A novel approach based on embedding Green's functions into fixed point iterations for solving boundary value problems

Abstract

In this paper, a novel numerical method based on embedding Green's functions into classical fixed point iterations for solving a class of boundary value problems (BVPs) was developed. The iterative algorithms are implemented by embedding suitable integral operators on them. Numerical examples were given to demonstrate the applicability and efficiency of the methods. Furthermore, we prove that the Picard-Ishikawa-Green method developed by embedding Green's functions into the Picard-Ishikawa hybrid iteration (G. A. Okeke, Convergence of the Picard-Ishikawa hybrid iterative process with applications, Afrika Matematica 30, 817-835 (2019)) converges faster than several methods. The results show that the new approach provides approximations that are highly accurate.