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3. FURTHER APPLICATIONS
The ideas and techniques developed in Chapter 2 have taken us beyond elementary
applications, and as far as some methods that enable us to construct (asymptotic)
solutions of a few types of ordinary differential equation. The aim now is to extend
these methods, in particular, to partial differential equations. The first reaction to this
proposal might be that the move from ordinary to partial differential equations is a very
big step—and it can certainly be argued thus if we compare the solutions, and methods
of solution, for these two categories of equation. However, in the context of singular
perturbation theory, this is a misleading position to adopt. Without doubt we must have
some skills in the methods of solution of partial differential equations (albeit usually in
a reduced, simplified form), but the fundamental ideas of singular perturbation theory
are essentially the same as those developed for ordinary differential equations. The only
adjustment, because the solution will now sit in a domain of two or more dimensions,
is that an appropriate scaling may apply, for example, in only one direction and not in
the others, or in time and not in space.
In this chapter we will examine some fairly straightforward problems that are repre-
sented by partial differential equations, starting with an example of a regular problem.
The approach that we adopt will emphasise how the methods for ordinary differential
equations carry over directly to partial differential equations. In addition, we will take
the opportunity to write a little more about more advanced aspects of the solution
of ordinary differential equations, in part as a preparation for the very powerful and
general methods introduced in Chapter 4.