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Q2.14 Breakdown as II. See Q2.11 (a) and (c); for these equations and bound-
ary conditions, and the asymptotic solutions already found for x = O(l), take
the domain now to be Hence show that the expansions are not uni-
formly valid as find the breakdown, rescale and then find the first
terms in the expansions valid for large x, matching as necessary.
Q2.15 Problem E 2.7 reconsidered. Find the first two terms in an asymptotic expansion,
valid for x = O(1) as of

with Show that, formally, this requires
two matched expansions, but that the asymptotic solution obtained for x =
O(1) correctly recovers the solution for D i.e. it is uniformly valid. (Note
the balance of terms, when scaled near x = 0!)
Q2.16 Scaling of equations. See Q2.11 and Q2.14; use the dominant terms only, valid
for x = O(1), together with appropriate scalings associated with the relevant
balance of terms, to analyse these equations. Compare your results with the
scalings obtained from the breakdown of the asymptotic expansions.
Q2.17 Boundary-layer problems I. Find the first two terms in asymptotic expansions,
valid for x = O(1) (away from the boundary layer) as for each of
these equations, with the given boundary conditions. Then, for each, find
the first term in the boundary-layer solution, matching as necessary. (You
may wish to use your expansions to construct composite expansions valid for
D, to this order.)
(a)
(b)
(c)

(d)
(e)
Q2.18 Boundary-layer problems II. See Q2.17; repeat for these more involved equations.
(a)

(b)
(c)
(d)
(e)

(f)