Mathematical Physics with Partial Differential Equations

By James Kirkwood

Mathematical Physics with Partial Differential Equations is for advanced undergraduate and beginning graduate students taking a course on mathematical physics taught out of math departments. The text presents some of the most important topics and methods of mathematical physics. The premise is to study in detail the three most important partial differential equations in the field – the heat equation, the wave equation, and Laplace’s equation. The most common techniques of solving such equations are developed in this book, including Green’s functions, the Fourier transform, and the Laplace transform, which all have applications in mathematics and physics far beyond solving the above equations. The book’s focus is on both the equations and their methods of solution. Ordinary differential equations and PDEs are solved including Bessel Functions, making the book useful as a graduate level textbook. The book’s rigor supports the vital sophistication for someone wanting to continue further in areas of mathematical physics.

• Examines in depth both the equations and their methods of solution
• Presents physical concepts in a mathematical framework
• Contains detailed mathematical derivations and solutions— reinforcing the material through repetition of both the equations and the techniques
• Includes several examples solved by multiple methods—highlighting the strengths and weaknesses of various techniques and providing additional practice

• Language: English
• Publisher: Academic Press
• Release date: Dec 1, 2011
• ISBN: 9780123869944

James Kirkwood

James Kirkwood (1924-1989) was a prominent figure in the theater world as well as the author of several novels. He's best remembered as the co-author of the long-running musical A Chorus Line and for P.S. Your Cat is Dead.

Book preview

Table of Contents

Cover image

Front-matter

Copyright

Preface

Chapter 1. Preliminaries

1-1. Self-Adjoint Operators

1-2. Curvilinear Coordinates

1-3. Approximate Identities and the Dirac-δ Function

1-4. The Issue of Convergence

1-5. Some Important Integration Formulas

Chapter 2. Vector Calculus

2-1. Vector Integration

2-2. Divergence and Curl

2-3. Green’s Theorem, the Divergence Theorem, and Stokes’ Theorem

Chapter 3. Green’s Functions

3-1. Construction of Green’s Function using the Dirac-δ Function

3-2. Construction of Green’s Function using Variation of Parameters

3-3. Construction of Green’s Function from Eigenfunctions

3-4. More General Boundary Conditions

3-5. The Fredholm Alternative (Or, what if 0 is an Eigenvalue?)

3-6. Green’s function for the Laplacian in Higher Dimensions

Chapter 4. Fourier Series

4-1. Basic Definitions

4-2. Methods of Convergence of Fourier Series

4-3. The Exponential Form of Fourier Series

4-4. Fourier Sine and Cosine Series

4-5. Double Fourier Series

Chapter 5. Three Important Equations

5-1. Laplace’s Equation

5-2. Derivation of the Heat Equation in One Dimension

5-3. Derivation of the Wave equation in One Dimension

5-4. An Explicit Solution of the Wave Equation

5-5. Converting Second-Order PDEs to Standard Form

Chapter 6. Sturm-Liouville Theory

6-1. The Self-Adjoint Property of a Sturm-Liouville Equation

6-2. Completeness of Eigenfunctions for Sturm-Liouville Equations

6-3. Uniform Convergence of Fourier Series

Chapter 7. Separation of Variables in Cartesian Coordinates

7-1. Solving Laplace’s Equation on a Rectangle

7-2. Laplace’s Equation on a Cube

7-3. Solving the Wave Equation in One Dimension by Separation of Variables

7-4. Solving the Wave Equation in Two Dimensions in Cartesian Coordinates by Separation of Variables

7-5. Solving the Heat Equation in One Dimension using Separation of Variables

7-6. Steady State of the Heat equation

7-7. Checking the Validity of the Solution

Chapter 8. Solving Partial Differential Equations in Cylindrical Coordinates Using Separation of Variables

8-1. The Solution to Bessel’s Equation in Cylindrical Coordinates

8-2. Solving Laplace’s Equation in Cylindrical Coordinates using Separation of Variables

8-3. The Wave Equation on a Disk (Drum Head Problem)

8-4. The Heat Equation on a Disk

Chapter 9. Solving Partial Differential Equations in Spherical Coordinates Using Separation of Variables

