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which clearly exhibits a catastrophic breakdown as Thus we have a gradual
acceleration of the process, until the time is reached and then—presumably—
combustion occurs. This breakdown is not simple: it is at a time given by
When logarithms (or exponentials) arise, we have learnt (§2.5) to
return to the original equation and seek a relevant scaling (although the presence
of a logarithm here indicates that ln terms are likely to appear in the asymptotic
expansion). Let us set then from (5.134), where
as thus we write and it is immediately clear from (5.132) that
we must choose The equation for is therefore

the original equation! In order to proceed, we need an appropriate solution—but this
is no longer required to satisfy the initial condition. The general solution to equation
(5.135) can be written as

and for small this gives and so to match we require the arbitrary
constant, B, to be zero (but we will return to this origin shift below). In passing, we
note that for close to unity, we obtain and so the state of full combustion
is attained as
Finally, we reconsider the matching of (5.136), with B = 0, to the expansion (5.134).
From (5.136) we obtain

which with gives

and the matching is not possible, as it stands, because of the presence of the ln
term. However, this suggests that the variable used in this region of rapid combustion
should include an origin shift. If we write, now, Then
(5.137) produces

and matching with (5.134) requires that Thus the
combustion occurs in an O(1) neighbourhood of the time the appear-
ance of shifts expressed in terms of ln are quite typical of these problems.