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<ul><li><p>PARTIALDIFFERENTIALEQUATIONSAN INTRODUCTION</p></li><li><p>PARTIALDIFFERENTIALEQUATIONSAN INTRODUCTION</p><p>WALTER A. STRAUSSBrown University</p></li><li><p>Publisher Laurie RosatoneAssociate Editor Shannon CorlissSenior Editorial Assistant Jeffrey BensonMarketing Manager Jaclyn ElkinsProduction Manager Dorothy SinclairSenior Production Editor Sandra DumasCreative Director Harry NolanSenior Designer Madelyn LesureProduction Management Services Aptara, Inc.Bicentennial Logo Design Richard J. Pacifico</p><p>This book was set in 11/12 Times Ten by Aptara, Inc. and printed and bound by Courier(Westford).The cover was printed by Courier(Stoughton).</p><p>This book is printed on acid free paper. </p><p>Copyright C 2008 John Wiley & Sons, Inc. All rights reserved. No part of this publication may bereproduced, 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 Sections107 or 108 of the 1976 United States Copyright Act, without either the prior written permission ofthe Publisher, or authorization through payment of the appropriate per-copy fee to the CopyrightClearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, website www.copyright.com.Requests to the Publisher for permission should be addressed to the Permissions Department,John Wiley & Sons, Inc., 111 River Street, Hoboken, NJ 07030-5774, (201)748-6011,fax (201)748-6008, website http://www.wiley.com/go/permissions.</p><p>To order books or for customer service, please call 1-800-CALL WILEY (225-5945).</p><p>ISBN-13 978-0470-05456-7</p><p>Printed in the United States of America</p><p>10 9 8 7 6 5 4 3 2 1</p></li><li><p>PREFACE</p><p>Our understanding of the fundamental processes of the natural world is basedto a large extent on partial differential equations. Examples are the vibrationsof solids, the flow of fluids, the diffusion of chemicals, the spread of heat,the structure of molecules, the interactions of photons and electrons, and theradiation of electromagnetic waves. Partial differential equations also play acentral role in modern mathematics, especially in geometry and analysis. Theavailability of powerful computers is gradually shifting the emphasis in partialdifferential equations away from the analytical computation of solutions andtoward both their numerical analysis and the qualitative theory.</p><p>This book provides an introduction to the basic properties of partial dif-ferential equations (PDEs) and to the techniques that have proved useful inanalyzing them. My purpose is to provide for the student a broad perspectiveon the subject, to illustrate the rich variety of phenomena encompassed byit, and to impart a working knowledge of the most important techniques ofanalysis of the solutions of the equations.</p><p>One of the most important techniques is the method of separation ofvariables. Many textbooks heavily emphasize this technique to the point ofexcluding other points of view. The problem with that approach is that onlycertain kinds of partial differential equations can be solved by it, whereasothers cannot. In this book it plays a very important but not an overridingrole. Other texts, which bring in relatively advanced theoretical ideas, requiretoo much mathematical knowledge for the typical undergraduate student. Ihave tried to minimize the advanced concepts and the mathematical jargonin this book. However, because partial differential equations is a subject atthe forefront of research in modern science, I have not hesitated to mentionadvanced ideas as further topics for the ambitious student to pursue.</p><p>This is an undergraduate textbook. It is designed for juniors and seniorswho are science, engineering, or mathematics majors. Graduate students, es-pecially in the sciences, could surely learn from it, but it is in no way conceivedof as a graduate text.