# How to find inner solution in the method of matched asymptotic expansions

Consider the equation of the form $$\epsilon y^{\prime \prime}+a y^{\prime}=0$$ on $$x\in[0,1]$$ with $$a\in\mathbb{R}$$, $$0<\epsilon\ll1$$,$$y(0) =\alpha$$, and $$y(1)=\beta$$. Show that if $$a >0$$ then the boundary layer is near $$x= 0$$ with the method of matched asymptotic expansions.

To find the inner solution, Wikipedia defines a new variable $$\tau=x/\epsilon$$, but why not $$x/\epsilon^{2}$$ or $$x/\epsilon^{3}$$? I already know it depends on the thickness of the boundary layer, but for my problem above, how can I determine the thickness? (How can I define a new variable to find the inner solution? Is $$\tau=x/\epsilon$$ okay?)

I got another question just now, i.e. how can I prove the boundary layer is near $$x=0$$? When I have the outer solution, there exists two boundary condition but I don't know which one should be used. How can I determine which one is the "outer"?

## 1 Answer

You are right that you can not necessarily just "see" what the thickness $$\tau$$, the exponent in $$\delta=ϵ^τ$$, of the boundary layer is. You would then need to make the general zooming coordinate transform $$x=x_0+δX$$ and consider the ODE for $$Y(X)=y(x)$$ which in the current case gives $$ϵY''(X)+aδY'(X)=0.$$ The enjoining reasoning is that $$Y$$, having the same values as $$y$$, is bounded and that this coordinate transform renders also the derivatives of $$Y$$ bounded relative to $$ϵ\to 0$$. For that being the case, the terms in the equation have to balance, $$ϵ$$ and $$δ$$ need to be on the same scale. This is easiest achieved by just setting $$δ=ϵ$$. Thus $$Y(X)=Ce^{-aX}+D$$, $$C\ne 0$$ (As $$C=0$$ implies that no boundary layer exists at this point).

Now one needs to ensure that the boundedness assumption is actually satisfied for this inner solution. If $$x_0$$ is an inner point, this requires $$\lim_{X\to\pm \infty}Y(X)$$ being finite/existing, which is not the case here. Thus $$x_0$$ has to be one of the boundary points, and only the limit towards the inside the interval has to be finite, which is feasible.

• Thanks. I got another question just now, i.e. how can I prove the boundary layer is near $x=0$? When I have the outer solution, there exists two boundary condition but I don't know which one should be used. How can I determine which one is the "outer"? Commented Nov 21, 2019 at 2:37
• If $a>0$, then $e^{-aX}$ goes to infinity for $X\to -\infty$. This means that $x_0$ has to be $0$, else the inner solution takes ever greater values for $ϵ→0$, it can not be connected to the outer solution, which is (relatively) independent of $ϵ$, in that direction.. Commented Nov 21, 2019 at 12:06
• Happy new year. :) Can you kindly take a look at this post and suggest some good books you know on perturbation theory. Commented Jan 1, 2020 at 12:39
• @H.R. I have looked into the online preview of Mark H. Holmes "Introduction to Perturbation Methods" for this question on multi-scale expansions, it looks solid, but I have no idea of how it compares to the other cited books. Commented Jan 1, 2020 at 13:08
• Thanks. :) It would help future readers if you just mention your experience with what you have read. Commented Jan 1, 2020 at 13:19