Parametric spline method for a class of singular two-point boundary value problems J. Rashidinia a , Z. Mahmoodi b, * , M. Ghasemi a a School of Mathematics, Iran University of Science and Technology, Narmak, Tehran, Iran b Department of Mathematics, Islamic Azad University of Tehran Central Branch, Tehran, Iran Abstract We develop a three-point formula based on cubic spline in compression, for a class of singular two-point boundary value problems. For different values of parameters we obtain the classes of methods. These methods are illustrated by three numerical examples, two linear and one non-linear. Ó 2006 Elsevier Inc. All rights reserved. Keywords: Spline in compression; Singular boundary value problem; Optimum second order method 1. Introduction Consider the class of singular two-point boundary value problems x a ðx a y 0 Þ 0 ¼ f ðx; y Þ; 0 < x < 1; ð1Þ y ð0Þ¼ A; y ð1Þ¼ B ð2Þ or y 0 ð0Þ¼ A; y ð1Þ¼ B; where a 2 [0, 1] and the constants A, B are finite. There is considerable amount of literature on singular bound- ary value problems. For a = 1 Russell and Shampine [1] have shown that in case, f(x, y)= ay + g(x): g 2 [0, 1), Eq. (1) possesses a unique solution if 1 < a < J 2 0 . For a 2 [0, 1) and in the linear case, Jamet [2] considered a standard three-point finite-difference method in uniform mesh and has shown that the order is o(h 1a ), in max- imum norm. Reddien [3] and Reddin and Schumaker [4] used certain projection methods and singular splines to solve the linear problem and also studied the existence and uniqueness of solution. The purpose of this paper is to give three-point finite-difference method, based on uniform mesh using parametric spline for the class of singular two-point boundary value problem (1). In Section 2, we first derive the formulation of our spline function approximations, in Section 3, we developed our methods. In Section 4 Truncation error 0096-3003/$ - see front matter Ó 2006 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2006.09.084 * Corresponding author. E-mail address: z_mahmoodi_a@yahoo.com (Z. Mahmoodi). Applied Mathematics and Computation 188 (2007) 58–63 www.elsevier.com/locate/amc