Partial Differential Equations: An Introduction, 2nd Edition
PARTIALDIFFERENTIALEQUATIONSAN INTRODUCTION
PARTIALDIFFERENTIALEQUATIONSAN INTRODUCTIONWALTER A. STRAUSSBrown University
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PREFACEOur 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.This book provides an introduction to the basic properties of partial differential 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.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.This is an undergraduate textbook. It is designed for juniors and seniorswho are science, engineering, or mathematics majors. Graduate students, especially in the sciences, could surely learn from it, but it is in no way conceivedof as a graduate text.The main prerequisite is a solid knowledge of calculus, especially multivariate. The other prerequisites are small amounts of ordinary differentialv
viPREFACEequations and of linear algebra, each much less than a semester’s worth. However, 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.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 spatial 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 phenomena.) Do not hesitate to present some facts without proofs, but provide themost critical proofs. Provide introductions to a variety of important advancedtopics.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.Each chapter is divided into sections, denoted A.B. An equation numbered (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].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.Walter A. Strauss
PREFACE TOSECOND EDITIONIn 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. Furthermore, 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.I would like to take this opportunity to thank all the people who havepointed out errors in the first edition or made useful suggestions, including 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 solutions manual, provided many suggestions and much insight regarding thetext itself.vii
CONTENTS(The starred sections form the basic part of the book.)Chapter 1 /Where PDEs Come From1.1*1.2*1.3*1.4*1.51.6What is a Partial Differential Equation?First-Order Linear EquationsFlows, Vibrations, and DiffusionsInitial and Boundary ConditionsWell-Posed ProblemsTypes of Second-Order Equations1610202528Chapter 2 /Waves and Diffusions2.1*2.2*2.3*2.4*2.5*The Wave EquationCausality and EnergyThe Diffusion EquationDiffusion on the Whole LineComparison of Waves and Diffusions3339424654Chapter 3 /Reflections and Sources3.13.23.33.43.5Diffusion on the Half-LineReflections of WavesDiffusion with a SourceWaves with a SourceDiffusion Revisited5761677180Chapter 4 /Boundary Problems4.1* Separation of Variables, The Dirichlet Condition4.2* The Neumann Condition4.3* The Robin Conditionviii848992
CONTENTSixChapter 5 /Fourier Series5.1*5.2*5.3*5.4*5.55.6The CoefficientsEven, Odd, Periodic, and Complex FunctionsOrthogonality and General Fourier SeriesCompletenessCompleteness and the Gibbs PhenomenonInhomogeneous Boundary Conditions104113118124136147Chapter 6 /Harmonic Functions6.1*6.2*6.3*6.4Laplace’s EquationRectangles and CubesPoisson’s FormulaCircles, Wedges, and Annuli152161165172(The next four chapters may be studied in any order.)Chapter 7 /Green’s Identities and Green’s Functions7.17.27.37.4Green’s First IdentityGreen’s Second IdentityGreen’s FunctionsHalf-Space and Sphere178185188191Chapter 8 /Computation of Solutions8.18.28.38.48.5Opportunities and DangersApproximations of DiffusionsApproximations of WavesApproximations of Laplace’s EquationFinite Element Method199203211218222Chapter 9 /Waves in Space9.19.29.39.49.5Energy and CausalityThe Wave Equation in Space-TimeRays, Singularities, and SourcesThe Diffusion and Schrödinger EquationsThe Hydrogen Atom228234242248254Chapter 10 /Boundaries in the Plane and in Space10.110.210.310.410.5Fourier’s Method, RevisitedVibrations of a DrumheadSolid Vibrations in a BallNodesBessel Functions258264270278282
