Applied partial differential equations logan solutions

Partial Differential Equation Problem

1) Problem 1 in Section 4.4 of APDE [15pts, look at example 8.12 in EPDE].
2) Problem 3 in Section 4.7 of APDE [20pts = 10pts for each half (find the formula and then show it is...)].
3) Problem 5 in Section 4.7 of APDE [15pts]. Here Q and u_0 are constants.
4) Problem 8.13 in EPDE

Advisory Board

Colin Adams, Williams College, Williamstown, MA, USA

Alejandro Adem, University of British Columbia, Vancouver, BC, Canada

Ruth Charney, Brandeis University, Waltham, MA, USA

Irene M. Gamba, The University of Texas at Austin, Austin, TX, USA

Roger E. Howe, Yale University, New Haven, CT, USA

David Jerison, Massachusetts Institute of Technology, Cambridge, MA, USA

Jeffrey C. Lagarias, University of Michigan, Ann Arbor, MI, USA

Jill Pipher, Brown University, Providence, RI, USA

Fadil Santosa, University of Minnesota, Minneapolis, MN, USA

Amie Wilkinson, University of Chicago, Chicago, IL, USA

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

and novel approaches. The books include motivation that guides the reader to

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

qc322@nyu.edu

J. David Logan

Applied Partial Differential Equations

qc322@nyu.edu

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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Springer is part of Springer Science+Business Media (www.springer.com)

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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

qc322@nyu.edu

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

qc322@nyu.edu

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.

qc322@nyu.edu

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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1.1 PDE Models 9

time snapshots. That is, for each fixed time t0, u(x, t0) is a function of x alone

and thus represents a time snapshot of the solution. In different words, u(x, t0)

is the trace of the solution surface u = u(x, t) taken in the t = t0-plane. In

some contexts, u(x, t0) is interpreted as a wave profile, or signal, at time t0. In

this way a solution u(x, t) of (1.6) can be regarded a continuous sequence of

evolving wave forms evolving in time.

Bibliographic Notes. There are dozens of excellent elementary PDE books

written at about the same level as this one. We especially mention Farlow

(1993) and Strauss (1992). A more advanced treatment is given by McOwen

(2003). Nonlinear PDEs at the beginning level are treated in detail in Debnath

(1997) or Logan (2008). PDE models occur in every area of the pure and applied

sciences. General texts involving modeling in engineering and science are Lin

& Segel (1989), Holmes (2011), and Logan (2013).

EXERCISES

1. Verify that a solution to the heat equation (1.1) on the domain −∞ < x < ∞, t > 0 is given by

u(x, t) = 1√ 4πkt

e−x 2/4kt.

For a fixed time, the reader should recognize this solution as a bell-shaped

curve. (a) Pick k = 0.5. Use software to sketch several time snapshots on

the same set of coordinate axes to show how the temperature profile evolves

in time. (b) What do the temperature profiles look like as t → 0? (c) Sketch the solution surface u = u(x, t) in a domain −2 ≤ x ≤ 2, 0.1 < t < 4. (d) How does changing the parameter k affect the solution?

2. Verify that u(x, y) = ln √

x2 + y2 satisfies the Laplace equation

uxx + uyy = 0

for all (x, y) �= (0, 0).

3. Find the general solution of the equation uxy(x, y) = 0 in terms of two

arbitrary functions.

4. Derive the solution u = u(x, y) = axy + bx+ cy + d (a, b, c, d constants),

of the PDE

u2xx + u 2 yy = 0.

Observe that the solution does not explicitly contain arbitrary functions.

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10 1. The Physical Origins of Partial Differential Equations

5. Find a function u = u(x, t) that satisfies the PDE

uxx = 0, 0 < x < 1, t > 0,

subject to the boundary conditions

u(0, t) = t2, u(1, t) = 1, t > 0.

