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84 2. Introductory applications

with we obtain or (constant). Thus the
expansion (2.74) becomes

and this is to be matched to the asymptotic expansion (2.73); from (2.75) we obtain

and from (2.73) we have

which match if Thus the solution valid for x = O(1), i.e. away from
the boundary layer near x = 0, incorporating the first exponentially small term, is

One final comment: this solution, (2.76), produces the value
so now the boundary value is in error by To correct this, a
further term is required; we must write (at the order of all the first exponentially small
terms)

where It is left as an exercise to show that
with the solution ensures that the boundary condition on
x = 1 is correct at this order. The inclusion of the term in the expansion
valid for x = O(1) forces, via the matching principle, a term of this same order in
the expansion for and so the pattern continues. (The appearance of all
these terms, in both expansions, can be seen by expanding the exact solution, (1.22),
appropriately.)

E2.13 A nonlinear boundary-layer problem
We consider the problem