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Preface
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computational techniques on other courses subsequently realize the scope
of partial differential equations beyond paper and pencil.
Our approach is different. We introduce analytical and computational
techniques in the same book and thus in the same course. The main reason
for doing this is that the computer, developed to assist scientists in solv-
ing partial differential equations, has become commonly available and is
currently used in all practical applications of partial differential equations.
Therefore, a modern introduction to this topic must focus on methods suit-
able for computers. But these methods often rely on deep analytical insight
into the equations. We must therefore take great care not to throw away
basic analytical methods but seek a sound balance between analytical and
computational techniques.
One advantage of introducing computational techniques is that nonlinear
problems can be given more attention than is common in a purely analytical
introduction. We have included several examples of nonlinear equations in
addition to the standard linear models which are present in any introduc-
tory text. In particular we have included a discussion of reaction-diffusion
equations. The reason for this is their widespread application as important
models in various scientific applications.
Our aim is not to discuss the merits of different numerical techniques.
There are a huge number of papers in scientific journals comparing different
methods to solve various problems. We do not want to include such discus-
sions. Our aim is to demonstrate that computational techniques are simple
to use and often give very nice results, not to show that even better results
can be obtained if slightly different methods are used. We touch briefly
upon some such discussion, but not in any major way, since this really be-
longs to the field of numerical analysis and should be taught in separate
courses. Having said this, we always try to use the simplest possible nu-
merical techniques. This should in no way be interpreted as an attempt to
advocate certain methods as opposed to others; they are merely chosen for
their simplicity.
Simplicity is also our reason for choosing to present exclusively finite
difference techniques. The entire text could just as well be based on finite
element techniques, which definitely have greater potential from an appli-
cation point of view but are slightly harder to understand than their finite
difference counterparts.
We have attempted to present the material at an easy pace, explaining
carefully both the ideas and details of the derivations. This is particularly
the case in the first chapters but subsequently less details are included and
some steps are left for the reader to fill in. There are a lot of exercises
included, ranging from the straightforward to more challenging ones. Some
of them include a bit of implementation and some experiments to be done
on the computer. We strongly encourage students not to skip these parts.
In addition there are some “projects.” These are either included to refresh