# Mathematics

Undergraduate Texts in Mathematics

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Undergraduate Texts in Mathematics

Series editors

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San Francisco State University, San Francisco, CA, USA

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University of California, Berkeley, CA, USA

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Colin Adams, Williams College, Williamstown, MA, USA

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Undergraduate Texts in Mathematics are generally aimed at third- and

fourth-year undergraduate mathematics students at North American universi-

ties. These texts strive to provide students and teachers with new perspectives

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an appreciation of interrelations among different aspects of the subject. They

feature examples that illustrate key concepts as well as exercises that strengthen

understanding.

For further volumes: http://www.springer.com/series/666

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J. David Logan

Applied Partial Differential Equations

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J. David Logan Department of Mathematics University of Nebraska-Lincoln Lincoln, NE, USA

ISSN 0172-6056 ISSN 2197-5604 (electronic) Undergraduate Texts in Mathematics ISBN 978-3-319-12492-6 ISBN 978-3-319-12493-3 (eBook) DOI 10.1007/978-3-319-12493-3 Springer Cham Heidelberg New York Dordrecht London

Library of Congress Control Number: 2014955188

Mathematics Subject Classification: 34-01, 00-01, 00A69, 97M50, 97M60

c© Springer International Publishing Switzerland 2015 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made.

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To Aaron, Rachel, and David

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Contents

Preface to the Third Edition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix

To Students . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi

1. The Physical Origins of Partial Differential Equations . . . . . . 1

1.1 PDE Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2 Conservation Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

1.3 Diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

1.4 Diffusion and Randomness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

1.5 Vibrations and Acoustics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

1.6 Quantum Mechanics* . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

1.7 Heat Conduction in Higher Dimensions . . . . . . . . . . . . . . . . . . . . . . 60

1.8 Laplace’s Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

1.9 Classification of PDEs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

2. Partial Differential Equations on Unbounded Domains . . . . . . 79

2.1 Cauchy Problem for the Heat Equation . . . . . . . . . . . . . . . . . . . . . 79

2.2 Cauchy Problem for the Wave Equation . . . . . . . . . . . . . . . . . . . . . 87

2.3 Well-Posed Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

2.4 Semi-Infinite Domains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

2.5 Sources and Duhamel’s Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

2.6 Laplace Transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

2.7 Fourier Transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

3. Orthogonal Expansions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

3.1 The Fourier Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

3.2 Orthogonal Expansions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

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viii Contents

3.3 Classical Fourier Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

4. Partial Differential Equations on Bounded Domains . . . . . . . . 155

4.1 Overview of Separation of Variables . . . . . . . . . . . . . . . . . . . . . . . . . 156

4.2 Sturm–Liouville Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167

4.3 Generalization and Singular Problems . . . . . . . . . . . . . . . . . . . . . . . 180

4.4 Laplace’s Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186

4.5 Cooling of a Sphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198

4.6 Diffusion in a Disk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202

4.7 Sources on Bounded Domains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207

4.8 Poisson’s Equation* . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216

5. Applications in the Life Sciences . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229

5.1 Age-Structured Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229

5.2 Traveling Waves Fronts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238

5.3 Equilibria and Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245

6. Numerical Computation of Solutions . . . . . . . . . . . . . . . . . . . . . . . 257

6.1 Finite Difference Approximations . . . . . . . . . . . . . . . . . . . . . . . . . . . 258

6.2 Explicit Scheme for the Heat Equation . . . . . . . . . . . . . . . . . . . . . . 260

6.3 Laplace’s Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268

6.4 Implicit Scheme for the Heat Equation . . . . . . . . . . . . . . . . . . . . . . 273

Appendix A. Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287

qc322@nyu.edu

Preface to the Third Edition

The goal of this new edition is the same as that for the original, namely, to

present a one-semester treatment of the basic ideas encountered in partial differ-

ential equations (PDEs). The text is designed for a 3-credit semester course for

undergraduate students in mathematics, science, and engineering. The prereq-

uisites are calculus and ordinary differential equations. The text is intimately

tied to applications in heat conduction, wave motion, biological systems, and

a variety other topics in pure and applied science. Therefore, students should

have some interest, or experience, in basic science or engineering.

