A family of variable mesh methods for the estimates of (d/d) and solution of non-linear two point boundary value problems with singularity

In this article, using three grid points, we discuss variable mesh methods of order two and three for the numerical solution of the nonlinear differential equation u ″ = f ( r , u , u ′ ) , 0 < r < 1 and the estimates of (d u /d r ) subject to the natural boundary conditions u ( 0 ) = A and u ( 1 ) = B . Both second- and third-order methods are compact and require two and three function evaluations, respectively. The proposed methods are successfully applied to the problems both in cartesian and polar coordinates. Numerical results are provided to illustrate the proposed methods and their convergence.