Green's function of a centered partial difference equation

R. I. Avery, Dakota State University, Madison, SD, U.S.A.
D. R. Anderson, Concordia College, Moorhead, MN, U.S.A.

E. J. Qualitative Theory of Diff. Equ., Spec. Ed. I, 2009 No. 4., pp. 1-12.

Abstract: Applying a variation of Jacobi iteration we obtain the Green's function for the centered partial difference equation $$\Delta_{ww} u(x_{w-1},y_z) + \Delta_{zz} u(x_w,y_{z-1}) + f(u(x_w,y_z))=0,$$ 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.

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