Contents

Cover

Half Title page

Title page

Copyright page

Preface

Chapter 1: First Ideas

1.1 Two Partial Differential Equations

1.2 Fourier Series

1.3 Two Eigenvalue Problems

1.4 A Proof of the Convergence Theorem

Chapter 2: Solutions of the Heat Equation

2.1 Solutions on an Interval [0, L]

2.2 A Nonhomogeneous Problem

2.3 The Heat Equation in Two Space Variables

2.4 The Weak Maximum Principle

Chapter 3: Solutions of the Wave Equation

3.1 Solutions on Bounded Intervals

3.2 The Cauchy Problem

3.3 The Wave Equation in Higher Dimensions

Chapter 4: Dirichlet and Neumann Problems

4.1 Laplace’s Equation and Harmonic Functions

4.2 The Dirichlet Problem for a Rectangle

4.3 The Dirichlet Problem for a Disk

4.4 Properties of Harmonic Functions

4.5 The Neumann Problem

4.6 Poisson’s Equation

4.7 Existence Theorem for a Dirichlet Problem

Chapter 5: Fourier Integral Methods of Solution

5.1 The Fourier Integral of a Function

5.2 The Heat Equation on the Real Line

5.3 The Debate over the Age of the Earth

5.4 Burgers’ Equation

5.5 The Cauchy Problem for the Wave Equation

5.6 Laplace’s Equation on Unbounded Domains

Chapter 6: Solutions Using Eigenfunction Expansions

6.1 A Theory of Eigenfunction Expansions

6.2 Bessel Functions

6.3 Applications of Bessel Functions

6.4 Legendre Polynomials and Applications

Chapter 7: Integral Transform Methods of Solution

7.1 The Fourier Transform

7.2 Heat and Wave Equations

7.3 The Telegraph Equation

7.4 The Laplace Transform

Chapter 8: First-Order Equations

8.1 Linear First-Order Equations

8.2 The Significance of Characteristics

8.3 The Quasi-Linear Equation

Chapter 9: End Materials

9.1 Notation

9.2 Use of MAPLE

9.3 Answers to Selected Problems

Index

Beginning Partial Differential Equations

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.

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Library of Congress Cataloging-in-Publication data:

O’Neil, Peter V.
Beginning partial differential equations / Peter V. O’Neil. — Third edition.
   pages cm
Includes bibliographical references and index.
ISBN 978-1-118-62994-9 (hardback)
1. Differential equations, Partial. I. Title.
QA377.O54 2014
515’.353—dc23

2013034307

Preface

This edition is based on four themes: methods of solution of initial-boundary value problems, properties and existence of solutions, applications of partial differential equations, and use of software to carry out computations and graphics.

The focus is on equations of diffusion processes and wave motion, and on Dirichlet and Neumann problems. Following an introductory chapter, we look at methods applied to these equations in bounded and unbounded media, and in one and several space dimensions. The topics are organized to make it easy to match problems in specific settings to methods for writing solutions. Methods include Fourier series and integrals, the use of characteristics, integral solutions, integral transforms, and special functions and eigenfunction expansions.

Properties of solutions that are considered include existence and uniqueness issues, maximum and mean value principles, integral representations, and sensitivity of solutions to initial and boundary conditions.

In addition to standard material for an introductory course, topics include traveling-wave solutions of Burger’s equation, damped wave motion, heat and wave equations with forcing terms, a general treatment of eigenfunction expansions, a complete solution of the telegraph equation using the Fourier transform, the use of characteristics to solve Cauchy problems and vibrating string problems with moving ends, double Fourier series solutions, and the Poisson-Kirchhoff integral solution of the wave equation in two dimensions. There are also proofs of important theorems, including an existence theorem for a Dirichlet problem and a convergence theorem for Fourier series.

Finally, there is a section on the use of MAPLE™ to carry out computations and experiment with graphics. MATLAB ®, MATHEMATICA ®, and other packages may also be used for these numerical aspects of partial differential equations.

