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40 1. Mathematical preliminaries

This concludes our presentation and discussion of all the elements that constitute
classical singular perturbation theory. However, this is far from a complete description
of all the techniques that are needed to solve differential equations. These will be
introduced as they are required; suffice it to record, at this stage, that one approach
that subsumes much that we have done thus far will be carefully developed later—the
method of multiple scales.

FURTHER READING
In this chapter we have introduced, with carefully chosen explanations and examples,
the basic ideas that are at the heart of singular perturbation theory. However, other
approaches are possible, emphasising one aspect or another, and a number of good texts
are available, which some readers may find both interesting and instructive. We offer
the following as some additional (but not essential) reading, with a few observations
about each. Of course, such a list is unlikely to be exhaustive, so I have included only
my own favourites; I apologise if yours has been omitted.
Any list of texts must include that written by van Dyke (1964), and especially its
annotated edition (1975); this provides an excellent introduction to the ideas, together
with their applications to many of the classical problems in fluid mechanics. Two
other good texts that present the material from a rather elementary stand-point are
Hindi (1991) and Bush (1992), although the former is somewhat sophisticated in
places; both these cover quite a wide range of applications. A text that also provides
an introduction, although perhaps in not so much detail as those already cited, is
Kevorkian & Cole (1996), but it offers an excellent introduction to the application of
these methods to various types of problem. (This publication is a revised and updated
edition of Kevorkian & Cole (1981), which was itself a rewritten and extended version
of Cole (1968), both of which are worth some exploration.) A nice introduction to
the subject, mainly following the work of Kaplun (see below), is given in Lagerstrom
(1988). Nayfeh (1973, 1981) provides many examples discussed in detail (and often
these are the same problem tackled in different ways); these two books are useful as
references to various applications, but perhaps are less useful as introductions to the
relevant underlying ideas.
A carefully presented discussion of how singular perturbation theory manifests itself
in ordinary differential equations, and a detailed description of methods of solution
with applications, can be found in O’Malley (1991). An outstanding collection of
many and varied problems, most of them physically interesting and important, is given
in Holmes (1995), but this text is probably best avoided by the novice. Eckhaus (1979)
presents a very formal and rigorous approach to the subject, and some interesting appli-
cations are included. Smith (1985) uses an instructive mixture of the formal approach
combined with examples and applications (although almost all of these relate to ordi-
nary, not partial, differential equations); a large number of references are also included.
A discussion of the matching principle, via expansion operators, can be found in the
papers by Fraenkel (1969).
We now turn to some texts that are more specialised in their content. A collection of
the work and ideas of Kaplun, particularly with reference to applications to problems in