On Some Operators Involving Hadamard Derivatives

TL;DR

This paper introduces a new Mittag-Leffler-type function and explores its properties related to Hadamard fractional derivatives, providing operational methods to solve related integro-differential equations with practical applications.

Contribution

It presents a novel Mittag-Leffler-type function and demonstrates its utility in solving integro-differential equations involving Hadamard derivatives.

Findings

New Mittag-Leffler-type function introduced

Operational methods effectively solve integro-differential equations

Application to modified Lamb-Bateman integral equation included

Abstract

In this paper we introduce a novel Mittag--Leffler-type function and study its properties in relation to some integro-differential operators involving Hadamard fractional derivatives or Hyper-Bessel-type operators. We discuss then the utility of these results to solve some integro-differential equations involving these operators by means of operational methods. We show the advantage of our approach through some examples. Among these, an application to a modified Lamb--Bateman integral equation is presented.