Correction method and extrapolation method for singular two-point boundary value problems

In this paper, we first discuss the constructions of three-point finite-difference approximation and a spline approximation for a class of singular two-point boundary value problems: x−α(xαu′)′=f(x,u), u′(0+)=0, u(1)=A, α⩾1. The asymptotic error expansions of the numerical solutions of these problems are obtained. From these asymptotic error expansions we find that the finite-difference solution and the spline approximation solution approximate the exact solution from two sides. So we can obtain correct solution of high-order accuracy. Richardsonʹs extrapolation can also be done and the accuracy of numerical solution can be improved greatly. We also present numerical examples which show the performance and efficiency of our methods.