9-1. An Example Where Legendre Equations Arise

9-2. The Solution to Bessel’s Equation in Spherical Coordinates

9-3. Legendre’s Equation and its Solutions

9-4. Associated Legendre Functions

9-5. Laplace’s Equation in Spherical Coordinates

Chapter 10. The Fourier Transform

10-1. The Fourier Transform as a Decomposition

10-2. The Fourier Transform from the Fourier Series

10-3. Some Properties of the Fourier Transform

10-4. Solving Partial Differential Equations using the Fourier Transform

10-5. The Spectrum of the Negative Laplacian in One Dimension

10-6. The Fourier Transform in Three Dimensions

Chapter 11. The Laplace Transform

11-1. Properties of the Laplace Transform

11-2. Solving Differential Equations using the Laplace Transform

11-3. Solving the Heat Equation using the Laplace Transform

11-4. The Wave Equation and the Laplace Transform

Chapter 12. Solving PDEs with Green’s Functions

12-1. Solving the Heat Equation using Green’s Function

12-2. The Method of Images

12-3. Green’s Function for the Wave Equation

12-4. Green’s Function and Poisson’s Equation

Appendix. Computing the Laplacian with the Chain Rule

References

Index

Front-matter

Mathematical Physics with Partial Differential Equations

Mathematical Physics with Partial Differential Equations

James R. Kirkwood

Sweet Briar College

B9780123869111000240/fm01-9780123869111.jpg is missing AMSTERDAM • BOSTON • HEIDELBERG • LONDON • NEW YORK • OXFORD • PARIS • SAN DIEGO • SAN FRANCISCO • SINGAPORE • SYDNEY • TOKYO B9780123869111000240/fm02-9780123869111.jpg is missing

Academic Press is an imprint of Elsevier

Copyright

Academic Press is an imprint of Elsevier

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© 2013 Elsevier Inc. All rights reserved

No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording, or any information storage and retrieval system, without permission in writing from the Publisher. Details on how to seek permission, further information about the Publisher’s permissions policies, and our arrangements with organizations such as the Copyright Clearance Center and the Copyright Licensing Agency, can be found at our website: www.elsevier.com/permissions

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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

B9780123869111000148/fm03-9780123869111.jpg is missing

Preface

James Radford Kirkwood

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 considered 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 calculus, 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 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.

Chapter 1. Preliminaries

In this section we describe the basic properties of linear operators, in particular self-adjoint operators. We define the Fourier coefficients and show why they are important. We solve the problem of coupled springs using eigenvectors.

Keywords

Self-adjoint operator, eigenvalues and eigenfunctions, Fourier coefficients, Bessel’s inequality, principle of superposition.

1-2

We define several coordinate systems and the formulas for functions associated with them, including the gradient, divergence, curl and Laplacian. This includes cylindrical and spherical coordinates and several lesser known systems such as parabolic and bipolar coordinates.

Keywords

Gradient, divergence, curl and Laplacian. Spherical and cylindrical coordinates, Jacobian.

1-3

We define the Dirac-δ function and approximate identities and describe a link between them. We give some properties used in physics such as the derivative of the Heaviside function is the Dirac-δ function. We derive some of the calculus of the Dirac-δ function and discuss the Dirac-δ function in spherical and cylindrical coordinates.

Keywords

Approximate identity, Dirac-δ function, distribution.

1-4

We define different notions of convergence of sequences of functions and describe some implications of different types of convergence. Some basic tests for convergence including the Weierstrass M-test are derived. Power series and Taylor series are developed.

Keywords

Pointwise convergence, uniform convergence, power series, Taylor series, Maclaurin series, uniformly Cauchy.

1-5

We evaluate integrals of functions that will be used later in the text when we study partial differential equations.

Keywords

Cauchy integral formula, Bessel function, power series expansion in integration, gamma function.

1-1

1-2. Curvilinear Coordinates

Many problems have a symmetry associated with them, and finding the solutions to such problems, as well as interpreting the solution, can often be simplified if we work in a coordinate system that takes advantage of the symmetry. In this section we describe the methods of transforming some important functions to other coordinate systems. The most common coordinate systems besides Cartesian coordinates are cylindrical and spherical coordinates, but the methods we develop are applicable to other systems as well. In our discussion, we include some of the less common systems. The less common systems will not be used in later sections but are included to reinforce the techniques of the transformations.

