Beginning Partial Differential Equations

453 Pages·2014·36.212 MB·English

by  Peter V. O'Neil

Beginning Partial Differential Equations
PURE AND APPLIED MATHEMATICS
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Beginning Partial Differential Equations
Third Edition
Peter V. O'Neil
The University of Alabama
at Birmingham
WILEY
Copyright© 2014 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 I Peter V. O'Neil.-Third edition.
pages em
Includes bibliographical references and index.
ISBN 978-1-118-62994-9 (hardback)
I. Differential equations, Partial. I. Title.
QA377.054 2014
515'.353-dc23 2013034307
Printed in the United States of America.
10 9 8 7 6 5 4 3 2 I
Contents
1 First Ideas 1
1.1 Two Partial Differential Equations 1
1.1.1 The Heat, or Diffusion, Equation 1
1.1.2 The Wave Equation 4
1.2 Fourier Series 10
1.2.1 The Fourier Series of a Function 10
1.2.2 Fourier Sine and Cosine Series 20
1.3 Two Eigenvalue Problems 28
1.4 A Proof of the Fourier Convergence Theorem 30
1.4.1 The Role of Periodicity 30
1.4.2 Dirichlet's Formula 33
1.4.3 The Riemann-Lebesgue Lemma 35 1.4.4 Proof of the Convergence Theorem
37
2 Solutions of the Heat Equation 39
2.1 Solutions on an Interval [0, L] 39
2.1.1 Ends Kept at Temperature Zero 39
2.1.2 Insulated Ends 44
2.1.3 Ends at Different Temperatures 46
2.1.4 A Diffusion Equation with Additional Terms 50
2.1.5 One Radiating End 54
2.2 A Nonhomogeneous Problem 64
2.3 The Heat Equation in Two Space Variables 71
2.4 The Weak Maximum Principle 75
3 Solutions of the Wave Equation 81
3.1 Solutions on Bounded Intervals 81
3.1.1 Fixed Ends 81
3.1.2 Fixed Ends with a Forcing Term 89
3.1.3 Damped Wave Motion 100
3.2 The Cauchy Problem 109
3.2.1 d'Alembert's Solution 110
3.2.1.1 Forward and Backward Waves 113
3.2.2 The Cauchy Problem on a Half Line 120
3.2.3 Characteristic Triangles and Quadrilaterals 123
3.2.4 A Cauchy Problem with a Forcing Term 127
3.2.5 String with Moving Ends 131
3.3 The Wave Equation in Higher Dimensions 137
3.3.1 Vibrations in a Membrane with Fixed Frame 137
3.3.2 The Poisson Integral Solution 140
3.3.3 Hadamard's Method of Descent 144
v
vi CONTENTS
4 Dirichlet and Neumann Problems 147
4.1 Laplace's Equation and Harmonic Functions 147
4.1.1 Laplace's Equation in Polar Coordinates 148
4.1.2 Laplace's Equation in Three Dimensions 151
4.2 The Dirichlet Problem for a Rectangle 153
4.3 The Dirichlet Problem for a Disk 158
4.3.1 Poisson's Integral Solution 161
4.4 Properties of Harmonic Functions 165
4.4.1 Topology of Rn 165
4.4.2 Representation Theorems 172
4.4.2.1 A Representation Theorem in R3 172
4.4.2.2 A Representation Theorem in the Plane 177
4.4.3 The Mean Value Property and the Maximum Principle 178
4.5 The Neumann Problem 187
4.5.1 Existence and Uniqueness 187
4.5.2 Neumann Problem for a Rectangle 190
4.5.3 Neumann Problem for a Disk 194
4.6 Poisson's Equation 197
4. 7 Existence Theorem for a Dirichlet Problem 200
5 Fourier Integral Methods of Solution 213
5.1 The Fourier Integral of a Function 213
5.1.1 Fourier Cosine and Sine Integrals 216
5.2 The Heat Equation on the Real Line 220
5.2.1 A Reformulation of the Integral Solution 222
5.2.2 The Heat Equation on a Half Line 224
5.3 The Debate over the Age of the Earth 230
5.4 Burger's Equation 233
5.4.1 Traveling Wave Solutions of Burger's Equation 235
5.5 The Cauchy Problem for the Wave Equation 239
5.6 Laplace's Equation on Unbounded Domains 244
5.6.1 Dirichlet Problem for the Upper Half Plane 244
5.6.2 Dirichlet Problem for the Right Quarter Plane 246
5.6.3 A Neumann Problem for the Upper Half Plane 249
6 Solutions Using Eigenfunction Expansions 253
6.1 A Theory of Eigenfunction Expansions 253
6.1.1 A Closer Look at Expansion Coefficients 260
6.2 Bessel Functions 266
6.2.1 Variations on Bessel's Equation 269
6.2.2 Recurrence Relations 272
6.2.3 Zeros of Bessel Functions 273
6.2.4 Fourier-Bessel Expansions 274
6.3 Applications of Bessel Functions 279
6.3.1 Temperature Distribution in a Solid Cylinder 279
6.3.2 Vibrations of a Circular Drum 282
6.3.3 Oscillations of a Hanging Chain 285
CONTENTS vii
6.3.4 Did Poe Get His Pendulum Right? 287
6.4 Legendre Polynomials and Applications 288
6.4.1 A Generating Function 291
6.4.2 A Recurrence Relation 292
6.4.3 Fourier-Legendre Expansions 294
6.4.4 Zeros of Legendre Polynomials 297
6.4.5 Steady-State Temperature in a Solid Sphere 298
6.4.6 Spherical Harmonics 301
7 Integral Transform Methods of Solution 307
7.1 The Fourier Transform 307
7.1.1 Convolution 311
7.1.2 Fourier Sine and Cosine Transforms 313
7.2 Heat and Wave Equations 318
7.2.1 The Heat Equation on the Real Line 318
7.2.2 Solution by Convolution 320
7.2.3 The Heat Equation on a Half Line 324
7.2.4 The Wave Equation by Fourier Transform 328
7.3 The Telegraph Equation 332
7.4 The Laplace Transform 334
7.4.1 Temperature Distribution in a Semi-Infinite Bar 334
7.4.2 A Diffusion Problem in a Semi-Infinite Medium 336
7.4.3 Vibrations in an Elastic Bar 337
8 First-Order Equations 341
8.1 Linear First-Order Equations 343
8.2 The Significance of Characteristics 349
8.3 The Quasi-Linear Equation 354
9 End Materials 361
9.1 Notation 361
9.2 Use of MAPLE 363
9.2.1 Numerical Computations and Graphing 363
9.2.2 Ordinary Differential Equations 367
9.2.3 Integral Transforms 368
9.2.4 Special Functions 369
9.3 Answers to Selected Problems 370
Index 434