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

Author:

Amechi Okeke Godwin1ORCID,Victor Udo Akanimo1,Rasulov Zaur2

Affiliation:

1. Functional Analysis and Optimization Research Group Laboratory (FANORG), Department of Mathematics, School of Physical Sciences Federal University of Technology, Owerri Imo State Nigeria

2. Mathematical Engineering Yildiz Technical University Istanbul Turkey

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.

Publisher

Wiley

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