Asymptotic Analysis and Singular Perturbation Theory

We seek an asymptotic expansion of a,
a(r, ε) ∼ a0(r) + εa1(r) + . . . as ε → 0.
Using this expansion in (4.5.2), expanding, and equating coefficients of ε 0 we find
that
Hence, if a0 = 0, we must have
′ 2
a0 (ϕ ) − 1 = 0.
(ϕ ′ ) 2 = 1.
Omitting the constants of integration, which can be absorbed into a, the solutions
are
ϕ(r) = ±r.
Equating coefficients of ε and simplifying the result, we find that
The solution is
where A is a constant.
We therefore obtain that
a ′ 0 + 1
2r a0 = 0.
a0(r) = A
,
r1/2 u0(r) ∼ A+
r 1/2 er/ε + A−
r 1/2 e−r/ε .
Matching this solution as r → 0 + with the the inner solution at r = 0, whose outer
expansion is given in (4.8), and using R = r/ε, U0 = λu0, we find that there are no
terms that grow exponentially as r → 0 + so A− = 0, and
ε
A+ = λ
2π .
Thus, the outer expansion of the inner solution (4.8) is valid as ε → 0 in the interior
0 < r < 1, and the leading order behavior of the solution is given by
ε
u(r, ε) ∼ λ
2πr er/ε as ε → 0. (4.11)
Here, the height λ(ε) of the interface at the origin remains to be determined. We
will find it by matching with the solution in the boundary layer.
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