Page 161 -

144 3. Further applications

which we choose to do, to give:

The behaviour of is discussed in Q3.18, where it is shown that the
asymptotic expansion is not uniformly valid as it breaks down where
Here, we will approach the problem of finding a solution by introducing
a strained coordinate.
The strained coordinate, is defined by

which, if this expansion is uniformly valid on the domain in (that corresponds to
may be inverted to find note that, if (3.89) is uniformly valid, then
for all x on the domain. The solution we seek is now written in terms of the
strained coordinate as

and the reason for using (3.89), rather than becomes evident when we
see that we transform of the original problem into functions of, and derivatives with
respect to, only Thus, with

the equation in (3.86) can be written

where, as in our previous convention, ‘= 0’ means zero to all orders in From equation
(3.91) we obtain

and so on. Because we have defined (3.89) with each the boundary condi-
tion on x = 1 becomes simply