In Figure 1-2-1a we give a diagram of how cylindrical coordinates are defined, and in Figure 1-2-1b we do the same for spherical coordinates. We note that while the convention we use for spherical coordinates is common, it is not universal. Some sources reverse the roles of θ and φ.

Our approach will be to describe the general case of converting from Cartesian coordinates ( x, y, z) to a system of coordinates ( u1, u2, u3). After making a statement that holds in the general case, to visualize that statement, we demonstrate how the statement applies to cylindrical coordinates.

General Case: We start with Cartesian coordinates ( x, y, z) and select the group of variables u1, u2, u3 so that each of x, y, z is expressible in terms of u1, u2, u3; that is, we have

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Cylindrical Case: The variables in cylindrical coordinates are r, θ, and z. The relations are

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We write the vector B978012386911100001X/si173.gif is missing in terms of u1, u2, u3; that is,

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In cylindrical coordinates, this is

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For some of our relations to be viable, the coordinates ( u1, u2, u3) must be orthogonal. This means that the pairs of surfaces ui= constant and uj= constant must meet at right angles. In the case of cylindrical coordinates, the surface r= constant is shown in Figure 1-2-2a , the surface θ= constant is shown in Figure 1-2-2b and the surface z= constant is shown in Figure 1-2-2c . Each pair does indeed meet at right angles. It is also possible to determine that the coordinates are orthogonal by analytical methods, as we now describe.

In the general case, the vector B978012386911100001X/si176.gif is missing will be tangent to the u1 curve, which is the intersection of the u2= constant and u3= constant surfaces. Similar relations hold for B978012386911100001X/si177.gif is missing and B978012386911100001X/si178.gif is missing .

In cylindrical coordinates,

B978012386911100001X/si179.gif is missingB978012386911100001X/si180.gif is missingB978012386911100001X/si181.gif is missing

We can show that a system of coordinates forms an orthogonal coordinate system by showing that the vectors B978012386911100001X/si182.gif is missing are orthogonal; that is, by showing their inner product is zero. In the cylindrical case,

B978012386911100001X/si183.gif is missingB978012386911100001X/si184.gif is missingB978012386911100001X/si185.gif is missing

Scaling Factors

We know that in an orthogonal coordinate system, the vectors B978012386911100001X/si186.gif is missing are mutually orthogonal. We create an orthonormal system of vectors B978012386911100001X/si187.gif is missing by setting

B978012386911100001X/si188.gif is missing

We define the scaling factors hi by B978012386911100001X/si189.gif is missing , so that

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In the case of cylindrical coordinates,

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so

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Also,

B978012386911100001X/si193.gif is missingB978012386911100001X/si194.gif is missingB978012386911100001X/si195.gif is missing

Back to the general case, we have

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For cylindrical coordinates, this is

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Volume Integrals

We now describe how to convert volume integrals to other coordinate systems.

General Case: Our aim is to determine an expression for an incremental volume element dV in a general coordinate system. The volume of the parallelepiped formed by three non-coplanar vectors B978012386911100001X/si198.gif is missing , and B978012386911100001X/si199.gif is missing is B978012386911100001X/si200.gif is missing . (See exercise 1.) For the Cartesian case, we compute an incremental volume element dV using

B978012386911100001X/si201.gif is missing

Then

B978012386911100001X/si202.gif is missing

For the case

B978012386911100001X/si203.gif is missingB978012386911100001X/si204.gif is missingB978012386911100001X/si205.gif is missing

since B978012386911100001X/si206.gif is missing because B978012386911100001X/si207.gif is missing is an orthonormal system.

Another way to do this computation is to use

B978012386911100001X/si208.gif is missing

and that

(1)

B978012386911100001X/si209.gif is missing

The determinant in equation (1) is called the Jacobian of x, y, z with respect to u1, u2, u3, and is denoted B978012386911100001X/si210.gif is missing . So we have

B978012386911100001X/si211.gif is missing

We now compute dV for cylindrical coordinates. We demonstrate two methods. First, we use

B978012386911100001X/si212.gif is missing

where u1= r, u2= θ, u3= z so that du1= dr, du2=dθ, u3= dz. We have previously found that h1=1, h2= r, h3=1, so

B978012386911100001X/si213.gif is missing

For the second method we compute the Jacobian. We have

B978012386911100001X/si214.gif is missingB978012386911100001X/si215.gif is missingB978012386911100001X/si216.gif is missing

Thus,

B978012386911100001X/si217.gif is missing

and dV= rdrdθdz.

