Higher order method for singular boundary-value problems by using spline function

Abstract

In problems of physics and engineering we often come across singular boundary-value problems which cannot be solved by the usual numerical methods. Special methods for solving such problems have been developed. These methods lead to banded systems, linear and non-linear depending upon the nature of boundary-value problem. In this paper, we discuss the construction of a spline function for a class of singular two-point boundary-value problems. x - α ( x α u ′ ) ′ = f ( x , u ) , 0 < x ⩽ 1 , u ( 0 ) = A , u ( 1 ) = B . Five-point finite difference method using the above splines, is obtained it is shown to be order- h 4 convergent for all α ∈ (0, 1). The method is illustrated computationally.