Page 164 -

147

and this series converges for

where is some constant independent of Hence the expansion converges for
—but is to be no smaller than which is larger
than Thus the asymptotic expansion with (which maps to
is not just uniformly valid—it is convergent! This is an altogether unlooked-
for bonus; the method of strained coordinates, in this example, has proved to be a very
significant improvement on our standard matched-expansions approach.

Some further examples that make use of a strained coordinate are set as exercises Q3.19–
3.25. Although our example, E3.7, has demonstrated, to the full, the advantages of the
strained coordinate method, not all problems are quite this successful. Many ordinary
differential equations of the type discussed in E3.7 do indeed possess convergent series
for the coordinate—so the complete solution is no longer simply asymptotic—but
other problems may produce a strained coordinate that is uniformly valid only (without
being convergent). In the exercises, the question of convergence is not explored (but,
of course, the interested and skilful reader may wish to investigate this aspect).
In this text, at this stage, we have introduced many of the ideas and techniques
of singular perturbation theory, and have applied them—in the main—to ordinary
and partial differential equations of various types. In the next chapter we present one
further technique for solving singular perturbation problems. This is a method which
subsumes most of what we have presented so far and is, probably, the single most
powerful approach that we have available. When this has been completed, we will
employ all our methods in the examination of a selection of examples taken from a
number of different scientific fields.

FURTHER READING
A few regular perturbation problems that are described by partial differential equations
are discussed in van Dyke (1964, 1975) and also in Hinch (1991). A discussion of wave
propagation and breakdown, and especially with reference to supersonic flow past thin
aerofoils, can be found in van Dyke (above), Kevorkian & Cole (1981, 1996) and in
Holmes (1995). Any good text on compressible fluid mechanics will cover these ideas,
and much more, for the interested reader; we can recommend Courant & Friedrichs
(1967), Ward (1955), Miles (1959), Hayes & Probstein (1966) and Cox & Crabtree
(1965), but there are many others to choose from. A nice collection of partial differential
equations that exhibit boundary-layer behaviour are presented in Holmes (1995). Most
of the standard texts that discuss more general aspects of singular perturbation theory
include Mathieu’s equation, and related problems, as examples; good, dedicated works
on ordinary differential equations will give a broad and general background to the