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80 2. Introductory applications

But we also have so this law is commutative. Furthermore, the
associative law is satisfied:

in addition, we have the identity transformation i.e. for all
Finally, we form

and so is both the left and right inverse of Thus the elements of for all real
form an infinite group, where is the parameter of this continuous group. (If n is
fractional, then we may have to restrict the parameter to
Although we have not used the full power of this continuous group—we eventually
select only one member for a given —there are other significant applications of
this fundamental property in the theory of differential equations. For example, if a
particular scaling transformation leaves the equation unchanged (except for a change
of the symbols!) i.e. the equation is invariant, then we may seek solutions which satisfy
the same invariance. Such solutions are, typically, similarity solutions (if they exist) of
the equation; this aspect of differential equations is generally outside the considerations
of singular perturbation theory (although these solutions may be the relevant ones in
certain regions of the domain, in particular problems).

2.6 EQUATIONS WHICH EXHIBIT A BOUNDARY-LAYER BEHAVIOUR
There are many problems, posed in terms of either ordinary or partial differential
equations, that have solutions which include a thin region near a boundary of the
domain which is required to accommodate the boundary value there. Such regions are
thin by virtue of a scaling of the variables in the appropriate parameter and, typically,
this involves large values of the derivatives near the boundary. The terminology—
boundary layer—is rather self-evident, although it was first associated with the viscous
boundary layer in fluid mechanics (which we will describe in Chapter 5). Here, we
will introduce the essential idea via some appropriate ordinary differential equations,
and make use of the relevant scaling property of the equation.
The nature of this problem is best described, first, by an analysis of equation (1.16):

with and (and we will assume that and are not functions
of and that In this presentation of the construction of the asymptotic
solution, valid as we will work directly from (2.63) (although the exact
solution is available in (1.22), which may be used as a check, if so desired). Because
we wish to incorporate an application of the scaling property, we need to know where
the boundary layer (interpreted as a scaled region) is situated: is it near x = 0 or near