Ultrafunctions and Applications

TL;DR

This paper introduces ultrafunctions, a new class of generalized functions based on non-Archimedean fields, designed to provide solutions to equations lacking solutions in traditional frameworks, with applications discussed.

Contribution

It presents the concept of ultrafunctions, expanding the theory of generalized functions using non-Archimedean fields, and demonstrates their applications in solving previously unsolvable equations.

Findings

Ultrafunctions extend the space of solutions for certain equations.

They enable solutions where traditional distributions fail.

Applications illustrate the effectiveness of ultrafunctions in practical problems.

Abstract

This paper deals with a new kind of generalized functions, called "ultrafunctions" which have been introduced recently and developed in some previous works. Their peculiarity is that they are based on a Non-Archimedean field namely on a field which contains infinite and infinitesimal numbers. Ultrafunctions have been introduced to provide generalized solutions to equations which do not have any solutions not even among the distributions. Some of these applications will be presented in the second part of this paper.