On a finite difference method for singular two-point boundary value problems

Abstract

The objective of this article is to provide a finite difference method for the solution of a general class of second-order two-point boundary value problems of the form - 1/w(x) (p(x)y'(x))' + q(x)y(x) = f(x); x ∈ (0, 1), lim x→0+ p(x)y'(x) = 0, y(1) = 0 with general conditions on the real-valued functions w(x), p(x), q(x) and f(x). The class of problems we consider here includes both limit-point and limit-circle cases. We obtain the rate of convergence of the method in the uniform norm and show the dependence of the rate of convergence on the properties of the data. In the particular case w(x) = p(x) = x α , α ≥ 0 the order of convergence reduces to O(h 2 ) which is developed in the literature.