A Note on Fourth Order Method for Doubly Singular Boundary Value Problems

We present a fourth order finite difference method for doubly singular boundary value problem with boundary conditions and , where , , and are finite constants. Here and is allowed to be discontinuous at the singular point . The method is based on uniform mesh. The accuracy of the method is established under quite general conditions and also corroborated through one numerical example. 1. Introduction Consider the following class of singular two point boundary value problems: with boundary conditions where , and , are finite constants. Here and is unbounded near . The condition says that the problem is singular and as is unbounded near , so the problem is doubly singular [1]. Let , , and satisfy the following conditions:？ (i) in ,(ii) ,(iii) , , , and(iv) and exists on ,？？where on .？ (i) in , is unbounded near ,(ii) ,(iii) , with ,(iv) and exists on , and(v) . ？ ？ is continuous on , exists, and is continuous and for all and for all real . Thomas [2] and Fermi [3] independently derived a boundary value problem for determining the electrical potential in an atom. The analysis leads to the nonlinear singular second order problem with a set of boundary conditions. The following are of our interest:(i)the neutral atom with Bohr radius given by ;(ii)the ionized atom given by .Furthermore, Chan and Hon [4] have considered the generalized Thomas-Fermi equation for parameter values , , , and . Such singular problems have been the concern of several researchers [5–8]. The existence-uniqueness of the solution of the boundary value problem (1.1) with boundary condition (1.2) or (1.3) is established in [1, 9–13]. Bobisud [1] has mentioned that in case , the condition is quite severe, that is, it is sufficient but not necessary for forcing the solution to be differentiable at . In fact if , then Thus if either or . But if has discontinuity at ; thus it is natural to consider the weaker boundary condition . There is a considerable literature on numerical methods for but to the best of our knowledge very few numerical methods are available to tackle doubly singular boundary value problems. Reddien [14] has considered the linear form of (1.1) and derived numerical methods for which is stronger assumption than . Some second order methods (Chawla and Katti [15], Pandey and Singh [16, 17]) as well as fourth order methods (Chawla et al. [18–21], Pandey and Singh [22–24]) have been developed for . Most of the researchers have developed methods for the function , and the boundary conditions and . Chawla [19] has given fourth order method for the problem (1.1)-(1.2) with and .