Page 106 -

89

the boundary layer is present, but only to correct the boundary value at —the
leading term (for x = O(1)) is uniformly valid. We call upon all these ideas in the next
example.

E2.14 A nonlinear, variable coefficient boundary-layer problem
We consider

with for Because the coefficient of is negative for
the boundary layer will be situated at the right-hand edge of the domain i.e.
near x = 1. Away from x = 1, we seek a solution

and so

etc., and we may use the boundary condition on x = 0:

Thus we obtain

and then the application of the boundary condition yields the solution

we note that remains real and positive as (The second term can also be
found, but it is a slightly tiresome exercise and its inclusion teaches us little about the
solution.) Clearly, as the right-hand boundary is approached, which does not
satisfy the given boundary condition and so a boundary layer is required.
(Of course, if then (2.90) would be a uniformly valid 1-term asymptotic
solution.)
We introduce (so that and write equation
(2.88) then becomes