Page 145 -

128 3. Further applications

the boundary- (or transition-) layer problem. We now turn to an examination of this
classical singular perturbation problem in the context of partial differential equations.

3.3 SINGULAR PROBLEMS II
There are two partial differential equation-types that are often encountered with small
parameters multiplying the highest derivative(s): the elliptic equation (e.g. Laplace’s
equation) and the parabolic equation (e.g. the heat conduction, or diffusion, equation).
These two, together with the wave equation (i.e. of hyperbolic type) discussed in §3.2,
complete the set of the three that constitutes the classification of second-order partial
differential equations. The two new equations are exemplified by

respectively. Of course, the use of singular perturbation methods to solve these partic-
ular examples is somewhat redundant, because we are able to solve them exactly (for
using standard techniques. Thus we will discuss two simple—but not completely
trivial—extensions of these basic equations.

E3.3 Laplace’s equation with nonlinearity
We are going to find an asymptotic solution, for of the equation

where u is prescribed on the boundary of the region. First, we will make a few general
observations about this problem and then obtain some of the details in a particular
case.
The appearance of in equation (3.44) suggests that we may seek a solution

which turns out to be consistent with the matching requirements to the boundary
layers; thus we obtain

and so on. Immediately we see that we can find which satisfies the given data
on x = 0 and x = a, but only in very special circumstances will this also satisfy the
data on y= 0 and y = b: the solution will (in general) require boundary layers near
y = 0 and near y = b. No such layers exist near x = 0 and x = a. All this follows
from the term the term simply contributes a (small) nonlinear adjustment