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96 2. Introductory applications

and this is completely determined, as is (2.106), so matching is used only to confirm
the correctness of these results (and this is left as an exercise).
The boundary layer near x = 0 is written in terms of with
again); equation (2.105) now becomes

The boundary layer is, not surprisingly, the same size at this end of the domain:
and then with we obtain

A bounded solution, as requires the choice and will ensure
that the boundary condition is satisfied i.e. The next term
in the asymptotic expansion satisfies

with (for boundedness) and (because i.e.

it is left as another exercise to confirm that this matches with (2.106).

In our examples so far (and see also Q2.21, 2.22), the character and position of the
boundary layer (or its interior counterpart, the transition layer) have been controlled
by the known function a (x), as in (or a (x) y in our most recent example).
We will now investigate how the same approach can be adopted when the relevant
coefficient is a function of y. In this situation, we do not know, a priori, the sign of
y—and this is usually critical. Typically, we make an appropriate assumption, seek a
solution and then test the assumption. For example, if the two boundary values for
y—we are thinking here of two-point boundary value problems—have the same sign,
then we may reasonably suppose that y retains this sign throughout the domain. On
the other hand, if the boundary values are of opposite sign, then the solution must
have at least one zero somewhere on the domain (and this indicates the existence of a
transition layer).

E2.18 A problem which exhibits either a boundary layer or a transition
layer: I
(An example similar to this one is discussed in great detail in Kevorkian & Cole, 1981
& 1996; see Q2.23.)