Chapter 1

First Ideas

We will begin a study of partial differential equations by deriving equations modeling diffusion processes and wave motion. These are widely applicable in the physical and life sciences, engineering, economics, and other areas. Following this, we will lay the foundations for the Fourier method, which is used to write solutions for many kinds of problems, and then solve two eigenvalue/eigenfunction problems that occur frequently when this method is used.

The chapter concludes with a proof of a theorem on the convergence of Fourier series.

1.1 Two Partial Differential Equations

1.1.1 The Heat, or Diffusion, Equation

We will derive a partial differential equation modeling heat flow in a medium. Although we will speak in terms of heat flow because it is familiar to us, the heat equation applies to general diffusion processes, which might be a flow of energy, a dispersion of insect or bacterial populations in controlled environments, changes in the concentration of a chemical dissolving in a fluid, or many other phenomena of interest. For this reason the heat equation is also called the diffusion equation.

Consider a bar of material of constant density, ρ, having uniform cross sections with area A. The lateral surface of the bar is insulated, so there is no heat loss across this surface.

Place an x-axis along the length, L, of the bar and assume that at a given time, the temperature is the same along any cross section perpendicular to this axis, although it may vary from one cross section to another. We will derive an equation for u(x, t), the temperature in the cross section of the bar at x, at time t. In the context of diffusion, u(x, t) is called a density distribution function.

Let c be the specific heat of the material of the bar. This is the amount of heat energy that must be supplied to a unit mass of the material to raise its temperature one degree. The segment of bar between x and x + Δx has mass ρAΔx, and it will take approximately ρcAu(x, t)Δx units of heat energy to change the temperature of this segment from zero to u(x, t), its temperature at time t.

The total heat energy in this segment at any time t > 0 is

This amount of heat energy within the segment at time t can increase in two ways: heat energy may flow into the segment across its ends (this change is the flux of the energy), and/or there may be a source or loss of heat energy within the segment. This can occur if there is, say, a chemical reaction or if the material is radioactive.

The rate of change of the temperature within the segment, with respect to time, is therefore

Assume for now that there is no source or loss of energy within the bar. Then

(1.1) equation

Now let F(x, t) be the amount of heat energy per unit area flowing across the cross section at x at time t, in the direction of increasing x. Then the flux of the energy into the segment between x and x + Δx at time t is the rate of flow into the segment across the section at x, minus the rate of flow out of the segment across the section at x + Δx (Figure 1.1):

Figure 1.1: Flux in segment = rate in minus rate out.

Write this as

(1.2) equation

Now recall Newton’s law of cooling, which states that heat energy flows from the warmer to the cooler region, and the amount of heat energy is proportional to the temperature difference (gradient). This means that

The positive constant of proportionality, K, is called the heat conductivity of the bar. The negative sign in this equation is due to the fact that energy flows from the warmer to the cooler segment. Substitute this expression for F(x, t) into equation 1.2 to obtain

Write this as

(1.3) equation

From equations 1.1 and 1.3 for the flux, we have

Divide out the common factor A and write this equation as

This equation must be valid for any choices of x and x + Δx, as long as

If the integrand were nonzero at some x, then, assuming continuity of this integrand (which is reasonable on physical grounds), it would be nonzero, therefore strictly positive or strictly negative on some interval (x, x + Δx). This would force this integral to be positive or negative, not zero, for this x and Δx, and this is a contradiction. We conclude that the integrand must be identically zero, hence

It is convenient to denote partial derivatives using subscripts. In this notation,

(1.4) equation

where k = K/cρ is called the diffusivity of the material of the bar. Equation 1.4 is the one-dimensional heat, or diffusion, equation. This equation, with appropriate boundary and initial conditions, models a wide range of diffusion phenomena, providing a setting for a mathematical analysis to draw conclusions about the behavior of the process under study.

If we allow for a source term Q(x, t), then the heat equation is

(1.5) equation

We say that is Because of the () term, is Both equations are partial differential equations because they contain at least one second derivative term, but no higher derivative. Both equations are also , which means they are linear in the unknown function and its derivatives. By contrast, the second-order partial differential equation

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