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xiv Preface

the inverse Reynolds number and the equations are based on the classical Navier-Stokes
equation of fluid mechanics. This analysis, coupled with small-Reynolds-number ap-
proximations that were developed at about the same time (1910), prepared the ground
for a century of singular perturbation work in fluid mechanics. But other fields over
the century also made important contributions, for example: integration of differential
equations, particularly in the context of quantum mechanics; the theory of nonlinear
oscillations; control theory; the theory of semiconductors. All these, and many others,
have helped to develop the mathematical study of singular perturbation theory, which
has, from the mid-1960s, been supported and made popular by a range of excellent
text books and research papers. The subject is now quite familiar to postgraduate stu-
dents in applied mathematics (and related areas) and, to some extent, to undergraduate
students who specialise in applied mathematics. Indeed, it is an essential tool of the
modern applied mathematician, physicist and engineer.
This book is based on material that has been taught, mainly by the author, to MSc
and research students in applied mathematics and engineering mathematics, at the
University of Newcastle upon Tyne over the last thirty years. However, the presentation
of the introductory and background ideas is more detailed and comprehensive than has
been offered in any particular taught course. In addition, there are many more worked
examples and set exercises than would be found in most taught programmes. The style
adopted throughout is to explain, with examples, the essential techniques, but without
dwelling on the more formal aspects of proof, et cetera; this is for two reasons. Firstly, the
aim of this text is to make all the material readily accessible to the reader who wishes
to learn and use the ideas to help with research problems and who (in all likelihood)
does not have a strong mathematical background (or who is not that concerned about
these niceties). And secondly, many of the results and solutions that we present cannot
be recast to provide anything that resembles a routine proof of existence or asymptotic
correctness. Indeed, in many cases, no such proof is available, but there is often ample
evidence that the results are relevant, useful and probably correct.
This text has been written in a form that should enable the relatively inexperienced
(or new) worker in the field of singular perturbation theory to learn and apply all the
essential ideas. To this end, the text has been designed as a learning tool (rather than
a reference text, for example), and so could provide the basis for a taught course. The
numerous examples and set exercises are intended to aid this process. Although it is
assumed that the reader is quite unfamiliar with singular perturbation theory, there
are many occasions in the text when, for example, a differential equation needs to be
solved. In most cases the solution (and perhaps the method of solution) are quoted, but
some readers may wish to explore this aspect of mathematical analysis; there are many
good texts that describe methods for solving (standard) ordinary and partial differential
equations. However, if the reader can accept the given solution, it will enable the main
theme of singular perturbation theory to progress more smoothly.
Chapter 1 introduces all the mathematical preliminaries that are required for the
study of singular perturbation theory. First, a few simple examples are presented that
highlight some of the difficulties that can arise, going some way towards explaining
the need for this theory. Then notation, definitions and the procedure of finding