Green's function of a centered partial difference equation - Richard Avery; Douglas Anderson (Electronic Journal of Qualitative Theory of Differential Equations, 2009)
Applying a variation of Jacobi iteration we obtain the Green's function for the centered partial difference equation $$Δ_{ww} u(x_{w-1},y_z) + Δ_{zz} u(x_w,y_{z-1}) + f(u(x_w,y_z))=0,$$ which is the result of applying the finite difference method to an associated nonlinear partial differential equation of the form $$u_{xx}+u_{yy} +h(u)=0.$$ 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.