Description
1. Consider the singular equation
ϵy′′ + (1 + x)
2
y
′ + y = 0
with y(0) = y(1) = 1 and with 0 < ϵ ≪ 1.
(a) Obtain a uniform approximation which is valid to leading order.
(b) Show that assuming the boundary layer to be at x = 1 is inconsistent. (Hint:
use the stretched inner variable ξ = (1 − x)/ϵ).
(c) Plot the uniform solution for ϵ = 0.01, 0.05, 0.1, 0.2.
2. Consider the singular equation:
ϵy′′ − x
2
y
′ − y = 0
with y(0) = y(1) = 1 and with 0 < ϵ ≪ 1.
(a) With the method of dominant balance, show that there are three distinguished
limits: δ = ϵ
1/2
, δ = ϵ, and δ = 1 (the outer problem). Write down each of the
problems in the various distinguished limits.
(b) Obtain the leading order uniform approximation. (Hint: there are boundary
layers at x = 0 and x = 1).
(c) Plot the uniform solution for ϵ = 0.01, 0.05, 0.1, 0.2.