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respectively, where all the evaluations on x = X have been mapped to by
allowing Taylor expansions about (and constructed for The subscripts
in x denote partial derivatives and the over-dot is the time derivative. The leading-order
problem is then described by the equations

with as where
Thus and then the complete solution for (obtained by using the
Laplace transform, for example) is

where erfc is the complementary error function:

All this appears to be quite satisfactory, at this stage.
The solution for (5.30), can now be used to initiate the procedure for finding
the next term. In particular, (5.29b) becomes

which means that

and this asymptotic expansion is not uniformly valid as indeed, we see that it
breaks down when But we still have so further, for
the general in equation (5.26) to be O(1) then, retaining T = O(1) which is
necessary in order to accommodate (5.25), we see that in this region. Thus
we define the new variables

which produces the problem in this region as