# Uniqueness of Asymptotic Matching

I'm studying the basics of Boundary Layer theory in which one makes a number of asymptotic expansions of the solution in various regions separated by a number of layers and then matches them to have an uniform approximation to the whole soultuion. Basically one finds an asymptotic expansion of the solution $y(x;\epsilon)$ inside the boundary and outside and matches them if there is an overlap region (and then repeat the process if there are many layers). Provided this is correct (?), my question is: how i can be sure that the matching procedure produces an unique approximated solution and what is the meaning of matching two solutions in a whole region?

I couldn't find any piece of literature that addresses this problem or even mention it so does it even makes sense and if so is there an obvious explanation of why?

• +1, this is a good question. I can't answer it, but It seems that the uniqueness comes from the way that the boundary conditions are applied when in the inner/outer regimes. For example consider a 2nd order system with conditions at both boundaries. When in the outer regime you're looking at a 1st (or 0th) order system and so you get a unique outer solution when you apply the relevant outer boundary condition. When in the inner regime you have a 2nd order system with only one boundary condition, so your inner solution is unique up to one degree of freedom. (...) – Antonio Vargas Nov 4 '15 at 12:56
• (...) The constraint of matching the inner solution with the outer solution, assuming such a matching is possible, should (but maybe won't?) fix that degree of freedom. – Antonio Vargas Nov 4 '15 at 12:56
• @AntonioVargas that is also my understanding, but ihave to confess that my formal education in differential eqautions is quite bad – tired Nov 4 '15 at 14:38

## 1 Answer

In general, asymptotic matching is used to find an approximation to a solution to a (system of) differential equation(s). The uniqueness of the underlying, 'real' solution can be obtained from general theory on existence and uniqueness of solutions to differential equations. In particular, under certain general conditions, an initial condition gives rise to a unique solution.

Asymptotic matching can now give more explicit knowledge about that unique solution. So, we know it exists, we know it is unique, but we don't know what it looks like -- and that's the point where you start to use asymptotic methods.

A restriction (or, actually, a feature) of matched asymptotics is that the underlying solution is approximated up to a certain order in the small parameter of the problem. That means that, for 'nearby' initial conditions (i.e. initial conditions so close to the 'real' initial condition that they are all identified on the 'level of magnification' employed by the asymptotic method), the associated solutions are indistinguishable from the matched asymptotics point of view. So, every solution approximation you construct using matched asymptotics is actually a 'tube' of nearby solutions. You can make this 'tube' smaller by zooming in, i.e. carrying out the matched asymptotic method to higher orders in the small parameter, but this still does not solve the problem entirely. In particular, you can have solutions which are exponentially close, for which you would need so-called asymptotics 'beyond all orders'.

For more information on techniques dealing with these issues of approximation and uniqueness, I highly recommend you consult

C. Kuehn, Multiple Time Scale Dynamics, Springer (2015), ISBN 978-3-319-12316-5 .

• This was very clear and thank you so much for the reference. – Fra Nov 6 '15 at 19:02