The main part of the text is the first four chapters, which cover the essential

concepts. Specifically, they treat first- and second-order equations on bounded

and unbounded domains and include transform methods (Laplace and Fourier),

characteristic methods, and eigenfunction expansions (separation of variables);

there is considerable material on the origin of PDEs in the natural sciences

and engineering. Two additional chapters, Chapter 5 and Chapter 6, are short

introductions to applications of PDEs in biology and to numerical computation

of solutions. The text offers flexibility to instructors who, for example, may want

to insert topics from biology or numerical methods at any time in the course. A

brief appendix reviews techniques from ordinary differential equations. Sections

marked with an asterisk (*) may safely be omitted. The mathematical ideas

are strongly motivated by physical problems, and the exposition is presented in

a concise style accessible to students in science and engineering. The emphasis

is on motivation, methods, concepts, and interpretation rather than formal

theory.

The level of exposition is slightly higher than students encounter in the

post-calculus differential equations course. The philosophy is that a student

should progress in the ability to read mathematics. Elementary texts contain

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x Preface to the Third Edition

many examples and detailed calculations, but advanced mathematics and sci-

ence books leave a lot to the reader. This text leaves some of the easy details

to the reader. Often, the arguments are derivations in lieu of carefully con-

structed proofs. The exercises are at varying levels and encourage students to

think about the concepts and derivations rather than just grind out lots of rou-

tine solutions. A student who reads this book carefully and who solves many

of the exercises will have a sound knowledge base to continue with a second-

year partial differential equations course where careful proofs are constructed

or with upper-division courses in science and engineering where detailed, and

often difficult, applications of partial differential equations are introduced.

This third edition, a substantial revision, contains many new and revised

exercises, and some sections have been greatly expanded with more worked

examples and additional explanatory material. A new, less dense, format makes

key results more apparent and the text easier to read for undergraduates. The

result is a text one-third longer. But the size and brevity of text, contrary to

voluminous other texts, struck a chord with many users and that has been

maintained. Many users provided suggestions that have become part of this

revision, and I greatly appreciate their interest and comments.

Elizabeth Loew, my editor at Springer, deserves special recognition for her

continuous and expert support. I have found Springer to be an extraordinary

partner in this project.

Finally, this book is very affectionately dedicated to my two sons and daugh-

ter, Aaron, David, and Rachel, who have often been my teachers with their

challenging and unique perspectives on life. For these gifts I greatly thank you.

I welcome suggestions, comments, and corrections. Contact information is

on my web site: http://www.math.unl.edu/~jlogan1, where additional items

can be found. Solutions to some of the exercises can be found on the Springer

web site.

J. David Logan

Willa Cather Professor

Lincoln, Nebraska

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To Students

Our understanding of the fundamental processes of the natural world is

based to a large extent on partial differential equations. W. A. Strauss

Partial differential equations (PDEs) is a topic worthy of your study. It

is a subject about differential equations involving unknown functions of sev-

eral variables; the derivatives are partial derivatives. As such, it is a subject

that is intimately connected with multivariable calculus. To be successful you

should have a good command of the concepts in the calculus of several vari-

ables. So keep a calculus text nearby and review concepts when needed. The

same comments apply to elementary ordinary differential equations (ODEs).

An appendix at the end of the book reviews basic solution techniques for ODEs.

If you wish to consult other sources, the texts by Farlow (1993) and Strauss

(1994) are good choices.

A mathematics book must be read with a pencil and paper in hand. Ele-

mentary books fill in most steps in the exposition, but more advanced books

leave many details to the reader. This book has enough detail so that you can

follow the discussion, but pencil and paper work is required in some portions.

Verifying all the statements and derivations in a text is a worthwhile endeavor

and will help you learn the material. Many students find that studying PDEs

provides an opportunity to hone their skills and reinforce concepts in calculus

and differential equations. Further, studying PDEs increases your understand-

ing of physical principles in a monumental way.

The exercises are the most important part of this text, and you should try

to solve most of them. Some require routine analytical calculations, but others

require careful thought. We learn mathematics by doing mathematics, even

when we are stymied by a problem. The effort put into a failed attempt will

help you sort out the concepts and reinforce the learning process. View the

exercises as a challenge and resist the temptation to give up. It is also a good

habit to write up your solutions in a clear, concise, logical form. Good writing

entails good thinking, and conversely.

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1 The Physical Origins of Partial

Differential Equations

Many important ideas in mathematics are developed within the framework

of physical science, and mathematical equations, especially partial differential

equations, provides the language to formulate these ideas. In reverse, advances

in mathematics provides the stimulus for new advancements in science. Over the

years mathematicians and scientists extended these methodologies to include

nearly all areas of science and technology, and a paradigm emerged called math-

ematical modeling. A mathematical model is an equation, or set of equations,

whose solution describes the physical behavior of the related physical system.