</p><p>The main prerequisite is a solid knowledge of calculus, especially mul-tivariate. The other prerequisites are small amounts of ordinary differential</p><p>v</p></li><li><p>vi PREFACE</p><p>equations and of linear algebra, each much less than a semesters worth. How-ever, since the subject of partial differential equations is by its very nature notan easy one, I have recommended to my own students that they should alreadyhave taken full courses in these two subjects.</p><p>The presentation is based on the following principles. Motivate withphysics but then do mathematics. Focus on the three classical equations:All the important ideas can be understood in terms of them. Do one spa-tial dimension before going on to two and three dimensions with their morecomplicated geometries. Do problems without boundaries before bringing inboundary conditions. (By the end of Chapter 2, the student will already havean intuitive and analytical understanding of simple wave and diffusion phe-nomena.) Do not hesitate to present some facts without proofs, but provide themost critical proofs. Provide introductions to a variety of important advancedtopics.</p><p>There is plenty of material in this book for a year-long course. A quartercourse, or a fairly relaxed semester course, would cover the starred sectionsof Chapters 1 to 6. A more ambitious semester course could supplement thebasic starred sections in various ways. The unstarred sections in Chapters 1to 6 could be covered as desired. A computational emphasis following thestarred sections would be provided by the numerical analysis of Chapter 8. Toresume separation of variables after Chapter 6, one would take up Chapter 10.For physics majors one could do some combination of Chapters 9, 12, 13, and14. A traditional course on boundary value problems would cover Chapters1, 4, 5, 6, and 10.</p><p>Each chapter is divided into sections, denoted A.B. An equation num-bered (A.B.C) refers to equation (C) in section A.B. A reference to equation(C) refers to the equation in the same section. A similar system is used fornumbering theorems and exercises. The references are indicated by brackets,like [AS].</p><p>The help of my colleagues is gratefully acknowledged. I especially thankYue Liu and Brian Loe for their extensive help with the exercises, as well asCostas Dafermos, Bob Glassey, Jerry Goldstein, Manos Grillakis, Yan Guo,Chris Jones, Keith Lewis, Gustavo Perla Menzala, and Bob Seeley for theirsuggestions and corrections.</p><p>Walter A. Strauss</p></li><li><p>PREFACE TOSECOND EDITION</p><p>In the years since the first edition came out, partial differential equations hasbecome yet more prominent, both as a model for scientific theories and withinmathematics itself. In this second edition I have added 30 new exercises. Fur-thermore, this edition is accompanied by a solutions manual that has answersto about half of the exercises worked out in detail. I have added a new sectionon water waves as well as new material and explanatory comments in manyplaces. Corrections have been made wherever necessary.</p><p>I would like to take this opportunity to thank all the people who havepointed out errors in the first edition or made useful suggestions, includ-ing Andrew Bernoff, Rustum Choksi, Adrian Constantin, Leonid Dickey,Julio Dix, Craig Evans, A. M. Fink, Robert Glassey, Jerome Goldstein, LeonGreenberg, Chris Hunter, Eva Kallin, Jim Kelliher, Jeng-Eng Lin, HowardLiu, Jeff Nunemacher, Vassilis Papanicolaou, Mary Pugh, Stan Richardson,Stuart Rogers, Paul Sacks, Naoki Saito, Stephen Simons, Catherine Sulem,David Wagner, David Weinberg, and Nick Zakrasek. My warmest thanks goto Julie and Steve Levandosky who, besides being my co-authors on the so-lutions manual, provided many suggestions and much insight regarding thetext itself.</p><p>vii</p></li><li><p>CONTENTS</p><p>(The starred sections form the basic part of the book.)