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90 2. Introductory applications

which gives, with

The solution of (2.92), which satisfies the boundary condition is

and is to be determined by matching. From (2.90) we have directly that
and from (2.93) we see that thus matching requires and the first
term in the boundary-layer solution is

and then a composite expansion can be written down, if that is required.

The fundamental ideas that underpin the notion of boundary-layer-type solutions, in
second-order ordinary differential equations, have been developed, but many variants
of this simple idea exist; see also Q2.17–2.20. These lead to adjustments in the for-
mulation, or to generalisations, or to a rather different structure (with corresponding
interpretation). We now describe a few of these possibilities, but what we present is
far from providing a comprehensive list; rather, we present some examples which em-
phasise the application of the basic technique of scaling to find thin layers where rapid
changes occur. In the next section, we briefly describe a number of different scenarios,
and present an example of each type.

2.8 BOUNDARY LAYERS AND TRANSITION LAYERS
Our first development from the simple notion of a boundary layer is afforded by an
extension of our discussion of the position of this layer, via equation (2.80); here,
we consider the case where Such a point is analogous to a
turning point (see Q2.24) and the solution valid near takestheform of a transition
layer. (The terminology ‘turning point’ is used to denote where the character of the
solution changes or ‘turns’, typically from oscillatory to exponential, in equations such
as The general approach is to seek a scaling–just as for a boundary
layer–but now at this interior point. Let us suppose that for given
constants and n, then equation (2.80) becomes

and we introduce