6. Verify that

u(x, t) = 1

2c

∫ x+ct x−ct

g(s)ds

is a solution to the wave equation utt = c 2uxx, where c is a constant and g is

a given continuously differentiable function. Hint: Here you will need to use

Leibniz’s rule for differentiating an integral with respect to a parameter

that occurs in the limits of integration:

d

dt

∫ b(t) a(t)

F (s)ds = F (b(t))b′(t)− F (a(t))a′(t).

7. For what values of a and b is the function u(x, t) = eat sin bx a solution to

the heat equation

ut = kuxx.

8. Find the general solution to the equation uxt + 3ux = 1. Hint: Let v = ux and solve the resulting equation for v; then find u.

9. Show that the nonlinear equation ut = u 2 x+uxx can be reduced to the heat

equation (1.1) by changing the dependent variable to w = eu.

10. Show that the function u(x, y) = arctan(y/x) satisfies the two-dimensional

Laplace’s equation uxx + uyy = 0.

11. Show that e−ξy sin(ξx), x ∈ R, y > 0, is a solution to uxx + uyy = 0 for any value of the parameter ξ. Deduce that

u(x, y) =

∫ ∞ 0

c(ξ)e−ξy sin(ξx)dξ

is a solution to the same equation for any function c(ξ) that is bounded

and continuous on [0,∞). Hint: The hypotheses on c allow you to bring a derivative under the integral sign. [This exercise shows that taking inte-

grals of solutions sometimes gives another solution; integration is a way of

superimposing, or adding, a continuum of solutions.]

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1.1 PDE Models 11

12. Linear, homogeneous PDEs with constant coefficients admit complex solu-

tions of the form

u(x, t) = Aei(kx−ωt),

which are called plane waves. The real and imaginary parts of this com-

plex function,

Re(u) = A cos(kx− ωt), Im(u) = A sin(kx− ωt),

give real solutions. The constant A is the amplitude, k is the wave num-

ber, and ω is the temporal frequency. When the plane wave form is

substituted into a PDE there results a dispersion relation of the form

ω = ω(k),

which states how the frequency depends upon the wave number. For the

following PDEs find the dispersion relation and determine the resulting

plane wave; sketch wave profiles at different times.

a) ut = Duxx.

b) utt = c 2uxx.

c) ut + uxxx = 0.

d) ut = iuxx. (Here, i is the complex number i 2 = −1.)

e) ut + cux = 0.

13. Second-order linear homogeneous equations with constant coefficients are

often classified by their dispersion relation ω = ω(k) (see Exercise 12).

If ω(k) is complex, the PDE is called diffusive, and if ω(k) is real and

ω′′(k) �= 0, the PDE is called dispersive. The diffusion equation is dif- fusive; the wave equation is neither diffusive or dispersive. The term dis-

persive means that the speed ω(k)/k of a plane wave u = Aei(kx−ω(k)t)

travels depends upon the wave number k. So waves of different wavelength

travel at different speeds, and thus disperse. Classify the PDEs in (a)–(e)

of Exercise 12 according to this scheme.

14. Find plane wave solutions to the Kuromoto–Sivashinsky equation

ut = −u− δuxx − uxxxx, δ > 0.

Find the dispersion relation and classify the equation according to the

scheme of the preceding exercise. Describe the solutions and plot δ as a

function of the wave number k to determine when the growth rate of a

solution is zero. For which wave numbers will the solution decay?

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12 1. The Physical Origins of Partial Differential Equations

1.2 Conservation Laws

Many PDEs come from a basic balance, or conservation law. A conservation

law is a mathematical formulation of the fact that the rate at which a quantity

changes in a given domain must equal the rate at which the quantity flows

across the boundary (in minus out) plus the rate at which the quantity is

created within the domain. For example, consider a population of a certain

animal species in a fixed geographical region. The rate of change of the animal

population must equal the rate at which animals migrate into the region, minus

the rate at which they migrate out, plus the birth rate, minus the death rate.

Such a statement is a verbal expression of a balance, or conservation, law. One

can make similar kinds of statements for many quantities—energy, the mass of

a chemical species, the number of automobiles on a freeway, and so on.

