Mathematical Physics with PartialDifferential Equations

Mathematical Physicswith Partial Differential

Equations

James R. KirkwoodSweet Briar College

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Library of Congress Cataloging-in-Publication Data

James R. Kirkwood

Mathematical physics with partial differential equations / James Kirkwood.

p. cm.

Includes bibliographical references and index.

ISBN 978-0-12-386911-1 (hardback)

1. Mathematical physics. 2. Differential equations, Partial. I. Title.

QC20.7.D5K57 2013

530.14--dc23

2011028883

British Library Cataloguing-in-Publication Data

A catalogue record for this book is available from the British Library.

For information on all Academic Press publications

visit our website at www.elsevierdirect.com

Printed in the United States of America

12 13 14 15 10 9 8 7 6 5 4 3 2 1

Contents

Preface xi

1. Preliminaries 11-1 Self-Adjoint Operators 1

Fourier Coefficients 5Exercises 11

1-2 Curvilinear Coordinates 14Scaling Factors 17Volume Integrals 18The Gradient 22The Laplacian 23Spherical Coordinates 25Other Curvilinear Systems 25Applications 31An Alternate Approach (Optional) 33Exercises 33

1-3 Approximate Identities and the Dirac-δ Function 34Approximate Identities 35The Dirac-δ Function in Physics 37Some Calculus for the Dirac-δ Function 40The Dirac-δ Function in Curvilinear Coordinates 42Exercises 44

1-4 The Issue of Convergence 45Series of Real Numbers 45Convergence versus Absolute Convergence 47Series of Functions 48Power Series 54Taylor Series 56Exercises 60

1-5 Some Important Integration Formulas 64Other Facts We Will Use Later 68Another Important Integral 69Exercises 70

2. Vector Calculus 732-1 Vector Integration 73

Path Integrals 74Line Integrals 77Surfaces 80Parameterized Surfaces 82

v

Integrals of Scalar Functions Over Surfaces 83Surface Integrals of Vector Functions 85Exercises 91

2-2 Divergence and Curl 93Cartesian Coordinate Case 94Cylindrical Coordinate Case 97Spherical Coordinate Case 100The Curl 104The Curl in Cartesian Coordinates 104The Curl in Cylindrical Coordinates 109The Curl in Spherical Coordinates 114Exercises 122

2-3 Green’s Theorem, the Divergence Theorem, andStokes’ Theorem 122

The Divergence (Gauss’) Theorem 127Stokes’ Theorem 135An Application of Stokes’ Theorem 140An Application of the Divergence Theorem 141Conservative Fields 142Exercises 148

3. Green’s Functions 155Introduction 155

3-1 Construction of Green’s Function Using the Dirac-δ Function 156Exercises 164

3-2 Construction of Green’s Function Using Variation ofParameters 164

Exercises 1683-3 Construction of Green’s Function from Eigenfunctions 168

Exercises 1713-4 More General Boundary Conditions 171

Exercises 1733-5 The Fredholm Alternative (or, What If 0 Is an Eigenvalue?) 173

Exercises 1803-6 Green’s Function for the Laplacian in Higher Dimensions 180

Exercises 186

4. Fourier Series 187Introduction 187

4-1 Basic Definitions 188Exercises 191

4-2 Methods of Convergence of Fourier Series 193Fourier Series on Arbitrary Intervals 203Exercises 204

4-3 The Exponential Form of Fourier Series 206Exercises 207

vi Contents

4-4 Fourier Sine and Cosine Series 208Exercises 210

4-5 Double Fourier Series 210Exercise 212

5. Three Important Equations 213Introduction 213

5-1 Laplace’s Equation 215Exercises 216

5-2 Derivation of the Heat Equation in One Dimension 216Exercise 218

5-3 Derivation of the Wave Equation in One Dimension 218Exercises 222

5-4 An Explicit Solution of the Wave Equation 222Exercises 227

5-5 Converting Second-Order PDEs to Standard Form 228Exercise 232

6. Sturm-Liouville Theory 233Introduction 233Exercises 234

6-1 The Self-Adjoint Property of a Sturm-Liouville Equation 234Exercises 236

6-2 Completeness of Eigenfunctions for Sturm-Liouville Equations 237Exercises 245

6-3 Uniform Convergence of Fourier Series 245

7. Separation of Variables in Cartesian Coordinates 251Introduction 251

7-1 Solving Laplace’s Equation on a Rectangle 251Exercises 256

7-2 Laplace’s Equation on a Cube 258Exercises 261

7-3 Solving the Wave Equation in One Dimension bySeparation of Variables 262

Exercises 2677-4 Solving the Wave Equation in Two Dimensions in Cartesian

Coordinates by Separation of Variables 269Exercises 271

7-5 Solving the Heat Equation in One Dimension UsingSeparation of Variables 271

The Initial Condition Is the Dirac-δ Function 274Exercises 276

7-6 Steady State of the Heat Equation 277Exercises 281

7-7 Checking the Validity of the Solution 283

viiContents

8. Solving Partial Differential Equations in CylindricalCoordinates Using Separation of Variables 287

Introduction 287An Example Where Bessel Functions Arise 287Exercises 292

8-1 The Solution to Bessel’s Equation in CylindricalCoordinates 292

Exercises 2948-2 Solving Laplace’s Equation in Cylindrical Coordinates

Using Separation of Variables 295Exercises 299

8-3 The Wave Equation on a Disk (Drum Head Problem) 299Exercises 303

8-4 The Heat Equation on a Disk 303Exercises 306

9. Solving Partial Differential Equations in SphericalCoordinates Using Separation of Variables 3079-1 An Example Where Legendre Equations Arise 3079-2 The Solution to Bessel’s Equation in

