Page 103 -

86 2. Introductory applications

with (2.78). From (2.79) we obtain

from (2.78) we have

and these match if we select and thus (2.79) becomes

The fundamental issue relating to boundary-layer-type problems, which we have
avoided thus far, addresses the question of where the boundary layer might be lo-
cated. In the examples discussed above, we allowed ourselves the advantage of knowing
where this layer was situated—and the consistency of the resulting asymptotic solution
confirmed that unique, well-defined solutions existed, so presumably we started with
the correct information. We now examine this important aspect of boundary-layer
problems.

2.7 WHERE IS THE BOUNDARY LAYER?
For this discussion, we consider the general second-order ordinary differential equation
in the form

with suitable boundary conditions and for note that the coefficient of must
be The coefficient a (x) will satisfy either a(x) > 0 or a(x) < 0, for the
term (or smaller) as for and for all solutions
that may be of interest. Of course, this describes only one class of such boundary-layer
problems, but this does cover by far the most common ones encountered in mathe-
matical modelling. (Some of these conditions, both explicitly written and implied, can
be relaxed; we will offer a few generalisations later.) The guiding principle that we will
adopt is to seek solutions of (2.80) which remain bounded as for
The starting point is the construction of the asymptotic solution valid for suitable
x = O(1), directly from the equation as written in (2.80)—but this will necessarily
generate a sequence of first-order equations. It is therefore impossible to impose the
two boundary conditions (as we have already demonstrated in our examples above); the
inclusion of a boundary layer remedies this deficiency in the solution. We introduce
the boundary layer in the most general way possible: define for some g (x)