Elements of Partial Differential Equations

292 Pages·2014·3.491 MB·English

by  Pavel Drábek, Gabriela Holubová

Pavel Drábek, Gabriela Holubová
Elements of Partial Differential Equations
De Gruyter Textbook

Pavel Drábek, Gabriela Holubová
Elements of
Partial Differential
Equations
2nd, Revised and Extended Edition
Mathematics Subject Classification 2010
35F10, 35J05, 35J08, 35K05, 35K08, 35L05, 35A09, 35B50, 35C05, 35C10
Authors
Prof. Dr. Pavel Drábek
University of West Bohemia
Department of Mathematics and NTIS
Univerzitni 8
30100 Pilsen
Czech Republic
[email protected]
Dr. Gabriela Holubová
University of West Bohemia
Department of Mathematics and NTIS
Univerzitni 8
30100 Pilsen
Czech Republic
[email protected]
ISBN 978-3-11-031665-0
e-ISBN 978-3-11-031667-4
Library of Congress Cataloging-in-Publication Data
A CIP catalog record for this book has been applied for at the Library of Congress.
Bibliographic information published by the Deutsche Nationalbibliothek
The Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie;
detailed bibliographic data are available in the Internet at http://dnb.dnb.de.
© 2014 Walter de Gruyter GmbH, Berlin/Boston
Printing and binding: CPI buch bücher.de GmbH, Birkach
♾Printed on acid-free paper
Printed in Germany
www.degruyter.com
Preface
The changes we made in this second edition of our book are inspired by our
teachingexperiencefromthePDEclass. Besidesminorrevisions,wealsoadded
some new material and slightly reorganized its exposition. We quote from the
Preface to the first edition in order to clarify the original motivation behind
the publication of this book:
“Nowadays, there are hundreds of books (textbooks as well as monographs)
devotedtopartialdifferentialequationswhichrepresentoneofthemostpower-
fultoolsofmathematicalmodelingofreal-worldproblems. Thesebookscontain
an enormous amount of material. This is, on the one hand, an advantage since
due to this fact we can solve plenty of complicated problems. However, on the
other hand, the existence of such an extensive literature complicates the orien-
tation in this subject for the beginners. It is difficult for them to distinguish
an important fact from a marginal information, to decide what to study first
and what to postpone for later time.
Ourbookisaddressednotonlytostudentswhointendtospecializeinmathe-
maticsduringtheirfurtherstudies,butalsotostudentsofengineering,economy
and applied sciences. To understand our book, only basic facts from calculus
and linear ordinary differential equations of the first and second orders are
needed. We try to present the first introduction to PDEs and that is why our
textisonaveryelementarylevel. Ouraimistoenablethereadertounderstand
what the partial differential equation is, where it comes from and why it must
be solved. We also want the reader to understand the basic principles which
are valid for particular types of PDEs, and to acquire some classical methods
of their solving. We limit ourselves only to the fundamental types of equations
and the basic methods. At present, there are many software packages that can
beusedforsolvingalotofspecialtypesofpartialdifferentialequationsandcan
be very helpful, but these tools are used only as a “black box” without deeper
understanding of the general principles and methods.
We would like to point out that we work with most of the notions only in-
tuitively and avoid intentionally some precise definitions and exact proofs. We
believe that this way of exposition makes the text more readable and the main
ideas and methods used become thus more lucid. Hence, the reader who is not
a mathematician, will be not disturbed by technical hypotheses, which in con-
crete models are usually assumed to be satisfied. On the other hand, we trust
that a mathematician will fill up these gaps easily or find the answers in more
vi Preface
specialized literature. In order to guide the reader, we put the most important
equations, formulas and facts in shaded boxes, so that he/she could better dis-
tinguish them from intermediate steps. Further, the text contains many solved
examples and illustrating figures. At the end of each chapter there are several
exercises. More complicated and theoretical ones are accompanied by hints,
the exercises based on calculations include the expected solution. We want
to emphasize that there may exist many different forms of a solution and the
reader can easily check the correctness of his/her results by substituting them
in the equation.
In any case, with the limited extent of this text, we do not claim our book
to be comprehensive. It is a selection which is subjective, but which – in our
opinion – covers the minimum which should be understood by everybody who
wants to use or to study the theory of PDEs more deeply. We deal only with
theclassical methods which area necessary startingpoint for furtherand more
advanced studies. Wedrawfrom our experience that springsnot only from our
teaching activities but also from our scientific work.
Finally, we would like to thank our colleagues Petr Girg, Petr Nečesal, Pavel
Krejčí, Alois Kufner, Herbert Leinfelder, Luboš Pick, Josef Polák and Robert
Plato, editor of De Gruyter, for careful reading of the manuscript and valuable
comments. Ourspecial thanksbelongtoJiříJarníkforcorrectingourEnglish.”
The exposition of the material in the second edition is as follows. In Chap-
ter 1, we present the basic conservation and constitutive laws. We also derive
some basic models. Chapter 2 is devoted to different notions of a solution, to
boundary and initial conditions and to the classification of PDEs. Chapter 3
deals with the first-order linear PDEs. Chapters 4–9 study the simplest pos-
sible forms of the wave, diffusion and Laplace (Poisson) equations and explain
standard methods how to find their solutions. In Chapter 10, we point out
the general principles of the above mentioned main three types of second-order
linear PDEs. Chapters 11–13 are devoted to the Laplace (Poisson), diffusion
and wave equations in higher dimensions. For the sake of brevity, we restrict
ourselves to equations in three dimensions with respect to the space variable.
Manystandardproceduresandmethodsfromourexpositionhavebeenwidely
used by many other authors. Let us mention, e.g., the textbooks and mono-
graphs by Strauss [21], Logan [15], Arnold [2], Bassanini and Elcrat [5], Evans
[7], Farlow [8], Franců [9], Míka and Kufner [16], Stavroulakis and Tersian [20]
and many others. We do not refer to all of them in our text in order not to
disturb the flow of ideas.
SomeoftheexerciseswerealsoborrowedfromthebooksbyAsmar[3],Barták
et al. [4], Keane [12], Logan [15], Snider [19], Stavroulakis and Tersian [20],
Strauss[21],Zauderer[24]. Wedonotlistthemexplicitlyasinthefirstedition.
Many of them appear in several books parallelly and it is thusdifficult to hunt
out the original source.
Preface vii
WewouldliketothankourcolleaguesMartinaLangerová,SarathSasi,Anoop
ThazheVeetil,KalappattilLakshmiandAthiyanathumPoytlReshmaforcare-
ful reading of the final version of this manuscript and for their valuable sug-
gestions. We also thank the editorial staff of De Gruyter for the agreeable
collaboration.
Pilsen, January 2014 Pavel Drábek and Gabriela Holubová

