Page 14 -

PREFACE

The theory of singular perturbations has been with us, in one form or another, for a little
over a century (although the term ‘singular perturbation’ dates from the 1940s). The
subject, and the techniques associated with it, have evolved over this period as a response
to the need to find approximate solutions (in an analytical form) to complex problems.
Typically, such problems are expressed in terms of differential equations which contain
at least one small parameter, and they can arise in many fields: fluid mechanics, particle
physics and combustion processes, to name but three. The essential hallmark of a
singular perturbation problem is that a simple and straightforward approximation (based
on the smallness of the parameter) does not give an accurate solution throughout the
domain of that solution. Perforce, this leads to different approximations being valid in
different parts of the domain (usually requiring a ‘scaling’ of the variables with respect to
the parameter). This in turn has led to the important concepts of breakdown, matching,
and so on.
Mathematical problems that make extensive use of a small parameter were probably
first described by J. H. Poincaré (1854–1912) as part of his investigations in celestial
mechanics. (The small parameter, in this context, is usually the ratio of two masses.)
Although the majority of these problems were not obviously ‘singular’—and Poincaré
did not dwell upon this—some are; for example, one is the earth-moon-spaceship
problem mentioned in Chapter 2. Nevertheless, Poincaré did lay the foundations for
the technique that underpins our approach: the use of asymptotic expansions. The
notion of a singular perturbation problem was first evident in the seminal work of L.
Prandtl (1874–1953) on the viscous boundary layer (1904). Here, the small parameter is