Figure 1.3 Tube with cross-sectional area A shown with arbitrary cross-

section at x (shaded). The lateral sides are insulated, and the physical quantities

vary only in the x-direction and in time. All quantities are constant over any

cross-section

To quantify such statements we require some notation. Let the state variable

u = u(x, t) denote the density of a given quantity (mass, energy, animals,

automobiles, etc.); density is usually measured in amount per unit volume, or

sometimes amount per unit length. For example, energy density is measured in

energy units per volume. We assume that any variation in the state be restricted

to one spatial dimension. That is, we assume a one-dimensional domain (say, a

tube, as in Figure 1.3 where each cross-section is labeled by the spatial variable

x; we require that there be no variation of u(x, t) within the cross-section at

x. Implicit is the assumption that the quantity in the tube is abundant and

continuous enough in x so that it makes sense to define its density at each

section of the tube. The amount of the quantity in a small section of width dx

is u(x, t)Adx, where A is the cross-sectional area of the tube. Further, we let

φ = φ(x, t) denote the flux of the quantity at x, at time t. The flux measures

the amount of the quantity crossing the section at x at time t, and its units

are given in amount per unit area, per unit time. Thus, Aφ(x, t) is the actual

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1.2 Conservation Laws 13

amount of the quantity that is crossing the section at x at time t. By convention,

flux is positive if the flow is to the right, and negative if the flow is to the left.

Finally, let f = f(x, t) denote the given rate at which the quantity is created, or

destroyed, within the section at x at time t. The function f is called a source

term if it is positive, and a sink if it is negative; it is measured in amount

per unit volume per unit time. Thus, f(x, t)Adx represents the amount of the

quantity that is created in a small width dx per unit time.

A conservation law is a quantitative relation between u, φ, and f . We can

formulate the law by considering a fixed, but arbitrary, section a ≤ x ≤ b of the tube (Figure 1.3) and requiring that the rate of change of the total amount

of the quantity in the section must equal the rate at which it flows in at x = a,

minus the rate at which it flows out at x = b, plus the rate at which it is created

within a ≤ x ≤ b. In mathematical symbols,

d

dt

∫ b a

u(x, t)Adx = Aφ(a, t) −Aφ(b, t) + ∫ b a

f(x, t)Adx. (1.7)

This equation is the fundamental conservation law; it is an integral expression

of the basic fact that there must be a balance between how much goes in, how

much goes out, and how much is changed. Because A is constant, it may be

canceled from the formula.

Equation (1.7) is an integral law. However, if the functions u and φ are

sufficiently smooth, then it may be reformulated as a PDE, which is a local

law. For example, if u has continuous first partial derivatives, then the time

derivative on the left side of (1.7) may be brought under the integral sign to

obtain d

dt

∫ b a

u(x, t)dx =

∫ b a

ut(x, t)dx.

If φ has continuous first partials, then the fundamental theorem of calculus can

be applied to write the change in flux as the integral of a derivative, or

φ(a, t)− φ(b, t) = − ∫ b a

φx(x, t)dx.

Therefore, (1.7) may be written

∫ b a

(ut(x, t) + φx(x, t)− f(x, t))dx = 0.

Because a ≤ x ≤ b can be any interval whatsoever, and because the integrand is continuous, it follows that the integrand must vanish identically, or

ut(x, t) + φx(x, t) = f(x, t). (1.8)

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14 1. The Physical Origins of Partial Differential Equations

Equation (1.8) is a local version of (1.7), obtained under the assumption that u

and φ are continuously differentiable; it is a PDE model describing the relation

between the density the quantity, its flux, and the rate at which the quantity

is created. We call the PDE (1.8) the fundamental conservation law. The

f -term is called the source term, and the φ-term is called the flux term. In

(1.8) we usually drop the understood notational dependence on x and t and

just write ut + φx = f for simplicity.

Before studying some examples, we make some general comments. The flux

φ and source f are functions of x and t, but their dependence on x and t may be

through dependence upon the density u itself. For example, the source term f

may be given as a function of density via f = f(u), where, of course, u = u(x, t).

