An Identity Involving Integration with Respect to Variable Order of Fractional Derivative

TL;DR

This paper presents a new identity involving the integration of Riemann--Liouville fractional derivatives with respect to their variable order, expanding the theoretical framework of fractional calculus introduced in the 1990s.

Contribution

It introduces a novel identity that integrates fractional derivatives over their order, advancing the mathematical understanding of variable-order fractional calculus.

Findings

Derived a new integral identity involving fractional derivatives

Extended fractional calculus to include integration over derivative order

Provides theoretical foundation for variable-order fractional calculus

Abstract

In 1993, Samko and Ross introduced the study of fractional integration and differentiation when the order is not a constant but a function. This suggestion gave rise to a number of further ideas and results. In particular, this implies a possibility of integration with respect to derivative's order. Here an identity is presented, in which an expression involving Riemann--Liouville fractional derivative is integrated with respect to the derivative's order.