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ix
Preface
the student’s memory of results needed in this course, or to extend the
theories developed in the present text.
Given the fact that we introduce both numerical and analytical tools, we
have chosen to put little emphasis on modeling. Certainly, the derivation
of models based on partial differential equations is an important topic, but
it is also very large and can therefore not be covered in detail here.
The first seven chapters of this book contain an elementary course in
partial differential equations. Topics like separation of variables, energy ar-
guments, maximum principles, and finite difference methods are discussed
for the three basic linear partial differential equations, i.e. the heat equa-
tion, the wave equation, and Poisson’s equation. In Chapters 8–10 more
theoretical questions related to separation of variables and convergence of
Fourier series are discussed. The purpose of Chapter 11 is to introduce
nonlinear partial differential equations. In particular, we want to illustrate
how easily finite difference methods adopt to such problems, even if these
equations may be hard to handle by an analytical approach. In Chapter 12
we give a brief introduction to the Fourier transform and its application to
partial differential equations.
Some of the exercises in this text are small computer projects involving
a bit of programming. This programming could be done in any language.
In order to get started with these projects, you may find it useful to pick
up some examples from our web site, http://www.ifi.uio.no/˜pde/, where
you will find some Matlab code and some simple Java applets.
Acknowledgments
It is a great pleasure for us to thank our friends and colleagues for a lot of
help and for numerous discussions throughout this project. In particular,
we would like to thank Bent Birkeland and Tom Lyche, who both partici-
pated in the development of the basic ideas underpinning this book. Also
we would like to thank Are Magnus Bruaset, Helge Holden, Kenneth Hvis-
tendahl Karlsen, Jan Olav Langseth, Hans Petter Langtangen, Glenn Terje
Lines, Knut Mørken, Bjørn Fredrik Nielsen, Gunnar Olsen, Klas Samuels-
son, Achim Schroll, Wen Shen, Jan Søreng, and ˚ Asmund Ødeg˚ard for read-
ing parts of the manuscript. Finally, we would like to thank Hans Birkeland,
Truls Flatberg, Roger Hansen, Thomas Skjønhaug, and Fredrik Tyvand for
doing an excellent job in typesetting most of this book.
Oslo, Norway, April 1998. Aslak Tveito
Ragnar Winther