An Integro-Differential Structure for Dirac Distributions

TL;DR

This paper introduces a new algebraic framework combining piecewise functions and distributions with differential and Rota-Baxter structures, facilitating symbolic computation and analysis of boundary problems.

Contribution

It develops an algebraic setting that unifies distributions and functions with differential structures, enabling symbolic computation of boundary problems.

Findings

Green's functions are expressed naturally within the new framework

The framework characterizes distributional differential equations from analysis

Supports symbolic computation involving distributions and differential operators

Abstract

We develop a new algebraic setting for treating piecewise functions and distributions together with suitable differential and Rota-Baxter structures. Our treatment aims to provide the algebraic underpinning for symbolic computation systems handling such objects. In particular, we show that the Green's function of regular boundary problems (for linear ordinary differential equations) can be expressed naturally in the new setting and that it is characterized by the corresponding distributional differential equation known from analysis.