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Although we have touched on methods for finding roots of equations, and on inte-
gration, the main thrust has been to develop basic techniques for solving differential
equations—the most important use, by far, of these methods. We shall devote the rest
of the text to extending and developing the methods for solving differential equations,
both ordinary and partial, and their applications to many practical problems that are
encountered in various branches of mathematical modelling. Many of the examples
and exercises in this chapter are, perforce, invented to make a point or to test ideas;
however, a few of the later exercises that are included at the end of this chapter (see
Q2.27–2.35) begin to employ the techniques in physically relevant problems. In the
next chapter, we will show how these ideas can be applied to a broader class of prob-
lems and, in particular, begin our discussion of partial differential equations. This will
allow us, in turn, to begin to extend the applications of singular perturbation theory
to more problems which arise within a physically relevant context.
FURTHER READING
A few of the existing texts include a discussion of the methods for finding roots of
equations, and for evaluating integrals of functions which contain a small parameter;
in particular, the interested reader is directed to Holmes (1995) and Hinch (1991).
Differential equations that give rise to regular problems are given little consideration—
they are quite rare, after all—but some can be found in Holmes (1995) and in Georgescu
(1995). We have already mentioned those texts that present a more formal approach to
perturbation theory (Eckhaus, 1979; Smith, 1985; O’Malley, 1991), but some further
developments along these lines are also given in Chang & Howes (1984).
The whole arena of scaling with respect to a parameter, and we should include here
the construction of non-dimensional variables, is fairly routine but very powerful.
These ideas play a rôle, not only in the identification of asymptotic regions (as we have
seen), but also in providing more general pointers to the construction of solutions. A
very thorough introduction to these ideas, and their connection with asymptotics, can
be found in Barenblatt (1996). A discussion of the applications of group theory to the
study of differential equations is likely to be available in any good, relevant text; one
such, which emphasises precisely the application to differential equations, is Dresner
(1999).
The nature of a boundary layer (which is, for our current interest, limited to a
property of certain ordinary differential equations) is described at length, and carefully,
in most available texts on singular perturbation theory. We can mention, as examples
of the extent and depth of what is discussed, the excellent presentations on this subject
given by Smith (1985) and Holmes (1995). The determination of the position of
a boundary layer is also covered in most existing texts, although O’Malley (1991)
probably provides the most detailed analysis. (This work also includes a number of
relevant references which the interested reader may wish to investigate.) An excellent
discussion of the interplay between boundary layers and transition layers (for nonlinear
equations) is given in Kevorkian & Cole (1981, 1996). (Those readers who wish
to examine techniques applicable to turning points, at this stage, are encouraged to
study Wasow (1965) and Holmes (1995); we will touch on these ideas in Chapter 4.)