On a singular perturbation problem with two second-order turning points

In this paper, we study the singular perturbation problem epsilon p(x)u(xx)+2(x-a)(b-x)q(x)u(x)-2su=0, 0 < x < 1, 0 < a < b < 1, where 0 < epsilon << I is a small positive parameter, p(x) and q(x) are sufficiently smooth and strictly positive functions. The main feature of this equation is that there are two second-order turning points in the interval (0, 1). Based on the rigorous results on singular perturbation problems with one second-order turning point in our previous work, we obtain a uniform asymptotic approximation for the general solution of the above equation by means of a matching technique. (c) 2005 Elsevier B.V. All rights reserved.