A novel Picard-Ishikawa-Green's iterative scheme for solving third-order boundary value problems

Abstract

The purpose of this paper is to introduce a novel fixed point iterative scheme based on Green's function, called the Picard-Ishikawa-Green's iterative scheme and use it in approximating the solution of boundary value problems (BVPs). It is proved that Picard-Ishikawa-Green's scheme converges strongly for an integral operator which represents the solution of BVP and the scheme is stable. Moreover, we prove that the integral operator is a contraction. Furthermore, it is shown that the novel scheme converges faster than all of Ishikawa-Green's, Khan-Green's, and Mann-Green's schemes. Finally, numerical examples are given to substantiate the validity of our results for third-order BVPs. Our results extend and generalize several other results in literature.