</p><p>Chapter 1/Where PDEs Come From1.1* What is a Partial Differential Equation? 11.2* First-Order Linear Equations 61.3* Flows, Vibrations, and Diffusions 101.4* Initial and Boundary Conditions 201.5 Well-Posed Problems 251.6 Types of Second-Order Equations 28</p><p>Chapter 2/Waves and Diffusions2.1* The Wave Equation 332.2* Causality and Energy 392.3* The Diffusion Equation 422.4* Diffusion on the Whole Line 462.5* Comparison of Waves and Diffusions 54</p><p>Chapter 3/Reflections and Sources3.1 Diffusion on the Half-Line 573.2 Reflections of Waves 613.3 Diffusion with a Source 673.4 Waves with a Source 713.5 Diffusion Revisited 80</p><p>Chapter 4/Boundary Problems4.1* Separation of Variables, The Dirichlet Condition 844.2* The Neumann Condition 894.3* The Robin Condition 92</p><p>viii</p></li><li><p>CONTENTS ix</p><p>Chapter 5/Fourier Series5.1* The Coefficients 1045.2* Even, Odd, Periodic, and Complex Functions 1135.3* Orthogonality and General Fourier Series 1185.4* Completeness 1245.5 Completeness and the Gibbs Phenomenon 1365.6 Inhomogeneous Boundary Conditions 147</p><p>Chapter 6/Harmonic Functions6.1* Laplaces Equation 1526.2* Rectangles and Cubes 1616.3* Poissons Formula 1656.4 Circles, Wedges, and Annuli 172</p><p>(The next four chapters may be studied in any order.)</p><p>Chapter 7/Greens Identities and Greens Functions7.1 Greens First Identity 1787.2 Greens Second Identity 1857.3 Greens Functions 1887.4 Half-Space and Sphere 191</p><p>Chapter 8/Computation of Solutions8.1 Opportunities and Dangers 1998.2 Approximations of Diffusions 2038.3 Approximations of Waves 2118.4 Approximations of Laplaces Equation 2188.5 Finite Element Method 222</p><p>Chapter 9/Waves in Space9.1 Energy and Causality 2289.2 The Wave Equation in Space-Time 2349.3 Rays, Singularities, and Sources 2429.4 The Diffusion and Schrodinger Equations 2489.5 The Hydrogen Atom 254</p><p>Chapter 10/Boundaries in the Plane and in Space10.1 Fouriers Method, Revisited 25810.2 Vibrations of a Drumhead 26410.3 Solid Vibrations in a Ball 27010.4 Nodes 27810.5 Bessel Functions 282</p></li><li><p>x CONTENTS</p><p>10.6 Legendre Functions 28910.7 Angular Momentum in QuantumMechanics 294</p><p>Chapter 11/General Eigenvalue Problems11.1 The Eigenvalues Are Minima of the Potential Energy 29911.2 Computation of Eigenvalues 30411.3 Completeness 31011.4 Symmetric Differential Operators 31411.5 Completeness and Separation of Variables 31811.6 Asymptotics of the Eigenvalues 322</p><p>Chapter 12/Distributions and Transforms12.1 Distributions 33112.2 Greens Functions, Revisited 33812.3 Fourier Transforms 34312.4 Source Functions 34912.5 Laplace Transform Techniques 353</p><p>Chapter 13/PDE Problems from Physics13.1 Electromagnetism 35813.2 Fluids and Acoustics 36113.3 Scattering 36613.4 Continuous Spectrum 37013.5 Equations of Elementary Particles 373</p><p>Chapter 14/Nonlinear PDEs14.1 Shock Waves 38014.2 Solitons 39014.3 Calculus of Variations 39714.4 Bifurcation Theory 40114.5 Water Waves 406</p><p>AppendixA.1 Continuous and Differentiable Functions 414A.2 Infinite Series of Functions 418A.3 Differentiation and Integration 420A.4 Differential Equations 423A.5 The Gamma Function 425</p><p>References 427</p><p>Answers and Hints to Selected Exercises 431</p><p>Index 446</p></li><li><p>1WHERE PDEsCOME FROM</p><p>After thinking about the meaning of a partial differential equation, we willflex our mathematical muscles by solving a few of them. Then we will seehow naturally they arise in the physical sciences. The physics will motivatethe formulation of boundary conditions and initial conditions.</p><p>1.1 WHAT IS A PARTIAL DIFFERENTIAL EQUATION?