In this context we say, for example, that Maxwell’s equations form a model

for electromagnetic phenomena. Like most mathematical models, Maxwell’s

equations are based on physical observations. But the model is so accurate,

we regard the model itself as describing an actual physical law. Other mod-

els, for example a model of how a disease spreads in a population, are more

conceptual. Such models often explain observations, but only in a highly lim-

ited sense. In general, a mathematical model is a simplified description, or

caricature, of reality expressed in mathematical terms. Mathematical modeling

involves observation, selection of relevant physical variables, formulation of the

equations, analysis of the equations and simulation, and, finally, validation of

the model to ascertain whether indeed it is predictive. The subject of partial

differential equations encompasses all types of models, from physical laws like

Maxwell’s equations in electrodynamics, to conceptual laws that describe the

spread of an plant invasive species on a savanna.

c© Springer International Publishing Switzerland 2015 1 J. D. Logan, Applied Partial Differential Equations, 3rd edition. Undergraduate Texts in Mathematics, DOI 10.1007/978-3-319-12493-3 1

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2 1. The Physical Origins of Partial Differential Equations

1.1 PDE Models

In this book we examine models that can be described by partial differential

equations. The focus is on the origin of such models and tools used for their

analysis. Of particular interest are models in diffusion and heat flow, wave prop-

agation, and transport of energy, chemicals, and other matter. It is impossible

to overestimate the role and importance of PDEs in science and engineering.

Readers should be familiar with systems governed by ordinary differential

equations (ODEs). For example, a typical ODE model in population ecology is

the logistic model du

dt = ru

( 1− u

K

) , t > 0,

which is a simple equation for population growth where the per capita rate

of change of population, u′(t)/u(t), is a decreasing function of the population.

Here t is time, and u = u(t) is the population of a given system of individuals.

We refer to u as the state and say that the evolution of the state variable is

governed by the model equation. The positive numbers r and K are given phys-

ical parameters that represent the relative growth rate and carrying capacity,

respectively; presumably, r and K can be measured for the population under

investigation. The solution to the logistic equation is easily found by separation

of variables to be

u(t) = u0K

u0 + (K − u0) e−rt , t > 0,

where u(0) = u0 is the initial population. The logistic model accurately

describes some populations having a sigmoid growth shape. In general, an ODE

model has the form

du

dt = F (t, u; r1, . . . , rn), t > 0,

where F is a given functional relation between t, u, and m parameters

r1, . . . , rm. Often the model includes an initial condition of the form u(0) = u0,

where u0 is a given state value at t = 0. More generally, an ODE model may

consist of a system of n ODEs for n state variables u1(t), . . . , un(t).

A PDE model differs from an ODE model in that the state variable u

depends on more than one independent variable. ODEs govern the evolution of

a system in time, and observations are made in time. PDEs model the evolution

of a system in both time and space; the system can be observed both in a time

interval and in a spatial region (which may be one-, two-, or three-dimensional).

PDE models may also be independent of time, but depend on several spatial

variables. Two examples of PDEs are

utt(x, t) − c2uxx(x, t) = 0, (wave equation) uxx(x, y) + uyy(x, y) = 0. (Laplace’s equation)

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1.1 PDE Models 3

The wave equation describes the propagation of waves in a one dimensional

medium. The unknown function u = u(x, t) is a function of position x and time

t. In Laplace’s equation, the unknown state is a function u = u(x, y), where x

and y are spatial variables. It models, for example, equilibrium temperatures

in a two-dimensional region of the plane with prescribed temperatures on its

boundary.

Example 1.1

(Heat flow) Consider the problem of determining the temperature in a thin,

laterally insulated, cylindrical, metal bar of length l and unit cross-sectional

area, whose two ends are maintained at a constant zero degrees, and whose

temperature initially (at time zero) varies along the bar and is given by a fixed

function φ(x). See Figure 1.1.

Figure 1.1 A laterally insulated metal bar with zero temperature at both

ends. Heat flows in the axial, or x-direction, and u(x, t) is the temperature of

the cross-section at x at time t. At time t = 0 the temperature at locations x

is given by φ(x)

How does the bar cool down? In this case, the state variable u is the tempera-

ture, and it depends upon both when the measurement is taken and where in

the bar it is taken. Thus, u = u(x, t), where t is time and 0 < x < l. The equa-

tion governing the evolution of the temperature u is called the heat equation

(we derive it in Section 1.3), and it has the form

ut = kuxx. (1.1)

Observe that the subscript notation is used to indicate partial differentiation,

and we rarely write the independent variables, preferring u to u(x, t). The equa-

tion states that the partial derivative of the temperature with respect to t must

