On new types of fractional operators and applications

TL;DR

This paper introduces two novel fractional integral operators based on special functions, explores their properties and relationships, and develops a new fractional derivative concept with applications to a fractional relaxation model.

Contribution

The paper presents new fractional operators defined via exponential-integral and special functions, and introduces a fractional derivative concept linked to these operators, with theoretical and applied insights.

Findings

Established properties of the new fractional operators

Proved convergence of the fractional derivative to the standard derivative as order approaches zero

Developed a fractional relaxation model with proven existence and uniqueness

Abstract

We introduce two kinds of fractional integral operators; the one is defined via the exponential-integral function $$ E_1(x)=\int_x^\infty \frac{e^{-t}}{t}\,dt,\quad x>0, $$ and the other is defined via the special function $$ \mathcal{S}(x)=e^{-x} \int_0^\infty \frac{x^{s-1}}{\Gamma(s)}\,ds,\quad x>0. $$ We establish different properties of these operators, and we study the relationship between the fractional integrals of first kind and the fractional integrals of second kind. Next, we introduce a new concept of fractional derivative of order $\alpha>0$, which is defined via the fractional integral of first kind. Using an approximate identity argument, we show that the introduced fractional derivative converges to the standard derivative in $L^1$ space, as $\alpha\to 0^+$. Several other properties are studied, like fractional integration by parts, the relationship between this fractional derivative and the fractional integral of second kind, etc. As an application, we consider a new fractional model of the relaxation equation, we establish an existence and uniqueness result for this model, and provide an iterative algorithm that converges to the solution.