Similarly, φ may depend on u. These dependencies lead to nonlinear models.

Next, we observe that (1.8) is a single equation, yet there are two unknowns, u

and φ (the form of the source f is assumed to be prescribed). This implies that

another equation is required that relates u and φ. Such equations are called

constitutive relations (or equations of state), and they arise from physical

assumptions about the medium itself.

The Method of Characteristics

In this section, in the context of the advection of materials through a

medium, we introduce the basic method for solving first order PDEs, the

method of characteristics.

Example 1.6

(Advection) A model where the flux is proportional to the density itself, that

is,

φ = cu,

where c is a constant, is called an advection model. Notice that c must have

velocity units (length per time). In this case the conservation law (1.8) becomes,

in the absence of sources (f = 0),

ut + cux = 0. (1.9)

Equation (1.9) is called the advection equation. The reader should verify,

using the chain rule, that the function

u(x, t) = F (x− ct) (1.10)

is a solution to (1.9) for any differentiable function F . Such solutions (1.10)

are called right-traveling waves because the graph of F (x− ct) is the graph of F (x) shifted to the right ct spatial units. So, as time t increases, the wave

profile F (x) moves to the right, undistorted, with its shape unchanged, at speed

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1.2 Conservation Laws 15

c. Figure 1.4 shows two ways of viewing a right-traveling wave. Intuitively, (1.9)

describes what we usually call advection. For example, a density cloud of smoke

carried by the bulk motion of the wind would represent an advection process.

Other common descriptive terms for this kind of movement are transport and

convection. �

Remark. The function u(x, t) = F (z), z = x − ct, where F is an arbitrary function, is called the general solution of the advection equation ut+cux = 0,

c > 0. So the general solution is a right traveling wave. �

a b

Figure 1.4 Two views of a traveling wave: (a) wave snapshots (profiles) at

two different times, and (b) moving forward in space-time

Remark. If the flux is a nonlinear function of the density, that is, φ = φ(u),

then the conservation law (1.8) (again take f = 0) takes the form

ut + φ(u)x = ut + φ ′(u)ux = 0. (1.11)

If φ(u) is not linear in u, then (1.11) is a model of nonlinear advection,

and such models are more difficult to analyze. Later in this section we examine

simple nonlinear models. Logan (2008, 2013) can be consulted for an detailed

treatment of nonlinear equations. �

Example 1.7

(Advection and decay) Recall from elementary differential equations that

decay (e.g., radioactive decay) is modeled by the law du/dt = −λu, where λ is the decay rate. Thus, a substance advecting through a tube at positive velocity

c (for example, a radioactive chemical dissolved in water flowing at speed c) is

modeled by the advection–decay equation

ut + cux = −λu. (1.12)

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16 1. The Physical Origins of Partial Differential Equations

Here, f = −λu is the source term (specifically, the decay term) and φ = cu is the flux term in the conservation law (1.8). �

Example 1.8

The pure initial value problem for the advection equation is

ut + cux = 0, x ∈ R, t > 0, (1.13) u(x, 0) = u0(x), x ∈ R, (1.14)

where u0(x) is a given initial density, or signal. From (1.10) it follows that the

solution to (1.13)–(1.14) is

u(x, t) = u0(x− ct).

Physically, the initial density signal moves to the right at speed c. Alternatively,

we think of the density signal moving along the family of parallel straight lines

ξ = x− ct = constant in space–time. These lines, called characteristics, are the curves that carry the signal. For the pure advection equation, the solution

moves in such a way that the strength u of the density remains constant along

any characteristic curve. �

Now we solve a general advection equation of the form

ut + cux + au = f(x, t), (1.15)

where a and c are constants and f is a given function. Because the advection

equation propagates signals at speed c, it is reasonable to transform this equa-

tion to a moving coordinate system. Thus, let ξ and τ be new independent

variables, called characteristic coordinates, defined by

ξ = x− ct, τ = t.

