Page 108 -

91

to give

where we have used the same notation as in (2.81). The balance that we seek is given
by the choice or provided that n < 1 (in order that this balance
does indeed produce the dominant terms, with F = O(1) or smaller). The procedure
then unfolds as for the boundary-layer problems, although we will now have solutions
in and in where together with a matched
solution where If then the balance of terms requires
and then either (n = 1) all three terms in the equation contribute to leading
order, or (n > 1) the balance is between and F. (This description assumes that
F = O(1); note also that the chosen behaviour of a(x) used here need only apply near
for this approach to be relevant.)

E2.15 An equation with a transition layer
Consider the equation

for where, for real solutions, we interpret the boundary
conditions are

We will find the first terms only in the asymptotic expansions valid away from x = 0
(where the coefficient is zero), in and then in and
finally valid near x = 0. For x = O(l), we write and so, from (2.94),
we obtain

we determine the arbitrary constant by imposing the boundary conditions (2.95a,b),
thereby producing solutions valid either on one side, or the other, of x = 0:

Note that these solutions do not hold in the neighbourhood of x = 0 i.e. we may
allow but with this solution would be valid for to
leading order, if the function given in (2.96) were continuous at x = 0, and then no
transition layer would be needed (to this order, at least).