Green's function of a centered partial difference equation

Abstract

Applying a variation of Jacobi iteration we obtain the Green's function for the centered partial difference equation Δwwu(xw-1, yz) + Δzzu(xw, yz-1) + f(u(xw, yz)) = 0, which is the result of applying the finite difference method to an associated nonlinear partial differential equation of the form uxx + uyy + h(u) = 0. We show that approximations of the partial differential equation can be found by applying fixed point theory instead of the standard techniques associated with solving a system of nonlinear equations.