We think of ξ as a moving coordinate that travels (or advects) with the signal. If

we denote u(x, t) in the new variables by U(ξ, τ) (that is, U(ξ, τ) = u(ξ+cτ, τ),

or u(x, t) = U(x− ct, t)), then the chain rule gives

ut = Uξξt + Uττt = −cUξ + Uτ

and

ux = Uξξx + Uττx = Uξ.

So equation (1.15) becomes

Uτ + aU = F (ξ, τ),

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1.2 Conservation Laws 17

where F (ξ, τ) = f(ξ + cτ, τ). This PDE contains derivatives with respect to

only one of its independent variables and therefore can be regarded as an ODE

with the other independent variable as a parameter. Thus it can be solved by

ODE methods, which are reviewed in the Appendix. It has the form of a linear

equation, and so it can be solved by multiplying by the integrating factor eaτ

and integrating with respect to τ . An example illustrates this procedure.

Example 1.9

Find the general solution of

ut + 2ux − u = t.

Let ξ = x−2t, τ = t. In these characteristic coordinates the equation becomes

Uτ − U = τ.

Multiplying by e−τ gives ∂

∂τ (Ue−τ ) = τe−τ .

Integrating,

Ue−τ =

∫ τe−τdτ = −(1 + τ)e−τ + g(ξ),

where g is an arbitrary function. Transforming back to xt variables then gives

the general solution

u(x, t) = −(1 + t) + g(x− 2t)et. �

Remark. A more general reaction–advection PDE

ut + cux = f(x, t, u),

where the source term depends on u, can in principle be solved by making the

same transformation ξ = x − ct, τ = t to turn it into a simpler equation of the form

Uτ = F (ξ, τ, U).

In these characteristic coordinates the PDE simplifies to the form of an ODE

with only one derivative. � The important point in the preceding discussion is that the advection oper-

ator ∂∂t + c ∂ ∂x simplifies to

∂ ∂τ in characteristic coordinates; thus, changing

independent variables is a strategy for handling equations having advection

operators. This solution technique is called the method of characteristics.

A similar characteristic method can be applied to solve the equation

ut + c(x, t)ux = f(x, t, u).

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18 1. The Physical Origins of Partial Differential Equations

In this case, we think of c(x, t) as the advection speed in a heterogeneous

medium; it replaces the constant c in the previous problem and now depends

on the location in the medium and on time. The characteristic coordinates are

given by ξ = ξ(x, t), τ = t, where ξ(x, t) = C is the general solution of the

ODE dx

dt = c(x, t).

In these new coordinates we see that the original PDE transforms into an

equation of the form

Uτ = F (ξ, τ, U),

where U = U(ξ, τ). (Verify this.) In theory this equation can be solved for U

and then we can substitute for ξ and τ in terms of x and t to obtain u = u(x, t).

Example 1.10

Consider the PDE

ut + 2tux = 0.

Here, c(x, t) = 2t. Setting dxdt = 2t and solving gives x−t2 = C. Thus, ξ = x−t2. The characteristic coordinates are

ξ = x− t2, τ = t,

and we find by the chain rule that

ut = Uξ(−2t) + Uτ , ux = Uξ.

Therefore ut +2tux = Uτ and the original PDE transforms into Uτ = 0. Hence

U = g(ξ), where g is an arbitrary function. The general solution to the given

PDE is thus u(x, t) = g(x− t2). Observe that the solution is constant along the set of characteristic curves (parabolas in space–time) x− t2 = C. �

Example 1.11

We solve the advection equation in the first quadrant with both initial and

boundary conditions. Consider the equation

ut + 2ux = 0, x > 0, t > 0,

subject to the initial and boundary conditions

u(x, 0) = e−x, u(0, t) = (1 + t2)−1.

We know the general solution is u(x, t) = F (x − 2t), where F is arbitrary. The idea is to let the PDE carry the boundary signals into the region; so

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1.2 Conservation Laws 19

we determine the arbitrary function F separately in x > 2t and in x < 2t.

The separating characteristic x = 2t is called the leading signal. For x > 2t,

ahead of the leading signal, we apply the initial condition at u(x, 0) because

the characteristics in that region come from the x-axis:

u(x, 0) = F (x) = e−x.

