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.



Asymptotic Matching of an logarithmic Outer solution to an exponential growing inner solution 
It's been a long time since I did any singular perturbations, but when I did, the text we used was [Kevorkian and Cole](http://rads.stackoverflow.com/amzn/click/0387942025); 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.


