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100 2. Introductory applications

and hence a transition layer exists at provided

If this condition is not satisfied, for given and then the adjustment to the given
boundary value must be through a boundary layer near x = 0. Thus, for example, with
and there is a transition layer at and the jump
across the layer is between ±3/2 (to leading order). On the other hand, the problem
with and does not admit a transition layer; the boundary layer near
x = 0 is used to accommodate the change in value from (to leading order)
to The dominant solution in the transition layer is given by (2.117) with
although is unknown at this stage. (The role of is to
determine the position of the transition layer, correct at

These two examples, E2.18 and E2.19 (and see also Q2.23), demonstrate the complex-
ity and richness of solutions that are available for this type of problem, depending on
the particular boundary values that are prescribed. All this can be traced to the nonlin-
earity associated with the term; if this term were simply then we would have
a fixed boundary layer, or fixed transition layers, independent of the specific boundary
values (as we have seen in our earlier examples). We conclude this section with an
example which shows how these ideas can be extended, fairly straightforwardly, to
higher-order equations. (The following example is based on the type of problem that
can arise when examining the displacement of a loaded beam.)

E2.20 A problem with two boundary layers
We consider

for (and the use of here is merely an algebraic convenience), with

Before we begin the detailed analysis of this problem, a couple of points should be
noted. First, the variable coefficient, in (1.121), is positive for
and so, second, this implies that we have available (locally) two exponential solutions.
These arise from, approximately,

and so we may select one exponential near x = 0 and the other near x = 1, ensuring
exponential decay as we move away from the boundaries. Thus we must anticipate