Differential chain of algebras of generalized functions

Abstract

In this paper, it is shown how the spaces of generalized functions associated with the construction of the generalized solution for nonlinear partial differential equations through the order completion method using convergence spaces, may be interpreted as a chain of algebras of generalized functions. In particular, we showed that the spaces of normal lower semi-continuous functions that contain the generalized solution of the nonlinear partial differential equation under consideration is a differential chain of algebras of generalized functions. Consequently, this generalized solution is shown to be a chain generalized solution. The relationships between the chain of normal lower semi-continuous functions and the chain of nowhere dense algebras, as well as the chain of almost everywhere algebras of generalized functions are shown. We further show that the chain generalized solutions of nonlinear partial differential equations obtained in the chain of normal lower semi-continuous functions corresponds to the chain generalized solution for nonlinear partial differential equation obtained in the chain of nowhere dense algebras of generalized functions as well as the chain of almost everywhere algebra of generalized functions.