An improved setting for generalized functions: fine ultrafunctions

TL;DR

This paper introduces fine ultrafunctions, an improved class of functions on Non Archimedean fields, enhancing previous definitions and demonstrating their application in solving ill-posed PDE problems.

Contribution

It develops the concept of fine ultrafunctions, advancing the theory and expanding their applicability to partial differential equations and ill-posed evolution problems.

Findings

Existence of ultrafunction solutions to ill-posed PDEs

Enhanced properties of fine ultrafunctions over previous definitions

Applications demonstrating ultrafunctions in PDE analysis

Abstract

Ultrafunctions are a particular class of functions defined on a Non Archimedean field E. They have been introduced and studied in some previous works. In this paper we develop the notion of fine ultrafunctions which improves the older definitions in many crucial points. Some applications are given to show how ultrafunctions can be applied in studing Partial Differential Equations. In particular, it is possible to prove the existence of ultrafunction solutions to ill posed evolution poblems.