Beginning Partial Differential Equations, Second Edition

484 Pages·2008·6.502 MB·English

by  Peter V. O'Neil(auth.)

Beginning Partial Differential Equations
PURE AND APPLIED MATHEMATICS
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A complete list of the titles in this series appears at the end of this volume.
Beginning Partial Differential Equations
Second Edition
Peter V. O'Neil
The University of Alabama
at Birmingham
iWILEY-
INTERSCIENCE
A JOHN WILEY & SONS, INC., PUBLICATION
Copyright © 2008 by John Wiley & Sons, Inc. All rights reserved.
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Library of Congress Cataloging-in-Publication Data:
O'Neil, Peter V.
Beginning partial differential equations / Peter V. O'Neil. — 2nd ed.
p. cm.
Includes bibliographical references and index.
ISBN 978-0-470-13390-3 (cloth)
1. Differential equations, Partial. I. Title.
QA3 77.054 2008
515'.353—dc22 2007039327
Printed in the United States of America.
10 9 8 7 6 5 4
Contents
First-Order Equations 1
1.1 Notation and Terminology 1
1.2 The Linear First-Order Equation 4
1.3 The Significance of Characteristics 12
1.4 The Quasi-Linear Equation 16
Linear Second-Order Equations 23
2.1 Classification 23
2.2 The Hyperbolic Canonical Form 25
2.3 The Parabolic Canonical Form 30
2.4 The Elliptic Canonical Form 33
2.5 Some Equations of Mathematical Physics 38
2.6 The Second-Order Cauchy Problem 46
2.7 Characteristics and the Cauchy Problem 49
2.8 Characteristics as Carriers of Discontinuities 56
Elements of Fourier Analysis 59
3.1 Why Fourier Series? 59
3.2 The Fourier Series of a Function 60
3.3 Convergence of Fourier Series 63
3.4 Sine and Cosine Expansions 81
3.5 The Fourier Integral 89
3.6 The Fourier Transform 95
3.7 Convolution 101
3.8 Fourier Sine and Cosine Transforms 106
V
VI CONTENTS
The Wave Equation 109
4.1 d'Alembert Solution of the Cauchy Problem 109
4.2 d'Alembert's Solution as a Sum of Waves 117
4.3 The Characteristic Triangle 126
4.4 The Wave Equation on a Half-Line 131
4.5 A Half-Line with Moving End 134
4.6 A Nonhomogeneous Problem on the Real Line 137
4.7 A General Problem on a Closed Interval 141
4.8 Fourier Series Solutions on a Closed Interval 150
4.9 A Nonhomogeneous Problem on a Closed Interval 159
4.10 The Cauchy Problem by Fourier Integral 168
4.11 A Wave Equation in Two Space Dimensions 173
4.12 The Kirchhoff-Poisson Solution 177
4.13 Hadamard's Method of Descent 182
The Heat Equation 185
5.1 The Cauchy Problem and Initial Conditions 185
5.2 The Weak Maximum Principle 188
5.3 Solutions on Bounded Intervals 192
5.4 The Heat Equation on the Real Line 210
5.5 The Heat Equation on the Half-Line 218
5.6 The Debate Over the Age of the Earth 224
5.7 The Nonhomogeneous Heat Equation 227
5.8 The Heat Equation in Two Space Variables 234
Dirichlet and Neumann Problems 239
6.1 The Setting of the Problems 239
6.2 Some Harmonic Functions 247
6.3 Representation Theorems 251
6.4 Two Properties of Harmonic Functions 257
6.5 Is the Dirichlet Problem Well Posed? 263
6.6 Dirichlet Problem for a Rectangle 266
6.7 Dirichlet Problem for a Disk 269
6.8 Poisson's Integral Representation for a Disk 272
6.9 Dirichlet Problem for the Upper Half-Plane 276
6.10 Dirichlet Problem for the Right Quarter-Plane 279
6.11 Dirichlet Problem for a Rectangular Box 282
6.12 The Neumann Problem 285
6.13 Neumann Problem for a Rectangle 288
6.14 Neumann Problem for a Disk 290
6.15 Neumann Problem for the Upper Half-Plane 294
6.16 Green's Function for a Dirichlet Problem 296
6.17 Conformal Mapping Techniques 303
6.17.1 Conformal Mappings 303
6.17.2 Bilinear Transformations 308
6.17.3 Construction of Conformal Mappings between Domains 313
6.17.4 An Integral Solution of the Dirichlet Problem for a Disk 320
6.17.5 Solution of Dirichlet Problems by Conformal Mapping 323
vi i
Existence Theorems 327
7.1 A Classical Existence Theorem 327
7.2 A Hubert Space Approach 336
7.3 Distributions and an Existence Theorem 344
Additional Topics 351
8.1 Solutions by Eigenfunction Expansions 351
8.2 Numerical Approximations of Solutions 370
8.3 Burger's Equation 377
8.4 The Telegraph Equation 383
8.5 Poisson's Equation 390
End Materials 395
9.1 Historical Notes 395
9.2 Glossary 398
9.3 Answers to Selected Problems 399
Index 473
Preface
This book is a first course in partial differential equations. The first chapter
