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and so

where we have retained the logarithmic term in (as we have learnt previously is
necessary; see §1.9). From (5.20) we obtain

or, reverting to the R-variable, simply

For (5.21) and (5.22) to match, we choose

We note that the left-hand side of equation (5.13) allows z to be shifted by a constant,
which at this order is B, and this contains a term In The asymptotic procedure for the
full equation may proceed provided (the right-hand side) 0 for r = O(1), and this
is the case even in the presence of this logarithmic term, since In as
Of course, the appearance of this term indicates the need to include logarithmic terms
throughout the asymptotic expansions.

This completes the description of the exterior problem so far as we are concerned
here; much more detail can be found in Lo (1983).

E5.4 Drilling by laser
This problem is a one-dimensional model for the process of drilling through a thick
block of material using a laser. The laser heats the material until it vaporises, and we
assume that the vapour is continuously removed; the essential character of this problem
is therefore one of heat transfer at a boundary—the bottom of the drill hole—which
is moving. In suitable non-dimensional variables, the temperature relative to ambient
conditions satisfies the classical heat conduction equation

with

(which describes the initial state and the condition at infinity, respectively), and