Applications of a distributional fractional derivative to Fourier analysis and its related differential equations

TL;DR

This paper explores a novel distributional fractional derivative based on a fractional Dirac delta, enabling advanced Fourier analysis and solutions to differential equations, with new results across these mathematical areas.

Contribution

It introduces a new fractional derivative definition using a fractional Dirac delta, expanding applications in Fourier analysis and differential equations.

Findings

Development of a distributional fractional derivative

Application to fractional Fourier series and transforms

New solutions to wave and related equations

Abstract

A new definition of a fractional derivative has recently been developed, making use of a fractional Dirac delta function as its integral kernel. This derivative allows for the definition of a distributional fractional derivative, and as such paves a way for application to many other areas of analysis involving distributions. This includes (but is not limited to): the fractional Fourier series (i.e. an orthonormal basis for fractional derivatives), the fractional derivative of Fourier transforms, and fundamental solutions to differential equations such as the wave equation. This paper observes new results in each of these areas.