Then

u(x, t) = e−(x−2t), x > 2t.

In the domain 0 < x < 2t we apply the boundary condition at u(0, t) because

the characteristics in that region come from the t-axis:

u(0, t) = F (−2t) = 1 1 + t2

.

To determine the form of F let s = −2t. Then t = −s/2 and

F (s) = 1

1 + s2/4 .

Therefore, the solution in x < 2t is

u(x, t) = 1

1 + (x− 2t)2/4 , 0 ≤ x < 2t.

Notice that the solution is continuous along the leading characteristic x = 2t,

but the derivatives have discontinuities, giving a non-smooth solution. This

phenomenon is common for first-order PDEs. Discontinuities are carried along

the characteristics. �

In Section 1.4 there is an expanded treatment of advection in a biological

context.

Nonlinear Advection*

In the last last few pages we studied two simple model advection equations,

ut + cux = 0 and ut + c(x, t)ux = 0. Both are first-order and linear. Now

we study the same type of equation when a nonlinear nonlinear flux φ(u) is

introduced. Then the conservation law becomes

ut + φ(u)x = 0.

Using the chain rule we find φ(u)x = φ ′(u)ux. Denoting c(u) = φ

′(u) gives,

after appending an initial condition, the IVP

ut + c(u)ux = 0, x ∈ R, t > 0, (1.16) u(x, 0) = φ(x), x ∈ R. (1.17)

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20 1. The Physical Origins of Partial Differential Equations

We think of u as a density and c(u) as the speed that waves propagate. In many

physical problems the speed that waves propagate increases with the density,

so we assume for now that c′(u) > 0.

Consistent with the solution method for linear advection equations, we

define the characteristic curves as integral curves of the differential equation

dx

dt = c(u). (1.18)

Then along a particular characteristic curve x = x(t) we have

du

dt (x(t), t) = ux(x(t), t)c(u(x(t)) + ut(x(t), t) = 0.

Therefore, like linear equations, u is constant along the characteristic curves.

The characteristics curves are straight lines because

d2x

dt2 =

d

dt

( dx

dt

) =

d

dt c(u(x(t)) = c′(u)

du

dt = 0.

In the nonlinear case, however, the speed of the characteristic curves as defined

by (1.18) depends on the value u of the solution at a given point. To find the

equation of the characteristic C through (x, t) we note that its speed is

dx

dt = c(u(ξ, 0)) = c(φ(ξ))

(see Figure 1.5). In the xt coordinate system, the speed of a signal is the

reciprocal of its slope. This results from applying (1.18) at (ξ, 0). Thus, after

integrating, the characteristic curve is given by

x = c(φ(ξ))t + ξ. (1.19)

Equation (1.19) defines ξ = ξ(x, t) implicitly as a function of x and t, and the

solution u(x, t) of the initial value problem (1.16) and (1.17) is given by

u(x, t) = φ(ξ) (1.20)

where ξ is defined by (1.19).

In summary, for the nonlinear advection equation (1.16):

(a) Every characteristic curve is a straight line.

(b) The solution u is constant on each such characteristic.

(c) The speed of each characteristic, is equal to the value of c(u) on that

characteristic.

(d) The speed c(u) is the speed that signals, or waves, are propagated in the

system.

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1.2 Conservation Laws 21

Figure 1.5 A diagram showing characteristics, or signals, moving at different

speeds; each characteristic carries a constant value of u determined by its initial

value at t = 0, at the point (ξ, 0). The equation of the characteristic shown,

from (ξ, 0) to (x, t), is given by (1.19). Its slope in the xt coordinate system is

the inverse of its speed

Figure 1.6 Initial wave profile in Example 1.12

Example 1.12

Consider the initial value problem

ut + uux = 0, x ∈ R, t > 0,

u(x, 0) = φ(x) =

⎧ ⎨ ⎩

2, x < 0,

2− x, 0 ≤ x ≤ 1, 1, x > 1.

