Page 26 -

9

we have seen in earlier examples, a suitable rescaling of x is useful. In this case we set
and so obtain

and now, for any X fixed, as we have

This is a second, and different, approximation to valid for xs which are proportional
to note that on X = 0, (1.25) gives the value which is the correct boundary value.
In summary, therefore, we have (from (1.23))

and (from (1.25))

These two together constitute an approximation to the exact solution, each valid for an
appropriate size of x. Further, these two expressions possess the comforting property
that they describe a smooth—not discontinuous—transition from one to the other,
in the following sense. The approximation (1.26) is not valid for small x, but as x
decreases we have

(which we already know is incorrect because correspondingly, (1.27) is not
valid for large but we see that

results (1.28) and (1.29) agree precisely. This is clearly demonstrated in figure 1, where
we have plotted the exact solution for (as an example) i.e.

for various As decreases, the dramatically different behaviours for x not too small,
and x small, are very evident. (Note that the solution for x not too small is