A new superconvergent method for systems of nonlinear singular boundary value problems

Abstract

A new superconvergent method based on a sextic spline is described and analysed for the solution of systems of nonlinear singular two-point boundary value problems (BVPs). It is well known that the optimal orders of convergence could not be achieved using standard formulation of a sextic spline for the solution of BVPs. Based on the method used in our earlier research papers [J. Rashidinia and M. Ghasemi, B-spline collocation for solution of two-point boundary value problems, J. Comput. Appl. Math. 235 (2011), pp. 2325–2342; J. Rashidinia, M. Ghasemi, and R. Jalilian, An o(h 6) numerical solution of general nonlinear fifth-order two point boundary value problems, Numer. Algorithms 55(4) (2010), pp. 403–428], we construct a new O(h 8) locally superconvergent method for the solution of general nonlinear two-point BVPs up to order 6. The error bounds and the convergence properties of the method have been proved theoretically. Then, the method is extended to solve the system of nonlinear two-point BVPs. Some test problems are given to demonstrate the applicability and the superconvergent properties of the proposed method numerically. It is shown that the method is very efficient and applicable for stiff BVPs too.