Finite difference methods and their convergence for a class of singular two point boundary value problems

TLDR: In this paper, the construction of three-point finite difference approximations and their convergence for the class of singular two-point boundary value problems is discussed, and three possibilities are investigated, their O(h2)-convergence established and illustrated by numerical examples.

Abstract: We discuss the construction of three-point finite difference approximations and their convergence for the class of singular two-point boundary value problems: (x?y?)?=f(x,y), y(0)=A, y(1)=B, 0<?<1. We first establish a certain identity, based on general (non-uniform) mesh, from which various methods can be derived. To obtain a method having order two for all ??(0,1), we investigate three possibilities. By employing an appropriate non-uniform mesh over [0,1], we obtain a methodM1 based on just one evaluation off. For uniform mesh we obtain two methodsM2 andM3 each based on three evaluations off. For ?=0,M1 andM2 both reduce to the classical second-order method based on one evaluation off. These three methods are investigated, theirO(h2)-convergence established and illustrated by numerical examples.