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4 1. The Physical Origins of Partial Differential Equations

equal the second partial derivative of the temperature with respect to x, multi-

plied by a constant k. The constant k, called the diffusivity, is a known param-

eter and a property of the bar; it can be determined in terms of the density,

specific heat, and thermal conductivity of the metal. Values for these physical

constants for different materials can be found in handbooks or online. Later we

observe that (1.1) comes from a basic physical law (energy conservation) and

an empirical observation (Fourier’s heat conduction law). The conditions that

the end faces of the bar are maintained at zero degrees can be expressed by the

equations

u(0, t) = 0, u(l, t) = 0, t > 0, (1.2)

which are called boundary conditions because they impose conditions on the

temperature at the boundary of the spatial domain. The stipulation that the

bar initially has a fixed temperature φ(x) degrees across its length is expressed

mathematically by

u(x, 0) = φ(x), 0 < x < l. (1.3)

This condition is called an initial condition because it specifies the state

variable at time t = 0. The entire set of equations (1.1)–(1.3)—the PDE and

the auxiliary conditions—form the mathematical model for heat flow in the

bar. Such a model in the subject of PDEs is called an initial boundary value

problem. The invention and analysis of such models are the subjects of this

book. �

In this heat flow model, the state variable u, the temperature, depends upon

two independent variables, a time variable t and a spatial variable x. Such a

model is an evolution model. Some physical systems do not depend upon

time, but rather only upon spatial variables. Such models are called steady

state or equilibriummodels. For example, if Ω is a bounded, two-dimensional

spatial domain representing a planar, laminar plate, and on the boundary of

Ω, denoted by ∂Ω, there is imposed a given, time-independent temperature,

then the steady-state temperature distribution u = u(x, y) inside Ω satisfies

the Laplace equation, a partial differential equation having the form

uxx + uyy = 0, (x, y) ∈ Ω. (1.4)

If we denote the fixed boundary temperature by f(x, y), then (1.4) along with

the boundary condition

u(x, y) = f(x, y), (x, y) ∈ ∂Ω, (1.5)

is an equilibrium model for temperatures in the plate. In PDEs these spatial

models are called boundary value problems. Solving Laplace’s equation

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1.1 PDE Models 5

(1.4) in a region Ω subject to a given condition (1.5) on the boundary is a

famous problem called the Dirichlet problem.

In general, a second-order evolution PDE in one spatial variable and time

is an equation of the form

G(x, t, u, ux, ut, uxx, utt, uxt) = 0, x ∈ I, t > 0, (1.6)

where I is a given spatial interval, which may be a bounded or unbounded.

The equation involves an unknown function u = u(x, t), the state variable, and

some of its partial derivatives. The order of a PDE equation is the order of

the highest derivative that occurs. The PDE is almost always supplemented

with initial and/or boundary conditions that specify the state u at time t = 0

and on the boundary. One or more parameters, which are not explicitly shown,

may also occur in (1.6).

PDEs are classified according to their order and other properties. For exam-

ple, as is the case for ODEs, they are classified as linear or nonlinear. Equation

(1.6) is linear if G is a linear function in u and in all of its derivatives; how

the independent variables x and t appear is not relevant. This means that the

unknown u and its derivatives appear alone and to the the first power. Other-

wise, the PDE is nonlinear. A linear equation is homogeneous if every term

contains u or some derivative of u. It is nonhomogeneous if there is a term

depending only on the independent variables, t and x.

Example 1.2

Both second-order equations

ut + uuxx = 0 and utt − ux + sinu = 0

are nonlinear, the first because of the product uuxx and the second because the

unknown u is tied up in the nonlinear sine function. The second-order equation

ut − sin(x2t)uxt = 0

is linear and homogeneous, and the equation

ut + 3xuxx = tx 2

is linear and nonhomogeneous. �

In many discussions it is convenient to introduce operator notation. For

example, we can write the heat equation

ut − kuxx = 0

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6 1. The Physical Origins of Partial Differential Equations

as

Lu = 0 where L = ∂

∂t − k ∂

2

∂x2 .

Here L is a differential operator, and we write its action on a function u as

as either Lu or L(u). It acts on twice continuously differentiable functions

u = u(x, t) to produce a new function. We say a differential operator L is

linear if, and only if, it satisfies the two conditions

L(u+ v) = Lu+ Lv, L(cu) = cLu

for all functions u and v, and all constants c. If L is a linear, then the equation

Lu = 0 is said to be homogeneous, and the equation Lu = f is nonhomo-

geneous.

