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

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In this paper, we study the singular perturbation problemεp(x)uxx+2(x- a)(b-x)q(x)ux-2su=0,0<x<1,0<a<b<1,where 0<ε≪1 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. © 2005 Elsevier B.V. All rights reserved.

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Yang, H. (2006). On a singular perturbation problem with two second-order turning points. In Journal of Computational and Applied Mathematics (Vol. 190, pp. 287–303). https://doi.org/10.1016/j.cam.2005.01.040

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