covers first-order equations, solution of linear and quasi-linear equations, and the
role of characteristics in the Cauchy problem. Chapter 2 is devoted to linear
second-order equations, classification, the second order Cauchy problem, and
the significance of characteristics in existence and uniqueness of solutions, and
as carriers of discontinuities. Chapter 3 is a review of Fourier series, integrals,
and transforms, and Chapters 4, 5 and 6 develop properties of solutions, and
techniques for finding solutions in particular cases, for the wave equation, the
heat equation, and Dirichlet and Neumann problems.
Chapters 7 and 8 are new to this edition and are independent of each other.
Chapter 7 begins with a classical proof of an existence theorem for the Dirichlet
problem. This existence question is then reformulated as a problem of repre-
senting a linear functional as an inner product in a Hilbert space, serving as
an introduction to the use of function spaces in the study of partial differential
equations. The chapter concludes with a brief introduction to distributions and
the formulation of another existence theorem.
Chapter 8 is a collection of independent additional topics, including the
solution of boundary value problems by eigenfunction expansions, numerical
methods, and explicit solutions of Burger's equation, the telegraph equation,
and Poisson's equation.
Particularly in working with solutions of wave and heat equations, it is often
instructive to use computational software to carry out numerical approxima-
tions, to gauge the effects of parameters on solutions, to construct graphs, and
to manipulate special functions such as Bessel functions. If such routines are
not available, parts of some exercises can be omitted.
ix
Beginning Partial Differential Equations, Second Edition
by Peter V. O'Neil
Copyright © 2008 John Wiley & Sons, Inc.
Chapter 1
First-Order Equations
1.1 Notation and Terminology
A partial differential equation is an equation that contains at least one partial
derivative. Examples are
du du o
- x— = xuy
ox ay
and
d2h d2h d2h
rl
We often use subscripts to denote partial derivatives. In this notation, u =
x
du/dx,u = d2u/dx2,u = d2u/dydx, and so on. The partial differential
xx xy
equations listed above can be written, respectively,
u — xu = xuy2
x y
and
h + h + h =f(x,y,z). (1.1)
xx yy zz
A solution of a partial differential equation is any function that satisfies the
equation. We will often seek solutions satisfying certain conditions and perhaps
having the independent variables confined to a specified set of values.
As an example of a solution, the equation
4u+3u + u = 0 (1.2)
x y
has the solution
u(x,y) = e-^4f(3x-4y),
in which / can be any differentiable function of a single variable. This can be
verified by substituting u(x, y) into the partial differential equation. Chain rule
1
2 CHAPTER 1. FIRST-ORDER EQUATIONS
differentiations yield
= - ^ - ^ / ( fe - 4y) + 3e-*/4/'(3* - 4j/)
and similarly,
u = -4e-x/4f'(3x - Ay).
y
Upon substitution into equation 1.2, we obtain
Au + 3u + u = -e~x/if(3x - Ay)
x y
+ \2e~x'if'{3x - Ay) - 12e~x/4f'{3x - Ay)
+ e-x/4f{3x-Ay) = 0.
Because of the freedom to choose /, equation 1.2 has infinitely many solutions.
The order of a partial differential equation is the order of the highest partial
derivative occurring in the equation. Equation 1.2 is of order one and equa-
tion 1.1 is of order two.
A partial differential equation is linear if it is linear in the unknown function
and its partial derivatives. An equation that is not linear is nonlinear. For
example,
2 _
X %i y^xy — ^
is linear, whereas
2 2
X Ίίχχ yi^xy ^^ ^
is nonlinear because of the u2 term, and
(u x)1/2 - 4uj, = xu
X y
is nonlinear because of the (u )ll2 term.
xx
A partial differential equation is quasi-linear if it is linear in its highest-order
derivative term(s). The second-order equation
u + 4yu - (u)3 + uu = cos(u).
xx yy x x y
is quasi-linear because it is linear in its second derivative (highest-order) terms
u and Uyy. This equation is not linear because of the cos{u),u u, and (u )3
xx x y x
terms. Any linear equation is also quasi-linear.
We now have the vocabulary to begin studying partial differential equations,
starting with first order.
Problems for Section 1.1
1. Show that
u{x,y,z) =
y/x2 + y2 + z2
is a solution of u + u + u = 0 for (x, y, z) φ (0,0,0).
xx yy zz