The initial curve is sketched in Figure 1.6. Since c(u) = u the characteristics

are straight lines emanating from (ξ, 0) with speed c(φ(ξ)) = φ(ξ). These are

plotted in Figure 1.7. For x < 0 the lines have speed 2; for x > 1 the lines have

speed 1; for 0 ≤ x ≤ 1 the lines have speed 2− x and these all intersect at (2, 1). Immediately one observes that a solution cannot exist for t > 1, because

the characteristics cross at that time and they carry different constant values of

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22 1. The Physical Origins of Partial Differential Equations

Figure 1.7 Characteristic diagram showing colliding characteristics

Figure 1.8 Solution surface with time profiles

u. Figure 1.8 shows several wave profiles that indicate steepening of the signal

as it propogates. At t = 1 the wave breaks, which is the first instant when the

solution would become multiple valued. To find the solution for t < 1 we note

that u(x, t) = 2 for x < 2t and u(x, t) = 1 for x > t + 1. For 2t < x < t + 1

equation (1.19) becomes

x = (2− ξ)t+ ξ,

which gives

ξ = x− 2t 1− t .

Equation (1.20) then yields

u(x, t) = 2− x 1− t , 2t < x < t+ 1, t < 1.

This explicit form of the solution also indicates the difficulty at the breaking

time t = 1. �

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1.2 Conservation Laws 23

In general the initial value problem (1.16)–(1.17) may have a solution only

up to a finite time tb, which is called the breaking time. Let us assume in

addition to c′(u) > 0 that the initial wave profile satisfies the conditions

φ(x) ≥ 0, φ′(x) < 0.

At the time when breaking occurs the gradient ux will become infinite. To

compute ux we differentiate (1.19) implicitly with respect to x to obtain

ξx = 1

1 + c′(φ(ξ))φ′(ξ)t .

Then from (1.20)

ux = φ′(ξ)

1 + c′(φ(ξ))φ′(ξ)t .

The gradient catastrophe will occur at the minimum value of t, which makes

the denominator zero. Hence

tb = min ξ

−1 φ′(ξ)c′(φ(ξ))

, tb ≥ 0.

In the last example, c(u) = u and φ(ξ) = 2−ξ. Hence φ′(ξ)c′(φ(ξ)) = (−1)(1) = −1 and tb = 1 is the time when breaking occurs.

In summary we showed that the nonlinear partial differential equation

ut + c(u)ux = 0, c ′(u) > 0

propagates the initial wave profile at a speed c(u), which depends on the value

of the solution u at a given point. Since c′(u) > 0, large values of u are propa-

gated faster than small values and distortion of the wave profile occurs. This is

consistent with our earlier remarks. Wave distortion can occur and shock waves,

or discontinuities, develop in materials because of the property of the medium

to transmit signals more rapidly at higher levels of stress or pressure. Mathe-

matically, distortion and the development of shocks or discontinuous solutions

are distinctively nonlinear phenomena caused by the advection term c(u)ux.

Example 1.13

(Implicit solution) When the advection speed is constant, we showed that

ut + cux = 0

has general solution given explicitly by u = F (x− ct), where F is an arbitrary function. A similar type implicit solution can occur for the nonlinear equation

ut + c(u)ux = 0.

qc322@nyu.edu

24 1. The Physical Origins of Partial Differential Equations

In an exercise, the reader is asked to show, using the chain rule, that the

expression

u = F (x− c(u)t),

defines the solution u = u(x, t) implicitly, when it exists. The arbitrary function

F is determined, for example, by an initial condition. �

Example 1.14

Consider the PDE

ut + u 2ux = 0.

The general solution is given implicitly by u = F (x − u2t), which is easily verified. If u(x, 0) = x, then F (x) = x and u = x− u2t. Solving for u gives

u = 1

2t

( −1±

√ 1 + 4tx

)

= 1

2t

( −1 +

√ 1 + 4tx

) ,

where we have taken the positive square root to meet the initial condition. The

solution is valid for t < −1/4x. (See the Exercises.) �

Example 1.15