Differential algebraic interpretation with dense singularities of the order completion method

Abstract

In this paper we demonstrate the spaces of generalized function in terms of chain of algebras. In this work, we focus on generalized algebraic functions with nowhere dense singularities as mentioned in algebraic notation of Rosinger. This Rosinger’s dense algebra distributions are embedded & ensures its consistency with partial differential equations. Furthermore, in most of the cases this embedding phenomena maintains the products of smooth functions. Vernaeve introduced the issues of embedding derivatives in to generalized algebraic function of nonlinear partial differential equations. In this paper it is observed that the embedding of the spaces are used to conserve algebraic as well as differential structures. These findings indicates the extent to which chains (differential/algebraic) are capable of handling the singularities on closed dense set. Lastly we revealed that almost all the properties of Rosinger’s algebra have the ability to solve nonlinear partial differential equations.