Page 118 -

101

boundary layers near both x = 0 and x = 1; we will assume that they exist—we
can always ignore them if they are not required (because the boundary values are
automatically satisfied).
Hence, for x = O(1) away from x = 0 and x = 1, we seek a solution with
where

but no boundary conditions are available (by virtue of the assumed existence of bound-
ary layers). For the boundary layer near x = 0, it is clear that we require
(but we do not know at this stage; since presumably
as although the size of in this limit is unknown). Thus equation
(2.121) becomes

We seek a solution where satisfies

provided as we will assume that this condition applies, and we
will check it shortly. The boundary conditions are and we must not
allow the term which grows exponentially away from X = 0, so and then

Immediately we observe that the term is unmatchable to
unless we select (and then this we do, so that

Correspondingly, for the boundary layer near x = 1, we write
and now choose to give

Thus the first term in the expansion, satisfies