On modelling of singularities and their products in Colombeau algebra G(R)

TL;DR

This paper develops a systematic approach to modeling singularities using Colombeau generalized functions, enabling the evaluation of products of such models when they correspond to distributions, thus advancing the mathematical tools for physical phenomena involving singularities.

Contribution

It introduces a systematic method for modeling singularities with Colombeau generalized functions and evaluates their products when associated with distributions, extending existing theories.

Findings

Established a framework for modeling singularities with Colombeau functions.

Evaluated products of generalized models with associated distributions.

Extended Mikusinski's balancing concept to Colombeau algebra.

Abstract

Modelling of singularities given by discontinuous functions or distributions by means of generalized functions has proved useful in many problems posed by physical phenomena. We introduce in a systematic way generalized functions of Colombeau that model singularities given by distributions with singular point support. Moreover, we evaluate various products of such generalized models whenever the results admit associated distributions. The results obtained follow the idea of a well-known result of Jan Mikusinski on balancing of singular distributional products.