</p><p>The key defining property of a partial differential equation (PDE) is that thereis more than one independent variable x, y, . . . . There is a dependent variablethat is an unknown function of these variables u(x, y, . . . ). We will oftendenote its derivatives by subscripts; thus u/x = ux , and so on. A PDE is anidentity that relates the independent variables, the dependent variable u, andthe partial derivatives of u. It can be written as</p><p>F(x, y, u(x, y), ux (x, y), uy(x, y)) = F(x, y, u, ux , uy) = 0. (1)This is the most general PDE in two independent variables of first order. Theorder of an equation is the highest derivative that appears. The most generalsecond-order PDE in two independent variables is</p><p>F(x, y, u, ux , uy, uxx , uxy, uyy) = 0. (2)A solution of a PDE is a function u(x, y, . . . ) that satisfies the equation</p><p>identically, at least in some region of the x, y, . . . variables.When solving an ordinary differential equation (ODE), one sometimes</p><p>reverses the roles of the independent and the dependent variablesfor in-stance, for the separable ODE</p><p>dudx</p><p>= u3. For PDEs, the distinction betweenthe independent variables and the dependent variable (the unknown) is alwaysmaintained.</p><p>1</p></li><li><p>2 CHAPTER 1 WHERE PDEs COME FROM</p><p>Some examples of PDEs (all of which occur in physical theory) are:</p><p>1. ux + uy = 0 (transport)2. ux + yuy = 0 (transport)3. ux + uuy = 0 (shock wave)4. uxx + uyy = 0 (Laplaces equation)5. utt uxx + u3 = 0 (wave with interaction)6. ut + uux + uxxx = 0 (dispersive wave)7. utt + uxxxx = 0 (vibrating bar)8. ut iuxx = 0 (i =</p><p>1) (quantum mechanics)Each of these has two independent variables, written either as x and y or</p><p>as x and t. Examples 1 to 3 have order one; 4, 5, and 8 have order two; 6 hasorder three; and 7 has order four. Examples 3, 5, and 6 are distinguished fromthe others in that they are not linear. We shall now explain this concept.</p><p>Linearity means the following. Write the equation in the form lu = 0,wherel is an operator. That is, if v is any function,lv is a new function. Forinstance, l = /x is the operator that takes v into its partial derivative vx .In Example 2, the operator l is l = /x + y/y. (lu = ux + yuy.) Thedefinition we want for linearity is</p><p>l(u + v) = lu + lv l(cu) = clu (3)for any functions u, v and any constant c. Whenever (3) holds (for all choicesof u, v, and c), l is called linear operator. The equation</p><p>lu = 0 (4)is called linear if l is a linear operator. Equation (4) is called a homogeneouslinear equation. The equation</p><p>lu = g, (5)where g = 0 is a given function of the independent variables, is called aninhomogeneous linear equation. For instance, the equation</p><p>(cos xy2)ux y2uy = tan(x2 + y2) (6)is an inhomogeneous linear equation.</p><p>As you can easily verify, five of the eight equations above are linearas well as homogeneous. Example 5, on the other hand, is not linear becausealthough (u + v)xx = uxx + vxx and (u + v)t t = utt + vt t satisfy property (3),the cubic term does not:</p><p>(u + v)3 = u3 + 3u2v + 3uv2 + v3 = u3 + v3.</p><p>Libin Daniel</p></li><li><p>1.1 WHAT IS A PARTIAL DIFFERENTIAL EQUATION? 3</p><p>The advantage of linearity for the equation lu = 0 is that if u and v areboth solutions, so is (u + v). If u1, . . . , un are all solutions, so is any linearcombination</p><p>c1u1(x) + + cnun(x) =n</p><p>j=1c j uj (x) (cj = constants).</p><p>(This is sometimes called the superposition principle.) Another consequenceof linearity is that if you add a homogeneous solution [a solution of (4)] to aninhomogeneous solution [a solution of (5)], you get an inhomogeneous solu-tion. (Why?) The mathematical structure that deals with linear combinationsand linear operators is the vector space. Exercises 510 are review problemson vector spaces.</p><p>Well study, almost exclusively, linear systems with constant coefficients.Recall that for ODEs you get linear combinations. The coefficients are thearbitrary constants. For an ODE of order m, you get m arbitrary consta...</p></li></ul>

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