Freiling G., Yurko V. Lectures on Differential Equations of Mathematical Physics: A First Course

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TURES ON DIFFERENTIAL EQUATIONS

OF MATHEMATICAL PHYSICS:

A FIRST COURSE

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TURES ON DIFFERENTIAL EQUATIONS

OF MATHEMATICAL PHYSICS:

A FIRST COURSE

G. FREILING

AND

V. YURKO

Nova Science Publishers, Inc.

New York

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Library of Congress Cataloging-in-Publication Data

Freiling, Gerhard, 1950

Lectures on the differential equations of mathematical physics : a ﬁrst course / Gerhard

Freiling and Vjatcheslav Yurko. p. cm.

ISBN 978-1-60741-907-5 (E-Book)

1. Differential equations. 2. Mathematical physics. I. Yurko, V. A. II. Title. QC20.7.D5F74 2008

530.15’535–dc22

2008030492

Published by Nova Science Publishers, Inc. ╬ New York

Contents

eface vii

Introduction 1

2. Hyperbolic Partial Differential Equations 15

3. Parabolic Partial Differential Equations 155

4. Elliptic Partial Differential Equations 165

5. The Cauchy-Kowalevsky Theorem 219

6. Exercises 225

eface

The

theory of partial differential equations of mathematical physics has been one of the

most important ﬁelds of study in applied mathematics. This is essentially due to the fre-

quent occurrence of partial differential equations in many branches of natural sciences and

engineering.

With much interest and great demand for applications in diverse areas of sciences, sev-

eral excellent books on differential equations of mathematical physics have been published

(see, for example, [1]-[6] and the references therein). The present lecture notes have been

written for the purpose of presenting an approach based mainly on the mathematical prob-

lems and their related solutions. The primary concern, therefore, is not with the general

theory, but to provide students with the fundamental concepts, the underlying principles,

and the techniques and methods of solution of partial differential equations of mathematical

physics. One of our main goal is to present a fairly elementary and complete introduction to

this subject which is suitable for the “ﬁrst reading” and accessible for students of different

specialities.

The material in these lecture notes has been developed and extended from a set of lec-

tures given at Saratov State University and reﬂects partially the research interests of the

authors. It is intended for graduate and advanced undergraduate students in applied math-

ematics, computer sciences, physics, engineering, and other specialities. The prerequisites

for its study are a standard basic course in mathematical analysis or advanced calculus,

including elementary ordinary differential equations.

Although various differential equations and problems considered in these lecture notes

are physically motivated, a knowledge of the physics involved is not necessary for under-

standing the mathematical aspects of the solution of these problems.

The book is organized as follows. In Chapter 1 we present the most important examples

of equations of mathematical physics, give their classiﬁcation and discuss formulations of

problems of mathematical physics. Chapter 2 is devoted to hyperbolic partial differential

equations which usually describe oscillation processes and give a mathematical descrip-

tion of wave propagation. The prototype of the class of hyperbolic equations and one of

the most important differential equations of mathematical physics is the wave equation.

Hyperbolic equations occur in such diverse ﬁelds of study as electromagnetic theory, hy-

drodynamics, acoustics, elasticity and quantum theory. In this chapter we study hyperbolic

equations in one-, two- and three-dimensions, and present methods for their solutions. In

Sections 2.1-2.5 we study the main classical problems for hyperbolic equations, namely,

the Cauchy problem, the Goursat problem and the mixed problems. We present the main

methods for their solutions including the method of travelling waves, the method of sep-

viii G.

Freiling and V. Yurko

aration of

variables, the method of successive approximations, the Riemann method, the

Kirchhoff method. Thus, Sections 2.1-2.5 contain the basic classical theory for hyperbolic

partial differential equations in the form which is suitable for the “ﬁrst reading”. Sections

2.6-2.9 are devoted to more speciﬁc modern problems for differential equations, and can be

ommited for the “ﬁrst reading”. In Sections 2.6-2.8 we provide an elementary introduction

to the theory of inverse problems. These inverse problems consist in recovering coefﬁcients

of differential equations from characteristics which can be measured. Such problems often

appear in various areas of natural sciences and engineering. Inverse problems also play

an important role in solving nonlinear evolution equations in mathematical physics such as

the Korteweg-de Vries equation. Interest in this subject has been increasing permanently

because of the appearance of new important applications, and nowadays the inverse prob-

lem theory develops intensively all over the world. In Sections 2.6-2.8 we present the main

results and methods on inverse problems and show connections between spectral inverse

problems and inverse problems for the wave equation. In Section 2.9 we provide the solu-

tion of the Cauchy problem for the nonlinear Korteweg-de Vries equation, for this purpose

we use the ideas and results from Sections 2.6-2.8 on the inverse problem theory.

In Chapter 3 we study parabolic partial differential equations which usually describe

various diffusion processes. The most important equation of parabolic type is the heat

equation or diffusion equation. The properties of the solutions of parabolic equations do

not depend essentially on the dimension of the space, and therefore we conﬁne ourselves

to considerations concerning the case of one spatial variable. Chapter 4 is devoted to el-

liptic equations which usually describe stationary ﬁelds, for example, gravitational, electro-

statical and temperature ﬁelds. The most important equations of elliptic type are the Laplace

and the Poisson equations. In this chapter we study boundary value problems for elliptic

partial differential equations and present methods for their solutions such that the Green’s

function method, the method of upper and lower functions, the method of integral equa-

tions, the variational method. In Chapter 5 we prove the general Cauchy-Kowalevsky theo-

rem which is a fundamental theorem on the existence of the solution of the Cauchy problem

for a wide class of systems of partial differential equations. Chapter 6 contains exercises

for the material covered in Chapters 1-4. The material here reﬂects all main types of equa-

tions of mathematical physics and represents the main methods for the solution of these

equations.

G. FREILING AND V. YURKO

Chapter

Intr

oduction

1.1. Some Examples of Equations of Mathematical Physics

The theory of partial differential equations arose from investigations of some important

problems in physics and mechanics. Most of the problems lead to second-order differential

equations which will be (mainly) under our consideration.

The relation

,. .. ,x

,u,

∂u

∂x

.. ,

∂u

∂x

∂

∂x

(i, j =

1,n)

= 0, (1.1.1)

which

connects

the real variables x

,. .. ,x

, an unknown function

u(x

,. .. ,x

) and its partial derivatives up to the second order, is called a partial differential

equation of the second order.

Equation (1.1.1) is called linear with respect to the second derivatives (or quasi-linear)

if it has the form

∑

i, j=1

i j

,. .. ,x

)

∂

∂x

+ F

.. ,x

,u,

∂u

∂x

.. ,

∂u

∂x

= 0. (1.1.2)

Equation

(1.1.2)

is called linear if it has the form

∑

i, j=1

i j

,. .. ,x

)

∂

∂x

∑

i=1

.. ,x

)

∂u

∂x

+ b

.. ,x

= f (x

,. .. ,x

). (1.1.3)

Let us give several examples of problems of mathematical physics which lead to partial

differential equations.