%0 Journal Article
%T Modified Differential Transform Method for Two Singular Boundary Values Problems
%A Yinwei Lin
%A Hsiang-Wen Tang
%A Cha¡¯o-Kuang Chen
%J Journal of Applied Mathematics
%D 2014
%I Hindawi Publishing Corporation
%R 10.1155/2014/138087
%X This paper deals with the two singular boundary values problems of second order. Two singular points are both boundary values points of the differential equation. The numerical solutions are developed by modified differential transform method (DTM) for expanded point. Linear and nonlinear models are solved by this method to get more reliable and efficient numerical results. It can also solve ordinary differential equations where the traditional one fails. Besides, we give the convergence of this new method. 1. Introduction In the present paper, we consider the following two singular boundary value problems (BVPs) of second order: where , and are known and continuous functions, , and is a nonlinear function of . The equation is singular at these two boundary values . Scientists and engineers have been interested in the singular equation because of its importance in many applications such as physical and mathematical models. There are many research directions on these equations. Some studied their qualitative properties [1, 2]. For example, Bartolucci and Montefusco [1] studied the concentration-compactness problem and the mass quantization properties. Some others used theorems to establish the existence and uniqueness of solution [3, 4]. For example, Guo et al. [3] got the existence and uniqueness of solution using a fixed point theorem. Recently great attention had been paid to numerical solutions [5¨C13]. For example, Duan and Rach [5] solved boundary value problems using a new modified Adomian decomposition method. Chowdhury and Hashim [7] used homotopy asymptotic method for finding the approximate solutions. Wazwaz [6, 8, 9] used Adomian decomposition method to get the numerical solutions. Puhov in 1976 [14] proposed the concept of DTM. DTM is the extension of Taylor series method and had been applied to solve analytic solutions of ordinary [15, 16], partial [17¨C19], differential-algebraic equations [20, 21], differential-difference equations [22, 23], and integrodifferential equations [24, 25]. Numerical solutions are also obtained [26]. Furthermore, Alquran and Al-Khaled [27] applied DTM to solve some eigenvalue problems. In the present paper, there are two singular points and these two singular points are just boundary values of the equation. Traditional DTM only can solve one-point singular BVP or two-point BVP (but not both the two points are just boundary values). We add some operation properties of DTM; then DTM can be used to calculate this type of problem. Furthermore, we have a convergent analysis of this method. 2. Modified Technique Now,
%U http://www.hindawi.com/journals/jam/2014/138087/