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108 2. Introductory applications

Q2.19 Two boundary layers. The function is defined by the problem

with Find the first two terms in an asymptotic expan-
sion valid for x = O(1), as away from x = 0 and x = 1. Hence show
that, in this problem, boundary layers exist both near x = 0 and near x = 1,
and find the first term in each boundary-layer solution, matching as necessary.
Q2.20 A boundary layer within a thin layer. Consider the equation

for with Find the first terms in each of three
regions, two of which are near x = 0, matching as necessary. (Here, only the
inner-most region is a true boundary layer; the other is simply a scaled-thin-
connecting region.)
Q2.21 Boundary layers or transition layers? Decide if these equations, on the given
domains, possess solutions which may include boundary layers or transition
layers; give reasons for your conclusions.
(a)
(b)
(c)
(d)
(e)
Q2.22 Transition layer near a fixed point. In these problems, a transition layer exists at
a fixed point, independent of the boundary values. Find, for (a), the first two
terms, and for (b) the first term only, in an asymptotic expansion (as
valid away from the transition layer and the first term only of an expansion
valid in this layer; match your expansions as necessary.
(a)
(b)
Q2.23 Boundary layer or transition layer? (This example is based on the one which is
discussed carefully and extensively in Kevorkian & Cole, 1981 & 1996.) The
equation is

for and given Suppose that a
transition layer exists near find the leading terms in the
asymptotic expansions valid outside the transition layer, and in the transition
layer. Hence deduce that a transition layer is required if and are of opposite