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asymptotic expansions (based on a parameter) are described. The notions of uniformity
and breakdown are introduced, together with the important concepts of scaling and
matching. Chapter 2 is devoted to routine and straightforward applications of the
methods developed in the previous chapter. In particular, we discuss how these ideas
can be used to find the roots of equations and how to integrate functions represented
by a number of matched asymptotic expansions. We then turn to the most significant
application of these methods: the solution of differential equations. Some simple regular
(i.e. not singular) problems are discussed first—these are rather rare and of no great
importance—followed by a number of examples of singular problems, including some
that exhibit boundary or transition layers. The role of scaling a differential equation is
given some prominence.
In Chapter 3, the techniques of singular perturbation theory are applied to more
sophisticated problems, many of which arise directly from (or are based upon) im-
portant examples in applied mathematics or mathematical physics. Thus we look at
nonlinear wave propagation, supersonic flow past a thin aerofoil, solutions of Laplace’s
equation, heat transfer to a fluid flowing through a pipe and an example taken from gas
dynamics. All these are classical problems, at some level, and are intended to show the
efficacy of these techniques. The chapter concludes with some applications to ordinary
differential equations (such as Mathieu’s equation) and then, as an extension of some
of the ideas already developed, the method of strained coordinates is presented.
One of the most general and most powerful techniques in the armoury of singular
perturbation theory is the method of multiple scales. This is introduced, explained and
developed in Chapter 4, and then applied to a wide variety of problems. These in-
clude linear and nonlinear oscillations, classical ordinary differential equations (such as
Mathieu’s equation—again—and equations with turning points) and the propagation
of dispersive waves. Finally, it is shown that the method of multiple scales can be used
to great effect in boundary-layer problems (first mentioned in Chapter 2).
The final chapter is devoted to a collection of worked examples taken from a wide
range of subject areas. It is hoped that each reader will find something of interest here,
and that these will show—perhaps more clearly than anything that has gone before—
the relevance and power of singular perturbation theory. Even if there is nothing of
immediate interest, the reader who wishes to become more skilled will find these a
useful set of additional examples. These are listed under seven headings: mechanical
& electrical systems; celestial mechanics; physics of particles & light; semi- and su-
perconductors; fluid mechanics; extreme thermal processes; chemical & biochemical
reactions.
Throughout the text, worked examples are used to explain and describe the ideas,
which are reinforced by the numerous exercises that are provided at the end of each of
the first four chapters. (There are no set exercises in Chapter 5, but the extensive ref-
erences can be investigated if more information is required.) Also at the end of each of
Chapters 1–4 is a section of further reading which, in conjunction with the references
cited in the body of the chapter, indicate where relevant reference material can be
found. The references (all listed at the end of the book) contain both texts and research
papers. Sections in each chapter are numbered following the decimal pattern, and