The order completion method: a differential-algebraic representation

Dennis Ferdinand Agbebaku, Jan Harm van der Walt

This paper deals with an interpretation of the Order Completion Method for systems of nonlinear partial differential equations (PDEs) in terms of suitable differential algebras of generalized functions. In particular, it is shown that certain spaces of generalized functions that appear in the Order Completion Method may be represented as differential algebras of generalized functions. This result is based on a characterization of order convergence of sequences of normal lower semi-continuous functions in terms of pointwise convergence of such sequences. It is further shown how the mentioned differential algebras are related to the nowhere dense algebras introduced by Rosinger, and the almost everywhere algebras considered by Verneave, thus unifying two seemingly different theories of generalised functions. Existence results for generalised solutions of large classes of nonlinear PDEs obtained through the Order Completion Method are interpreted in the context of the earlier nowhere dense and almost everywhere algebras.