  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.
