A new fixed point iteration method for nonlinear third-order BVPs

Abstract

In this article, we shall present a novel approach based on embedding Green's function into Ishikawa fixed point iterative procedure for the numerical solution of a broad class of boundary value problems of third order. A linear integral operator expressed in terms of Green's function is constructed, then the well-known Ishikawa fixed point iterative scheme is applied to obtain a new iterative scheme. The aim of our alternative strategy is to overcome the major deficiency of other iterative schemes that usually result in the deterioration of the error as the domain increases. Furthermore the proposed strategy will improve the rate of convergence of other existing methods that are based on Picard's and Mann's iterative schemes. Convergence results of the iterative algorithm have been proved. A number of numerical examples shall be solved to illustrate the method and demonstrate its reliability and accuracy. Moreover, we shall compare our results with both the analytical and the numerical solutions obtained by other methods that exist in the literature.