Discrete analogues of Green's identities through the graph—theoretic field model

Abstract

In this paper, we derive discrete versions of Green's identities (which appear in the study of potential field theory) as direct consequences of applying Tellegen's Theorem to the Graph—Theoretic Field Model (GTFM) of a field. The procedure herein is in marked contrast to the existing procedures where Green's Identities are derived from the Divergence Theorem by using some strictly mathematical operations. In particular, Green's third identity, which is the starting point formulation for the Boundary Integral Method, is singled out for special attention in terms of its discrete counterpart in the Graph—Theoretic Field Model. The first discrete identity is used to establish certain properties of solutions for the GTFM and a limiting process is applied to the three discrete identities to derive the traditional vector-calculus forms of Green's identities.