# 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