A second order spline finite difference method for singular two-point boundary value problems. (Q1399737)

A second order spline finite difference method on a non-uniform mesh for problems of the form \[ x^{-\alpha}(x^{\alpha}u')'=f(x,u), 0<x\leq 1, u(0)=A, u(1)=B \] with \(0<\alpha<1\) is presented. Assumptions on \(f\) are continuity on \([0,1]\times \mathbb{R}\), \(\partial f/ \partial u\) exists, is continuous and \(\geq 0\). A spline with the three-point finite difference method is constructed for a non-uniform mesh. \(O(h^2)\) convergence under appropriate conditions is shown. Two numerical examples are presented.