A Fractional Calculus on Arbitrary Time Scales: Fractional Differentiation and Fractional Integration

TL;DR

This paper develops a unified framework for fractional calculus on arbitrary time scales, generalizing classical derivatives and integrals to non-integer orders across diverse mathematical contexts.

Contribution

It introduces a novel fractional derivative and integral on arbitrary time scales, bridging discrete, continuous, and hybrid calculus.

Findings

Defines a general fractional derivative on arbitrary time scales.

Recovers classical derivatives as special cases when the order is one.

Provides tools for fractional calculus applicable to various time scales.

Abstract

We introduce a general notion of fractional (noninteger) derivative for functions defined on arbitrary time scales. The basic tools for the time-scale fractional calculus (fractional differentiation and fractional integration) are then developed. As particular cases, one obtains the usual time-scale Hilger derivative when the order of differentiation is one, and a local approach to fractional calculus when the time scale is chosen to be the set of real numbers.