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

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.