Spherical Coordinates 3109-3 Legendre’s Equation and Its Solutions 315

Exercises 3189-4 Associated Legendre Functions 319

Exercise 3229-5 Laplace’s Equation in Spherical Coordinates 322

Exercise 325

10. The Fourier Transform 327Introduction 327

10-1 The Fourier Transform as a Decomposition 32810-2 The Fourier Transform from the Fourier Series 32910-3 Some Properties of the Fourier Transform 331

Exercises 33410-4 Solving Partial Differential Equations Using the

Fourier Transform 335Exercises 341

10-5 The Spectrum of the Negative Laplacian inOne Dimension 343

10-6 The Fourier Transform in Three Dimensions 346Exercise 350

11. The Laplace Transform 351Introduction 351Exercises 352

11-1 Properties of the Laplace Transform 352Exercises 356

viii Contents

11-2 Solving Differential Equations Using the Laplace Transform 356Exercises 360

11-3 Solving the Heat Equation Using the Laplace Transform 361Exercises 366

11-4 The Wave Equation and the Laplace Transform 368Exercises 373

12. Solving PDEs with Green’s Functions 37512-1 Solving the Heat Equation Using Green’s Function 375

Green’s Function for the Nonhomogeneous Heat Equation 377Exercises 379

12-2 The Method of Images 379Method of Images for a Semi-infinite Interval 379Method of Images for a Bounded Interval 383Exercises 389

12-3 Green’s Function for the Wave Equation 390Exercises 397

12-4 Green’s Function and Poisson’s Equation 398Exercises 401

Appendix: Computing the Laplacian with the Chain Rule 403References 413Index 415

ixContents

Preface

The major purposes of this book are to present partial differential equations

(PDEs) and vector analysis at an introductory level. As such, it could be con-

sidered a beginning text in mathematical physics. It is also designed to provide

a bridge from undergraduate mathematics to the first graduate mathematics

course in physics, applied mathematics, or engineering. In these disciplines, it

is not unusual for such a graduate course to cover topics from linear algebra,

ordinary and partial differential equations, advanced calculus, vector analysis,

complex analysis, and probability and statistics at a highly accelerated pace.

In this text we study in detail, but at an introductory level, a reduced list

of topics important to the disciplines above. In partial differential equations,

we consider Green’s functions, the Fourier and Laplace transforms, and how

these are used to solve PDEs. We also study using separation of variables to

solve PDEs in great detail. Our approach is to examine the three prototypical

second-order PDEs—Laplace’s equation, the heat equation, and the wave

equation—and solve each equation with each method. The premise is that in

doing so, the reader will become adept at each method and comfortable with

each equation.

The other prominent area of the text is vector analysis. While the usual

topics are discussed, an emphasis is placed on understanding concepts rather

than formulas. For example, we view the curl and gradient as properties of a

vector field rather than simply as equations. A significant—but optional—

portion of this area deals with curvilinear coordinates to reinforce the idea of

conversion of coordinate systems.

Reasonable prerequisites for the course are a course in multivariable cal-

culus, familiarity with ordinary differential equations including the ability to

solve a second-order boundary problem with constant coefficients, and some

experience with linear algebra.

In dealing with ordinary differential equations, we emphasize the linear

operator approach. That is, we consider the problem as being an eigenvalue/

eigenvector problem for a self-adjoint operator. In addition to eliminating

some tedious computations regarding orthogonality, this serves as a unifying

theme and an introduction to more advanced mathematics.

The level of the text generally lies between that of the classic encyclopedic

texts of Boas and Kreysig and the newer text by McQuarrie, and the partial

differential equations books of Weinberg and Pinsky. Topics such as Fourier

series are developed in a mathematically rigorous manner. The section on

completeness of eigenfunctions of a Sturm-Liouville problem is considerably

xi

more advanced than the rest of the text, and can be omitted if one wishes to

merely accept the result.

The text can be used as a self-contained reference as well as an introductory

text. There was a concerted effort to avoid situations where filling in details

of an argument would be a challenge. This is done in part so that the text

could serve as a source for students in subsequent courses who felt “I know I’m

supposed to know how to derive this, but I don’t.” A couple of such examples

are the fundamental solution of Laplace’s equation and the spectrum of the

Laplacian.

I want to give special thanks to Patricia Osborn of Elsevier Publishing

whose encouragement prompted me to turn a collection of disjointed notes

into what I hope is a readable and cohesive text, and also to Gene Wayne of

Boston University who provided valuable suggestions.

James Radford Kirkwood

xii Preface