Method of matched asymptotic expansions

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In mathematics, in particular in solving differential equations with perturbation theory, the method of matched asymptotic expansions is an approach to finding an approximate solution to a problem when a naïve perturbation approach fails. To do this one identifies a portion of the domain—a boundary layer—for which the perturbation contributes greatly, scales the domain so that region with a change of variables, solves that "inner" problem to an order of ε, and finally matches that solution with the original "outer" solution.

Consider the equation

ε y'' + (1 + ε) y' + y = 0

where

y (0) = 0

and

y (1) = 1

and $0<\epsilon\ll 1$

From perturbation theory we make the assumption that ε is small so we find the solution to the problem

y 0' + y 0 = 0

which is

y 0 = c e - t

for some

c

. Applying the boundary condition that

y (1) = 0

we find that

c = e

and so

y 0 = (e) e - t = e 1 - t

. But this does not follow the other boundary condition and does not account for the ε.

We note that at t=0 we have y=0, so there the equation begins to look like $ε y'' + (1 + ε) y' = 0$ or equivalently $y'' + (1 + 1 / ε) y' = 0$ where the $1 / ε$ term is clearly large, and so the $y''$ must be large as well.

Consider the change of variables $ξ = t / ε$ . Then $d y / d x = 1 / ε d y / d ξ$ and $d 2 y / d x 2 = 1 / ε 2 d 2 y / d ξ 2$ . Plugging into the equation, and canceling the ε terms yields...