Asymptotic Matching of an logarithmic Outer solution to an exponential growing inner solution

Question

Hi,

I'm studying an ODE with a small parameter $\epsilon$ and I'm trying to find the solution in terms of a zeroth-order term and a boundary layer. The zeroth-order term has a logarithmic behavior near $x=0$ while the boundary layer term has an exponential (special function Bi) behavior at $+\infty$. To get all the constants I need to do some sort of asymptotic matching between the two solutions but I'm a bit at a loss as to whether this is possible.

More specifically,

the solutions that span the outer solution are the Bessel functions $J_0$ and $y_0$ and the inner solutions are the integral of a sum of the Airy functions $Ai$ and $Bi$

The problem of-course pertains to the $y_0$ and the $Bi$. The only way out that I see now is to set the coefficients in front of both functions to zero due to this problem and only use the other two. But then there's a problem with the boundary conditions.

Any advice or a reference would be greatly appreciated.

Cheers,

Yossi.

Answer 1

It's been a long time since I did any singular perturbations, but when I did, the text we used was Kevorkian and Cole; it definitely covers this type of problem. I think that Bender and Orszag's Advanced Mathematical Methods for Scientists and Engineers (sorry, MO won't let me add another link) has a section on it also, but not in as much depth.

FWIW, I remember Bender and Orszag being a better book in general.

Answer 2

Your question is a bit vague. Indeed, the standard procedure would involve rejecting the singular solutions and then using a combination of inner and outer expansions to satisfy the boundary conditions. There are various specific methods of achieving the goal (Vishik-Lyusternik and the matched asymptotic expansions are the most popular), but typically, one or several boundary conditions are only satisfied asymptotically (i.e. with an error vanishing as the small parameter tends to its limit, with the error often being exponentially small). Therefore, if some (or all) of your boundary conditions are satisfied only approximately (i.e. only in the limit of vanishing small parameter), this may in principle be the "feature" of the method that you are using. Otherwise, if the boundary conditions cannot be satisfied in this sense, it usually signals of the presence of yet another boundary layer which needs to be accounted for.

Kevorkian and Cole is an excellent source; you may also find helpful Van Dyke's "Perturbation methods in fluid mechanics" (a bit terse), Nayfeh's "Perturbation methods" (textbook) and de Jager and Furu's "The theory of singular perturbations" (good alternative).