Complex-valued fractional derivatives on time scales

TL;DR

This paper introduces a new fractional derivative concept applicable to arbitrary time scales, extending classical derivatives to noninteger orders and establishing its fundamental properties with illustrative examples.

Contribution

It presents the first definition of fractional derivatives on arbitrary time scales, broadening the scope of fractional calculus.

Findings

The new operator generalizes classical derivatives on time scales.

Main properties such as linearity and product rule are established.

Several examples demonstrate the applicability of the fractional derivative on different time scales.

Abstract

We introduce a notion of fractional (noninteger order) derivative on an arbitrary nonempty closed subset of the real numbers (on a time scale). Main properties of the new operator are proved and several illustrative examples given.