On an internal boundary layer problem

In this paper, we consider the boundary value problem εy″+ a(x)y′ + b(x)y = 0, x ∈ [x_-,x₊], x_- <0 <x₊, y(x_-) = A, y(x₊) = B, where A and B are two prescribed constants, and 0 <ε ≪ 1 is a small positive parameter. As x → 0, it is assumed that a(x) ~ αx and b(x) ~ β, where α > 0 and β/α ≠ 1,2,3,... . Under certain smoothness conditions on a(x) and b(x), an asymptotic solution is constructed, which holds uniformly for x ∈ [x_-,x₊]. This result is proved rigorously by using the method of successive approximation. © 2002 Elsevier Science B.V. All rights reserved.