One cannot overstate the significance of the partition of PDEs into the two

categories of linear and nonlinear. Linear equations have algebraic structure to

their solution sets: the sum of two solutions to a homogeneous linear equation is

again a solution, as are constant multiples of solutions. Another way of saying

this is that solutions superimpose. Thus, if u1, u2, . . . , un are solutions to

Lu = 0, and c1, c2,. . . ,cn are constants, then the linear combination

c1u1 + c2u2 + · · ·+ cnun

is also a solution to Lu = 0. As we see later, this superposition principle

extends in many cases to infinite sums and even to a continuum of solutions.

For example, if u(x, t, ξ) is a one-parameter family of solutions to Lu = 0, for

all ξ in an interval J , then we can often prove ∫

J

c(ξ)u(x, t, ξ) dξ

is a solution to Lu = 0 for special conditions on the distributed ‘constants’

(i.e., the function) c(ξ). These superposition principles are essential in this

text. Every concept we use involves superposition in one way or another.

Another result based on linearity is that the real and imaginary parts of

a complex-valued solution w to a homogeneous differential equation Lw = 0

are both real solutions. Specifically, if w is complex-valued function, then w =

u + iv, where u = Rew and v = Imw are real-valued functions. Then, by

linearity,

Lw = L(u+ iv) = Lu+ iLv = 0.

This implies Lu = 0 and Lv = 0, because if a complex function is indentically

zero then both its real and imaginary parts are zero.

Nonlinear equations do not share these properties. Nonlinear equations are

harder to solve, and their solutions are more difficult to analyze. Even when

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1.1 PDE Models 7

nature presents us with a nonlinear model, we often approximate it with a more

manageable linear one.

Equally important in classifying PDEs is the specific nature of the physi-

cal phenomena that they describe. For example, a PDEs can be classified as

wave-like, diffusion-like, or equilibrium, depending on whether it models wave

propagation, a diffusion process, or an equilibrium state. For example, Laplace’s

equation (1.4) is a second-order, linear equilibrium equation; the heat equation

(1.1) is a second-order, linear diffusion equation because heat flow is a diffusion

process. In the last section of this chapter we give a more precise, mathematical

characterization of these properties.

By a solution to the PDE (1.6) we mean a function u = u(x, t) defined

on the space–time domain t > 0, x ∈ I, that satisfies, upon substitution, the equation (1.6) identically on that domain. Implicit in this definition is the

stipulation that u possess as many continuous partial derivatives as required by

the PDE. For example, a solution to a second-order equation should have two

continuous partial derivatives so that it makes sense to calculate the derivatives

and substitute them into the equation. Whereas the general solution to an ODE

involves arbitrary constants, the general solution to a PDE involves arbitrary

functions. Sometimes the general solution to a PDE can be found, but it is

usually not necessary to have it to solve most problems of interest.

Example 1.3

One should check, by direct substitution, that both functions

u1(x, t) = x 2 + 2t and u2(x, t) = e

−t sinx

are solutions to the heat equation

ut − uxx = 0.

There are many other solutions to this equation. Auxiliary conditions, like

initial and boundary conditions, generally single out the appropriate solution

to a problem. �

Example 1.4

Consider the first-order, linear, nonhomogeneous PDE

ux = t sinx.

This equation can be solved by direct integration. We integrate with respect to

x, holding t fixed, to get

u(x, t) = −t cosx+ ψ(t),

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8 1. The Physical Origins of Partial Differential Equations

where ψ is an arbitrary function of t. In PDEs, integration with respect to one

variable produces an arbitrary function of the other variable, not an arbitrary

constant as in one-dimensional calculus. This last equation defines the general

solution. One can check that it is a solution for any differentiable function

ψ(t). Usually, PDEs have arbitrary functions in the expression for their general

solutions; the number of such functions often agrees with the order of the

equation. �

Example 1.5

The second-order PDE for u = u(x, t),

utt − 4u = 0

is just an like an ODE with x as a parameter. So the ‘constants’ depend on x.

The solution is

u(x, t) = φ(x)e−2t + ψ(x)e2t,

where φ and ψ are arbitrary functions of x. �

Figure 1.2 A solution surface u = u(x, t). A cross-section u(x, t0) of the

surface at time t0 is interpreted as a wave profile at t = t0

Geometrically, a solution u = u(x, t) can be thought of as a surface in xtu-

space. Refer to Figure 1.2. The surface lies over the space–time domain: x ∈ I, t > 0. Alternately, one could regard the solution as a continuous sequence of

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