xCONTENTS10.6 Legendre Functions10.7 Angular Momentum in Quantum Mechanics289294Chapter 11 /General Eigenvalue Problems11.111.211.311.411.511.6The Eigenvalues Are Minima of the Potential EnergyComputation of EigenvaluesCompletenessSymmetric Differential OperatorsCompleteness and Separation of VariablesAsymptotics of the Eigenvalues299304310314318322Chapter 12 /Distributions and �s Functions, RevisitedFourier TransformsSource FunctionsLaplace Transform Techniques331338343349353Chapter 13 /PDE Problems from Physics13.113.213.313.413.5ElectromagnetismFluids and AcousticsScatteringContinuous SpectrumEquations of Elementary Particles358361366370373Chapter 14 /Nonlinear PDEs14.114.214.314.414.5Shock WavesSolitonsCalculus of VariationsBifurcation TheoryWater ous and Differentiable FunctionsInfinite Series of FunctionsDifferentiation and IntegrationDifferential EquationsThe Gamma Function414418420423425References427Answers and Hints to Selected Exercises431Index446
1WHERE PDEsCOME FROMAfter 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.1.1WHAT IS A PARTIAL DIFFERENTIAL EQUATION?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 u x , 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 asF(x, y, u(x, y), u x (x, y), u y (x, y)) F(x, y, u, u x , u y ) 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 isF(x, y, u, u x , u y , u x x , u x y , u yy ) 0.(2)A solution of a PDE is a function u(x, y, . . . ) that satisfies the equationidentically, at least in some region of the x, y, . . . variables.When solving an ordinary differential equation (ODE), one sometimesreverses the roles of the independent and the dependent variables—for indu u 3 . For PDEs, the distinction betweenstance, for the separable ODEdxthe independent variables and the dependent variable (the unknown) is alwaysmaintained.1
2CHAPTER 1WHERE PDEs COME FROMSome examples of PDEs (all of which occur in physical theory) are:1.2.3.4.5.6.7.8.u x u y 0 (transport)u x yu y 0 (transport)u x uu y 0 (shock wave)u x x u yy 0 (Laplace’s equation)u tt u x x u 3 0 (wave with interaction)u t uu x u x x x 0 (dispersive wave)u tt u x x x x 0 (vibrating bar) u t iux x 0 (i 1) (quantum mechanics)Each of these has two independent variables, written either as x and y oras 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.Linearity means the following. Write the equation in the form lu 0,where l 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 u x yu y .) Thedefinition we want for linearity isl(u v) lu lvl(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 equationlu 0(4)is called linear if l is a linear operator. Equation (4) is called a homogeneouslinear equation. The equationlu g,(5)where g 0 is a given function of the independent variables, is called aninhomogeneous linear equation. For instance, the equation(cos x y 2 )u x y 2 u y tan(x 2 y 2 )(6)is an inhomogeneous linear equation.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)x x u x x vx x and (u v)tt u tt vtt satisfy property (3),the cubic term does not:(u v)3 u 3 3u 2 v 3uv 2 v 3 u 3 v 3 .
1.1WHAT IS A PARTIAL DIFFERENTIAL EQUATION?3The 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 linearcombinationn c1 u 1 (x) · · · cn u n (x) c j u j (x)(cj constants).j 1(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 solution. (Why?) The mathematical structure that deals with linear combinationsand linear operators is the vector space. Exercises 5–10 are review problemson vector spaces.We’ll 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 constants.Let’s look at some PDEs.Example 1.Find all u(x, y) satisfying the equation uxx 0. Well, we can integrateonce to get ux constant. But that’s not really right since there’s anothervariable y. What we really get is ux (x, y) f (y), where f (y) is arbitrary.Do it again to get u(x, y) f (y)x g(y). This is the solution formula.Note that there are two arbitrary functions in the solution. We see thisas well in the next two examples. Example 2.Solve the PDE u x x u 0. Again, it’s really an ODE with an extravariable y. We know how to solve the ODE, so the solution isu f (y) cos x g(y) sin x,where again f (y) and g(y) are two arbitrary functions of y. You can easilycheck this formula by differentiating twice to verify that u x x u. Example 3.Solve the PDE uxy 0. This isn’t too hard either. First let’s integrate inx, regarding y as fixed. So we getu y (x, y) f (y).Next let’s integrate in y regarding x as fixed. We get the solutionu(x, y) F(y) G(x),where F f.
