Beginning Partial Differential Equations PURE AND APPLIED MATHEMATICS A Wiley Series of Texts, Monographs, and Tracts Founded by RICHARD COURANT Editors Emeriti: MYRON B ALLEN III, PETER HILTON, HARRY HOCHSTADT, ERWIN KREYSZIG, PETER LAX, JOHN TOLAND A complete list of the titles in this series appears at the end of this volume 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 Published by John Wiley & Sons, Inc., Hoboken, New Jersey All rights reserved Published simultaneously in Canada No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning, or otherwise, except as permitted under Section 107 or 108 of the 1976 United States Copyright Act, without either the prior written permission of the Publisher, or authorization through payment of 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Contents First Ideas 1.1 Two Partial Differential Equations 1.1.1 The Heat, or Diffusion, Equation 1.1.2 The Wave Equation 1.2 Fourier Series 1.2.1 The Fourier Series of a Function 1.2.2 Fourier Sine and Cosine Series 1.3 Two Eigenvalue Problems 1.4 A Proof of the Fourier Convergence Theorem 1.4.1 The Role of Periodicity 1.4.2 Dirichlet's Formula 1.4.3 The Riemann-Lebesgue Lemma Solutions of the Heat Equation 2.1 Solutions on an Interval [0, L] 2.1.1 Ends Kept at Temperature Zero 2.1.2 Insulated Ends 2.1.3 Ends at Different Temperatures 2.1.4 A Diffusion Equation with Additional Terms 2.1.5 One Radiating End 2.2 A Nonhomogeneous Problem 2.3 The Heat Equation in Two Space Variables 2.4 The Weak Maximum Principle Solutions of the Wave Equation 3.1 Solutions on Bounded Intervals 3.1.1 Fixed Ends 3.1.2 Fixed Ends with a Forcing Term 3.1.3 Damped Wave Motion 3.2 The Cauchy Problem 3.2.1 d'Alembert's Solution 3.2.1.1 Forward and Backward Waves 3.2.2 The Cauchy Problem on a Half Line 3.2.3 Characteristic Triangles and Quadrilaterals 3.2.4 A Cauchy Problem with a Forcing Term 3.2.5 String with Moving Ends 3.3 The Wave Equation in Higher Dimensions 3.3.1 Vibrations in a Membrane with Fixed Frame 3.3.2 The Poisson Integral Solution 3.3.3 Hadamard's Method of Descent v 1 10 10 20 28 30 30 33 35 1.4.4 Proof of 37 39 39 39 44 46 50 54 64 71 75 81 81 81 89 100 109 110 113 120 123 127 131 137 137 140 144 vi CONTENTS Dirichlet and Neumann Problems 4.1 Laplace's Equation and Harmonic Functions 4.1.1 Laplace's Equation in Polar Coordinates 4.1.2 Laplace's Equation in Three Dimensions 4.2 The Dirichlet Problem for a Rectangle 4.3 The Dirichlet Problem for a Disk 4.3.1 Poisson's Integral Solution 4.4 Properties of Harmonic Functions 4.4.1 Topology of Rn 4.4.2 Representation Theorems 4.4.2.1 A Representation Theorem in R 4.4.2.2 A Representation Theorem in the Plane 4.4.3 The Mean Value Property and the Maximum Principle 4.5 The Neumann Problem 4.5.1 Existence and Uniqueness 4.5.2 Neumann Problem for a Rectangle 4.5.3 Neumann Problem for a Disk 4.6 Poisson's Equation Existence Theorem for a Dirichlet Problem Fourier Integral Methods of Solution 5.1 The Fourier Integral of a Function 5.1.1 Fourier Cosine and Sine Integrals 5.2 The Heat Equation on the Real Line 5.2.1 A Reformulation of the Integral Solution 5.2.2 The Heat Equation on a Half Line 5.3 The Debate over the Age of the Earth 5.4 Burger's Equation 5.4.1 Traveling Wave Solutions of Burger's Equation 5.5 The Cauchy Problem for the Wave Equation 5.6 Laplace's Equation on Unbounded Domains 5.6.1 Dirichlet Problem for the Upper Half Plane 5.6.2 Dirichlet Problem for the Right Quarter Plane 5.6.3 A Neumann Problem for the Upper Half Plane Solutions Using Eigenfunction Expansions 6.1 A Theory of Eigenfunction Expansions 6.1.1 A Closer Look at Expansion Coefficients 6.2 Bessel Functions 6.2.1 Variations on Bessel's Equation 6.2.2 Recurrence Relations 6.2.3 Zeros of Bessel Functions 6.2.4 Fourier-Bessel Expansions 6.3 Applications of Bessel Functions 6.3.1 Temperature Distribution in a Solid Cylinder 6.3.2 Vibrations of a Circular Drum 6.3.3 Oscillations of a Hanging Chain 147 147 148 151 153 158 161 165 165 172 172 177 178 187 187 190 194 197 200 213 213 216 220 222 224 230 233 235 239 244 244 246 249 253 253 260 266 269 272 273 274 279 279 282 285 CONTENTS 6.3.4 Did Poe Get His Pendulum Right? 6.4 Legendre Polynomials and Applications 6.4.1 A Generating Function 6.4.2 A Recurrence Relation 6.4.3 Fourier-Legendre Expansions 6.4.4 Zeros of Legendre Polynomials 6.4.5 Steady-State Temperature in a Solid Sphere 6.4.6 Spherical Harmonics Integral Transform Methods of Solution 7.1 The Fourier Transform 7.1.1 Convolution 7.1.2 Fourier Sine and Cosine Transforms 7.2 Heat and Wave Equations 7.2.1 The Heat Equation on the Real Line 7.2.2 Solution by Convolution 7.2.3 The Heat Equation on a Half Line 7.2.4 The Wave Equation by Fourier Transform 7.3 The Telegraph Equation 7.4 The Laplace Transform 7.4.1 Temperature Distribution in a Semi-Infinite Bar 7.4.2 A Diffusion Problem in a Semi-Infinite Medium 7.4.3 Vibrations in an Elastic Bar First-Order Equations 8.1 Linear First-Order Equations 8.2 The Significance of Characteristics 8.3 The Quasi-Linear Equation End Materials 9.1 Notation 9.2 Use of MAPLE 9.2.1 Numerical Computations and Graphing 9.2.2 Ordinary Differential Equations 9.2.3 Integral Transforms 9.2.4 Special Functions 9.3 Answers to Selected Problems Index vii 287 288 291 292 294 297 298 301 307 307 311 313 318 318 320 324 328 332 334 334 336 337 341 343 349 354 361 361 363 363 367 368 369 370 434 9.3 ANSWERS TO SELECTED PROBLEMS 427 11 10 ' 37! 0.5 p $ Figure 9.59: u(p, t.p) in problem 15 17 L 00 u(p, t.p) = Cn (~) n Pn(cos(t.p)) , n=O where Cn 2n+ = ! -1 (cos(2 arccos(O))Pn(~) d~ Figure 9.60 is a graph of the surface u(p, t.p) 19 =L 00 u(p, t.p) ! where Cn Cn (~f Pn(cos(t.p)), n=O _1 (arccos(~)+ (arccos (~)))Pn(~) d~ = -2n+ 21 Figure 9.61 is a graph of u(p, t.p) Chapter 7: Integral 'fransform Methods of Solution 7.1 The Fourier Transform ~ f(w) = w2 _ (-sin(a)cos(aw) +wsin(aw)cos(a)) ~ -4iw f(w) = (w2 + 1)2 CHAPTER END MATERIALS 428 0.5 -{).5 -1 Figure 9.60: u(p,