In multivariable calculus, one shows that if B978012386911100001X/si218.gif is missing , then

(2)

B978012386911100001X/si219.gif is missing

where B978012386911100001X/si220.gif is missing is the matrix from which the Jacobian is formed.

There are different ways that equation (2) is expressed in other sources. One other way is

B978012386911100001X/si221.gif is missing

Example:

Evaluate

B978012386911100001X/si222.gif is missing

where A is the circle x²+ y² ≤ 9, by changing to polar coordinates.

In Cartesian coordinates,

B978012386911100001X/si223.gif is missing

To convert to polar coordinates, we compute

B978012386911100001X/si224.gif is missing

The region A in polar coordinates is 0 ≤ r ≤ 3, 0 ≤ θ ≤ 2 π. So, in this case, equation (2) says

B978012386911100001X/si225.gif is missing

Example:

We compute

B978012386911100001X/si226.gif is missing

where A is the region bounded by 1 ≤ xy ≤ 9, and the lines y= x and y=4 x.

We seek a coordinate system ( u, v) so that the transformed region of integration will be a rectangle a ≤ u ≤ b, c ≤ v ≤ d. The graph of the region A is shown in Figure 1-2-3 .

If we let u= xy, then 1 ≤ u ≤ 9. If we let B978012386911100001X/si227.gif is missing , then 1 ≤ v ≤ 4. Now

B978012386911100001X/si228.gif is missingB978012386911100001X/si229.gif is missing

We compute the Jacobian. We have

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Then

B978012386911100001X/si231.gif is missing

and

B978012386911100001X/si232.gif is missingB978012386911100001X/si233.gif is missingB978012386911100001X/si234.gif is missing

In the next section, we will use the following forms of the change of variables equation.

a. If φ −1 exists and is differentiable for φ( a) ≤ x ≤ φ( b), then

B978012386911100001X/si235.gif is missing

b.

B978012386911100001X/si236.gif is missing

The Gradient

Next, we determine the gradient of a function f, denoted ∇ f. In Cartesian coordinates,

B978012386911100001X/si237.gif is missing

To compute ∇ f in the general case, we set B978012386911100001X/si238.gif is missing and write df in two different ways.

First, B978012386911100001X/si239.gif is missing and using that B978012386911100001X/si240.gif is missing we get

B978012386911100001X/si241.gif is missing

(3)

B978012386911100001X/si242.gif is missing

Second,

(4)

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From (3) and (4) , we get

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so

B978012386911100001X/si245.gif is missing

Thus,

B978012386911100001X/si246.gif is missing

Accordingly, we write ∇ as the operator

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For the cylindrical coordinate case, we again have

B978012386911100001X/si248.gif is missing

so in cylindrical coordinates,

B978012386911100001X/si249.gif is missingB978012386911100001X/si9987.gif is missing

The Laplacian

The final function that we consider in this section is the Laplacian, one of the most important operators in mathematics and physics. The Laplacian of the function f, denoted Δ f (some authors use B978012386911100001X/si8799.gif is missing ) in Cartesian coordinates, is defined by

B978012386911100001X/si250.gif is missing

The notation ∇ ² for the Laplacian is suggestive because ∇·∇ f gives the Laplacian of f. We have

B978012386911100001X/si251.gif is missingB978012386911100001X/si1087.gif is missing

but computing ∇·∇ f is not as straightforward as it might seem. This is because B978012386911100001X/si252.gif is missing is not a simple expression. In fact,

B978012386911100001X/si253.gif is missing

We demonstrate the validity of

B978012386911100001X/si254.gif is missing

for cylindrical coordinates with i=2. We have

B978012386911100001X/si255.gif is missing

so

B978012386911100001X/si256.gif is missing

Also,

B978012386911100001X/si257.gif is missing

Now

B978012386911100001X/si258.gif is missing

since h1= h3=1. Now h2= r and u1= r so B978012386911100001X/si259.gif is missing . Also, u3= z so B978012386911100001X/si260.gif is missing . Thus,

B978012386911100001X/si261.gif is missing

so the formula holds in this case.

In fact, it will be simpler to derive expressions for ∇ ²f and B978012386911100001X/si262.gif is missing after we have studied the Divergence