High accuracy variable mesh method for nonlinear two-point boundary value problems in divergence form

In this article, using three grid points, we discuss variable mesh method of order three for the numerical solution of nonlinear two-point boundary value problems: (p(x)y′)′=f(x,y),y(0)=A,y(1)=B.We first establish an identity from which general three-point finite difference approximation of various order can be obtained. We obtain a family of third order discretization using variable mesh for the differential equations. We select the free parameter available in this discretization which leads to the simplest third order method. Numerical results are provided to illustrate the proposed method and their convergence.