Optimal homotopy analysis method with Green's function for a class of nonlocal elliptic boundary value problems

TL;DR

This paper introduces an optimal homotopy analysis method with Green's function to accurately solve nonlocal elliptic boundary value problems, offering advantages over existing methods in convergence, flexibility, and physical fidelity.

Contribution

The paper develops a novel OHAM with Green's function for nonlocal elliptic problems, eliminating the need for additional constants and providing convergence guarantees.

Findings

Method achieves accurate solutions without extra constants.

Convergence and error analysis confirm reliability.

Approach handles nonlocal problems without altering physical structure.

Abstract

In this paper, we present the optimal homotopy analysis method (OHAM) with Green's function technique to acquire accurate numerical solutions for the nonlocal elliptic problems. We first transform the nonlocal boundary value problems into an equivalent integral equation, and then use an OHAM with convergence control parameter $c_0$. To demonstrate convergence and accuracy characteristics of the OHAM method, we compare the OHAM and Adomian decomposition method (ADM) with Green's function. The numerical experiments confirm the reliability of the approach as it handles such nonlocal elliptic differential equations without imposing limiting assumptions that could change the physical structure of the solution. We also discuss the convergence and error analysis of proposed method. In summary: $(i)$ the present approach does not require any additional computational work for unknown constants unlike ADM and VIM \cite{khuri2014variational} $(ii)$ guarantee of convergence $(iii)$ flexibility on choice of initial guess of solution and $(iv)$ useful analytic tool to investigate a class of nonlocal elliptic boundary value problems.