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192 4. The method of multiple scales
of (This procedure is a very neat way to obtain higher-order terms in the
WKB approach.)
Q4.16 WKB (exponential case). Consider the equation
where for and Introduce and
and hence find, completely, the first term of a uniformly valid asymp-
totic expansion (which will be the counterpart of equation (4.47)).
Q4.17 Eigenvalues. Use the method of multiple scales, in the WKB form, to find the
leading approximation to the eigenvalues of the problem
with and a > 0 for (You should in-
troduce and Evaluate your results, explicitly,
for the cases: (a) (b) a (X) = 1 + X. [Written in the form
it is evident that this is the problem of find-
ing approximations to the large eigenvalues.]
Q4.18 A turning-point problem I. Consider
where with f > 0 and analytic throughout the given domain.
Show that the relevant scaling (cf. §2.7) in the neighbourhood of the turning
point (at x= 1) is and then introduce the more useful fast
scale and use x as the slow scale. Show that
where = constant and Ai(X) is the
(bounded) Airy function (a solution of Determine h(x) and
write down the first term of the asymptotic expansion of
Q4.19 A turning-point problem II. Show that the equation
where and has turning points at x = 0 and at x = 1. [Hint:
write Use the WKB approach (for to find the first term
in each of the asymptotic expansions valid in x <0, 0 < x <1, x > 1. Also
write down the leading term in the asymptotic expansions valid near x = 0
and near x = 1.
Q4.20 A higher-order turning-point. Show that the equation