Contents
Preface v
1 Motivation, Derivation of Basic Mathematical Models 1
1.1 Conservation Laws ....................................... 1
1.1.1 Evolution Conservation Law ......................... 3
1.1.2 Stationary Conservation Law ........................ 5
1.1.3 Conservation Law in One Dimension .................. 5
1.2 Constitutive Laws ........................................ 6
1.3 Basic Models ............................................ 7
1.3.1 Convection and Transport Equation .................. 7
1.3.2 Diffusion in One Dimension ......................... 9
1.3.3 Heat Equation in One Dimension .................... 10
1.3.4 Heat Equation in Three Dimensions .................. 10
1.3.5 String Vibrations and Wave Equation in One Dimension 11
1.3.6 Wave Equation in Two Dimensions – Vibrating
Membrane ........................................ 15
1.3.7 Laplace and Poisson Equations – Steady States ........ 16
1.4 Exercises ................................................ 18
2 Classification, Types of Equations, Boundary and Initial
Conditions 21
2.1 Basic Types of Equations ................................. 21
2.2 Classical, General, Generic and Particular Solutions ........... 23
2.3 Boundary and Initial Conditions ........................... 26
2.4 Well-Posed and Ill-Posed Problems ......................... 28
2.5 Classification of Linear Equations of the Second Order ........ 29
2.6 Exercises ................................................ 32
3 Linear Partial Differential Equations of the First Order 37
3.1 Equations with Constant Coefficients ....................... 37
3.1.1 Geometric Interpretation – Method of Characteristics ... 38
3.1.2 Coordinate Method ................................ 42
3.1.3 Method of Characteristic Coordinates ................. 43