .I Introduction 1
1.2 Derivation of the Conduction of Heat in a One-Dimensional
Rod 2
1.3 Boundary Conditions 10
1.4 Equilibrium Temperature Distribution 12
1.5 Derivation of the Heat Equation in Two or Three
Dimensions 17
Appendix 1.5 Review of Gradient and a Derivation of
Fourier's Law of Heat Conduction 26
CHAPTER 2 METHOD OF SEPARATION OF VARIABLES 28
2.1 Introduction 28
2.2 Linearity 29
2.3 Heat Equation with Zero Temperatures at Finite Ends 31
Appendix 2.3 Orthogonality of Functions 49
2.4 Worked Examples with the Heat Equation (Other Boundary
Value Problems) 50
2.5 Laplace's Equation: Solutions and Qualitative
Properties 61
CHAPTER 3 FOURIER SERIES 75
3.1 Introduction 75
3.2 Statement of Convergence Theorem 77
3.3 Fourier Cosine and Sine Series 81
3.4 Term-by-term Differentiation of Fourier Series 99
3.5 Term-by-term Integration of Fourier Series 108
CHAPTER 4 VIBRATING STRINGS AND MEMBRANES 113
4.1 Introduction 113
4.2 Derivation of a Vertically Vibrating String 113
4.3 Boundary Conditions 776
4.4 Vibrating String with Fixed Ends 120
4.5 Vibrating Membrane 126
CHAPTER 5 STURM-LIOUVILLE EIGENVALUE PROBLEMS 129
5.1 Introduction 129
5.2 Examples 130
5.3 Sturm-Liouville Eigenvalue Problems 133
5.4 Worked Example-Heat Flow in a Nonuniform Rod without
Sources 14 7
5.5 Self-Adjoint Operators and Sturm-Liouville Eigenvalue
Problems 145
Appendix 5.5 Matrix Eigenvalue Problem and Orthogonality
of Eigenvectors 754
5.6 Rayleig h Quotient 759
5.7 Worked Example-Vibrations of a Nonuniform String 164
5.8 Boundary Conditions of the Third Kind 167
5.9 Large Eigenvalues (Asymptotic Behavior) 180
5.10 Approximation Properties 184
CHAPTER 6 PARTIAL DIFFERENTIAL EQUATIONS WITH AT LEAST
THREE INDEPENDENT VARIABLES 190
6.1 Introduction 190
6.2 Separation of the Time Variable 19 7
6.3 Vibrating Rectangular Membrane 194
Appendix 6.3 Outline of Alternative Method to Separate
Variables 202
6.4 Statements and Illustrations of Theorems for the Eigenvalue
+
Problem V2+ A+ = 0 203
6.5 Self-Adjoint Operators and Multidimensional Eigenvalue
Problems 208
Appendix 6.5 Gram-Schmidt Method 27 1
Contents
6.6 Rayleigh Quotient 2 13
6.7 Vibrating Circular Membrane and Bessel Functions 2 14
6.8 More on Bessel Functions 229
6.9 Laplace's Equation in a Circular Cylinder 236
CHAPTER 7 NONHOMOGENEOUS PROBLEMS 246
7.1 Introduction 246
7.2 Heat Flow with Sources and Nonhomogeneous Boundary
Conditions 246
7.3 Method of Eigenfunction Expansion with Homogeneous
Boundary Conditions (Differentiating Series of
Eigenfunctions) 252
7.4 Method of Eigenfunction Expansion Using Green's Formula
(with or without Homogeneous Boundary Conditions) 257
7.5 Forced Vibrating Membranes and Resonance 262
7.6 Poisson's Equation 270
CHAPTER 8 GREEN'S FUNCTIONS FOR TIME-INDEPENDENT
PROBLEMS 277
8.1 Introduction 277
8.2 One-Dimensional Heat Equation 277
8.3 Green's Functions for Boundary Value Problems for Ordinary
Differential Equations 28 1
Appendix 8.3 Establishing Green's Formula with Dirac Delta
Functions 298
8.4 Fredholm Alternative and Modified Green's Functions 299
8.5 Green's Functions for Poisson's Equation 309
8.6 Perturbed Eigenvalue Problems 329
8.7 Summary 334
CHAPTER 9 INFINITE DOMAIN PROBLEMS-FOURIER TRANSFORM
SOLUTIONS OF PARTIAL DIFFERENTIAL
EQUATIONS 335
9.1 Introduction 335
9.2 Heat Equation on an Infinite Domain 335
9.3 Complex Form of Fourier Series 338
9.4 Fourier Transform Pair 34 1
Appendix 9.4 Derivation of the Inverse Fourier Transform of
a Gaussian 344
9.5 Fourier Transform and the Heat Equation 349
