Solving a Class of Singular Two-Point Boundary Value Problems Using New Modified Decomposition Method

We introduce an effective methodology for solving a class of linear as well as nonlinear singular two-point boundary value problems. This methodology is based on a modification of Adomian decomposition method (ADM) and a new two fold integral operator. We use all the boundary conditions to derive an integral equation before establishing the recursive scheme for the solution components of solution. Thus, we develop modified recursive scheme without any undetermined coefficients while computing the successive solution components. This modification also avoids solving a sequence of nonlinear algebraic or transcendental equations for the undetermined coefficients. However, most of earlier recursive schemes using ADM do require computation of undetermined coefficients. The approximate solution is obtained in the form of series with easily calculable components. Numerical examples are included to demonstrate the accuracy, applicability, and generality of the present technique. The results reveal that the method is very effective, straightforward, and simple. 1. Introduction We consider the following class of singular two-point boundary value problems [1–5]: subject to the boundary conditions where , and？？ are any finite constants and . We assume that, for , the function and are continuous and . In particular, the problem (1) arises very frequently in applied sciences and in physiological studies, for example, in the study of steady-state oxygen diffusion in a spherical cell with Michaelis-Menten uptake kinetics [6] and distribution of heat sources in the human head [7]. In particular, when and (1) is known as Thomas-Fermi equation [8], given the singular equation Recently, there has been much interest in the study of singular two-point boundary value problems of type (1), (see, e.g., [1–5, 8–15]) and many of the references therein. The main difficulty of problem (1) is that the singularity behavior occurs at . A lot of methods have been applied to tackle this singular boundary value problem. In [9], a standard three-point finite difference scheme was considered with uniform mesh for the solution of problem for . In [1, 5], the finite difference methods were used to obtain the numerical solutions. In [4], a numerical method based on Green’s function was used to obtain numerical solution of the same problem. A novel approach that combines a modified decomposition method with the cubic B-spline collocation technique is presented in [14] to obtain approximate solution with high accuracy. Recently, in [15], a new modified decomposition method was applied to tackle