A Green's function formulation for finite-differences schemes

The finite-differences (FD) method has been used with remarkable success in solving a wide range of problems in virtually all areas of engineering. Our aim in this paper is to show how FD schemes can be derived from an integral formulation of boundary-value problems from Green's functions. For this purpose, we confine our attention to a simple second-order model representing diffusion and non-linear reaction in a catalytic slab. The classical FD discretization is obtained by forcing the integral equation formulation of the boundary-value problem to hold at the discretization points. Under the Green's function formulation, Dirichlet boundary conditions are incorporated as in classical FD. Interestingly, Neumann boundary conditions modify the discretization at the boundary node, and numerical results show that such modification improves the performance of the FD method.