Lectures on differential equations of mathematical physics

314 Pages·2008·1.789 MB·English

by  Freiling G., Yurko V.

L D E
ECTURES ON IFFERENTIAL QUATIONS
M P :
OF ATHEMATICAL HYSICS
A F C
IRST OURSE
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L D E
ECTURES ON IFFERENTIAL QUATIONS
M P :
OF ATHEMATICAL HYSICS
A F C
IRST OURSE
G. FREILING
AND
V. YURKO
NovaSciencePublishers,Inc.
NewYork
©2008 by Nova Science Publishers, Inc.

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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
Preface vii
1. Introduction 1
2. HyperbolicPartialDifferentialEquations 15
3. ParabolicPartialDifferentialEquations 155
4. EllipticPartialDifferentialEquations 165
5. TheCauchy-KowalevskyTheorem 219
6. Exercises 225
v

Preface
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-
quentoccurrenceofpartialdifferentialequationsinmanybranchesofnaturalsciencesand
engineering.
Withmuchinterestandgreatdemandforapplicationsindiverseareasofsciences,sev-
eralexcellentbooksondifferentialequationsofmathematicalphysicshavebeenpublished
(see, for example, [1]-[6] and the references therein). The present lecture notes have been
writtenforthepurposeofpresentinganapproachbasedmainlyonthemathematicalprob-
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,
andthetechniquesandmethodsofsolutionofpartialdifferentialequationsofmathematical
physics. Oneofourmaingoalistopresentafairlyelementaryandcompleteintroductionto
thissubjectwhichissuitableforthe“firstreading”andaccessibleforstudentsofdifferent
specialities.
Thematerialintheselecturenoteshasbeendevelopedandextendedfromasetoflec-
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, computersciences, physics, engineering, andotherspecialities. Theprerequisites
for its study are a standard basic course in mathematical analysis or advanced calculus,
includingelementaryordinarydifferentialequations.
Althoughvariousdifferentialequationsandproblemsconsideredintheselecturenotes
are physically motivated, a knowledge of the physics involved is not necessary for under-
standingthemathematicalaspectsofthesolutionoftheseproblems.
Thebookisorganizedasfollows. InChapter1wepresentthemostimportantexamples
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,elasticityandquantumtheory. Inthischapterwestudyhyperbolic
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.FreilingandV.Yurko
aration of variables, the method of successive approximations, the Riemann method, the
Kirchhoffmethod. Thus,Sections2.1-2.5containthebasicclassicaltheoryforhyperbolic
partial differential equations in the form which is suitable for the “first reading”. Sections
2.6-2.9aredevotedtomorespecificmodernproblemsfordifferentialequations,andcanbe
ommitedforthe“firstreading”. InSections2.6-2.8weprovideanelementaryintroduction
tothetheoryofinverseproblems. Theseinverseproblemsconsistinrecoveringcoefficients
ofdifferentialequationsfromcharacteristicswhichcanbemeasured. Suchproblemsoften
appear in various areas of natural sciences and engineering. Inverse problems also play
animportantroleinsolvingnonlinearevolutionequationsinmathematicalphysicssuchas
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-
lemtheorydevelopsintensivelyallovertheworld. InSections2.6-2.8wepresentthemain
results and methods on inverse problems and show connections between spectral inverse
problemsandinverseproblemsforthewaveequation. InSection2.9weprovidethesolu-
tionoftheCauchyproblemforthenonlinearKorteweg-deVriesequation,forthispurpose
weusetheideasandresultsfromSections2.6-2.8ontheinverseproblemtheory.
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-
lipticequationswhichusuallydescribestationaryfields,forexample,gravitational,electro-
staticalandtemperaturefields. ThemostimportantequationsofelliptictypearetheLaplace
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,thevariationalmethod. InChapter5weprovethegeneralCauchy-Kowalevskytheo-
remwhichisafundamentaltheoremontheexistenceofthesolutionoftheCauchyproblem
for a wide class of systems of partial differential equations. Chapter 6 contains exercises
forthematerialcoveredinChapters1-4. Thematerialherereflectsallmaintypesofequa-
tions of mathematical physics and represents the main methods for the solution of these
equations.
G. FREILING AND V. YURKO