9.6 Fourier Sine and Cosine Transforms-The Heat Equation on
Semi-infinite Intervals 359
9.7 Worked Examples Using Transforms 369
Contents vii
CHAPTER 10 GREEN'S FUNCTIONS FOR TIME-DEPENDENT
PROBLEMS 389
10.1 Introduction 389
10.2 Green's Functions for the Wave Equation 389
10.3 Green's Functions for the Heat Equation 404
CHAPTER 11 THE METHOD OF CHARACTERISTICS FOR LINEAR
AND QUASI-LINEAR WAVE EQUATIONS 417
11 .1 Introduction 417
11.2 Characteristics for First-Order Wave Equations 418
11.3 Method of Characteristics for the One-Dimensional Wave
Equation 423
11.4 Semi-infinite Strings and Reflections 429
11.5 Method of Characteristics for a Vibrating String of Fixed
Length 434
11.6 The Method of Characteristics for Quasi-Linear Partial
Differential Equations 437
CHAPTER 12 A BRIEF INTRODUCTION TO LAPLACE TRANSFORM
SOLUTION OF PARTIAL DIFFERENTIAL
EQUATIONS 450
12.1 Introduction 450
12.2 Elementary Properties of the Laplace Transform 45 7
12.3 Green's Functions for Initial Value Problems for Ordinary
Differential Equations 459
12.4 An Elementary Signal Problem for the Wave
Equation 462
12.5 A Signal Problem for a Vibrating String of Finite
Length 465
12.6 The Wave Equation and Its Green's Function 468
12.7 Inversion of Laplace Transforms Using Contour Integrals
in the Complex Plane 47 1
12.8 Solving the Wave Equation Using Laplace Transforms
(with Complex Variables) 475
CHAPTER 13 AN ELEMENTARY DISCUSSION OF FINITE DIFFERENCE
NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL
EQUATIONS 478
13.1 Introduction 478
13.2 Finite Differences and Truncated Taylor Series 479
13.3 Heat Equation 484
viii Contents
13.4 Two-Dimensional Heat Equation 506
13.5 Wave Equation 509
13.6 Laplace's Equation 512
SELECTED ANSWERS TO STARRED EXERCISES 519
BIBLIOGRAPHY 531
INDEX 533
Contents
Preface
This text discusses elementary partial differential equations in the engineering
and physical sciences. It is suited for courses whose titles include Fourier series,
orthogonal functions, or boundary value problems. It may also be used in courses
on Green's functions or transform methods.
Simple models (heat flow, vibrating strings and membranes) are emphasized.
Equations are formulated carefully from physical principles, motivating most
mathematical topics. Solution techniques are developed patiently. Mathematical
results frequently are given physical interpretations. Proofs of theorems (if given
at all) are presented after explanations based on illustrative examples. Numerous
exercises of varying difficulty form an essential part of this text. Answers are
provided for those exercises marked with a star (*).
Standard topics such as the method of separation of variables, Fourier
series, and orthogonal functions are developed with considerable detail. In addition,
there is a variety of clearly presented topics, such as differentiation and integration
of Fourier series, zeros of Sturm-Liouville eigenfunctions, Rayleigh quotient,
multidimensional eigenvalue problems, Bessel functions for a vibrating circular
membrane, eigenfunction expansions for nonhomogeneous problems, Green's
functions, Fourier and Laplace transform solutions, method of characteristics,
and numerical methods. Some optional advanced material of interest is also
included (for example, asymptotic expansion of large eigenvalues, calculation of
perturbed frequencies using the Fredholm alternative, and the dynamics of shock
waves).
The text has evolved from the author's experiences teaching this material
to different types of students at various institutions (M.I.T., U .C.S.D., Rutgers,