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186 4. The method of multiple scales

The boundary conditions, at this order, give

and so we have the solutions

which completely determines the first term of the asymptotic expansion:

as given in equation (2.76). The other terms in the expansion follow directly by im-
posing uniformity at each order; the terms for are used to remove the ex-
ponentially small contributions that appear in the boundary condition on x = 1.

The method of multiple scales is therefore equally valid, and beneficial, for the analysis
of boundary-layer problems. However, the example that we have presented is particu-
larly straightforward (and, of course, we have available the exact solution). We conclude
this chapter by applying the method to a more testing example (taken from Q2.17(a)).

E4.11 A nonlinear boundary-layer problem
We consider the problem

with for The slow scale is clearly x, but for the fast
scale we will use the most general formulation of the boundary-layer variable; see §2.7.
Thus we introduce

and then we write so that equation (4.89) becomes

with

The terms that are exponentially small on x = 1, as are and so we
seek a solution