%0 Journal Article
%T New Approach for Solving a Class of Doubly Singular Two-Point Boundary Value Problems Using Adomian Decomposition Method
%A Randhir Singh
%A Jitendra Kumar
%A Gnaneshwar Nelakanti
%J Advances in Numerical Analysis
%D 2012
%I Hindawi Publishing Corporation
%R 10.1155/2012/541083
%X We propose two new modified recursive schemes for solving a class of doubly singular two-point boundary value problems. These schemes are based on Adomian decomposition method (ADM) and new proposed integral operators. We use all the boundary conditions to derive an integral equation before establishing the recursive schemes for the solution components. Thus we develop recursive schemes without any undetermined coefficients while computing successive solution components, whereas several previous recursive schemes have done so. This modification also avoids solving a sequence of nonlinear algebraic or transcendental equations for the undetermined coefficients with multiple roots, which is required to complete calculation of the solution by several earlier modified recursion schemes using the ADM. The approximate solution is computed in the form of series with easily calculable components. The effectiveness of the proposed approach is tested by considering four examples and results are compared with previous known results. 1. Introduction Consider the following class of doubly singular two-point boundary value problems: with boundary conditions where , , , , and are any finite constants. The condition characterizes that problem (1.1) is singular and in addition to this if is allowed to be discontinuous at , then problem (1.1) is called doubly singular Bobisud [1]. Consider problem (1.1) with the following conditions on , , and . Type 1. Dirichlet boundary conditions: , , with in and ; in and ; £¿(i) ; (ii) £¿ exists and continuous, for all and all real . Type 2. Mixed type boundary conditions: ,£¿£¿ £¿ ,£¿£¿ with in ; £¿ in (0,1], and ; £¿ the same as ( ). In recent years, the study of such singular boundary value problems (1.1) has attracted the attention of several researchers [2¨C9]. In particular, if ,£¿£¿ and problem (1.1) with Type 1 boundary conditions is known as Thomas-Fermi equation. Thomas [10] and Fermi [11] independently derived a boundary value problem for determining the electrical potential in an atom. The analysis leads to the nonlinear SBVP with boundary conditions given by ,£¿£¿ . Chan and Hon [12] considered the generalized Thomas-Fermi equation: , , with , where , , , , which is doubly singular problem. Problem (1.1) with where arises in the study of the distribution of heat sources in the human head [13] with , , . There is a huge literature available on numerical methods for problem (1.1) with , but very few numerical methods are available to tackle doubly singular boundary value problems. Reddien [14] studied the linear form of problem (1.1) and
%U http://www.hindawi.com/journals/ana/2012/541083/