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for a transition layer at some and determining becomes an
essential element in the construction of the solution.
For x = O(1) we have, as before (see (2.111)),

but this can hold only for but where is
introduced below); for we have the corresponding solution

Near we write with which essentially
repeats (2.112) i.e. and so

and this gives the same general solution, to leading order, as before (see (2.113),
et seq.):

However, for a transition layer, we do not have any boundary conditions; here, we
must match (2.117) to both (2.115) and (2.116).
From (2.115) and (2.116) we obtain

respectively, both for X = O(1); from (2.117), with we have

as We observe, immediately, that a property of this transition layer is to admit
only a change in value across it from to (which will fix the value of
and that the matching excludes (so this cannot be determined at this stage). Now
(2.119) does match with (2.118) when we choose

which requires that