A Uniform Mesh Finite Difference Method for a Class of Singular Two-Point Boundary Value Problems

A new finite difference method based on uniform mesh is given for the (weakly) singular two-point boundary value problem: $(x^\alpha y')' = f(x,y)$, $y(0) = A$, $y(1) = B$, $0 < \alpha < 1$. Under quite general conditions on $f'$ and $f''$, we show that our present method based on uniform mesh provides $O(h^2 )$-convergent approximations for all $\alpha \in (0,1)$. Our method is based on one evaluation of f and for $\alpha = 0$ it reduces to the classical second order method for $y'' = f(x,y)$.