4CHAPTER 1WHERE PDEs COME FROMMoral A PDE has arbitrary functions in its solution. In these examples thearbitrary functions are functions of one variable that combine to produce afunction u(x, y) of two variables which is only partly arbitrary.A function of two variables contains immensely more information thana function of only one variable. Geometrically, it is obvious that a surface{u f (x, y)}, the graph of a function of two variables, is a much more complicated object than a curve {u f (x)}, the graph of a function of one variable.To illustrate this, we can ask how a computer would record a functionu f (x). Suppose that we choose 100 points to describe it using equally spacedvalues of x: x1 , x2 , x3 , . . . , x100 . We could write them down in a column, andnext to each xj we could write the corresponding value u j f (x j ). Now howabout a function u f (x, y)? Suppose that we choose 100 equally spacedvalues of x and also of y: x1 , x2 , x3 , . . . , x100 and y1 , y2 , y3 , . . . , y100 . Eachpair xi , y j provides a value u ij f (xi , y j ), so there will be 1002 10,000lines of the formxiyju ijrequired to describe the function! (If we had a prearranged system, we wouldneed to record only the values uij .) A function of three variables describeddiscretely by 100 values in each variable would require a million numbers!To understand this book what do you have to know from calculus? Certainly all the basic facts about partial derivatives and multiple integrals. Fora brief discussion of such topics, see the Appendix. Here are a few things tokeep in mind, some of which may be new to you.1. Derivatives are local. For instance, to calculate the derivative( u/ x)(x0 , t0 ) at a particular point, you need to know just the valuesof u(x, t0 ) for x near x0 , since the derivative is the limit as x x0 .2. Mixed derivatives are equal: u x y u yx . (We assume throughout this book,unless stated otherwise, that all derivatives exist and are continuous.)3. The chain rule is used frequently in PDEs; for instance, g [ f (g(x, t))] f (g(x, t)) ·(x, t). x x4. For the integrals of derivatives, the reader should learn or review Green’stheorem and the divergence theorem. (See the end of Section A.3 in theAppendix.) b(t)5. Derivatives of integrals like I (t) a(t) f (x, t) d x (see Section A.3).6.7.8.9.Jacobians (change of variable in a double integral) (see Section A.1).Infinite series of functions and their differentiation (see Section A.2).Directional derivatives (see Section A.1).We’ll often reduce PDEs to ODEs, so we must know how to solve simpleODEs. But we won’t need to know anything about tricky ODEs.
1.1WHAT IS A PARTIAL DIFFERENTIAL EQUATION?5EXERCISES1. Verify the linearity and nonlinearity of the eight examples of PDEs givenin the text, by checking whether or not equations (3) are valid.2. Which of the following operators are linear?(a) lu u x xu y(b) lu u x uu y(c) lu u x u 2y(d) lu u x u y 1(e) lu 1 x 2 (cos y)u x u yx y [arctan(x/y)]u3. For each of the following equations, state the order and whether itis nonlinear, linear inhomogeneous, or linear homogeneous; providereasons.(a) u t u x x 1 0(b) u t u x x xu 0(c) u t u x xt uu x 0(d) u tt u x x x 2 0(e) iu t u x x u/x 0 1/2 1/2(f) u x (1 u 2x ) u y (1 u 2y ) 0y(g) u x e u y 0 (h) u t u x x x x 1 u 04. Show that the difference of two solutions of an inhomogeneous linearequation lu g with the same g is a solution of the homogeneousequation lu 0.5. Which of the following collections of 3-vectors [a, b, c] are vectorspaces? Provide reasons.(a) The vectors with b 0.(b) The vectors with b 1.(c) The vectors with ab 0.(d) All the linear combinations of the two vectors [1, 1, 0] and [2, 0, 1].(e) All the vectors such that c a 2b.6. Are the three vectors [1, 2, 3], [ 2, 0, 1], and [1, 10, 17] linearly dependent or independent? Do they span all vectors or not?7. Are the functions 1 x, 1 x, and 1 x x2 linearly dependent orindependent? Why?8. Find a vector that, toget
differential equations away from the analytical computation of solutions and toward both their numerical analysis and the qualitative theory. This book provides an introduction to the basic properties of partial dif-ferential equations (PDEs) and to the techniques that have proved useful in analyzing them.
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