A Conformable Fractional Calculus on Arbitrary Time Scales

TL;DR

This paper introduces a new conformable fractional calculus on arbitrary time scales, unifying fractional differentiation and integration with existing calculus as a special case, expanding the mathematical toolkit for dynamic systems.

Contribution

It develops a conformable fractional calculus framework on arbitrary time scales, generalizing existing calculus and including Hilger time-scale calculus as a special case.

Findings

Established a conformable fractional derivative and integral on arbitrary time scales.

Unified fractional calculus with classical calculus at =1.

Provided foundational tools for further research in dynamic systems on time scales.

Abstract

A conformable time-scale fractional calculus of order $\alpha \in ]0,1]$ is introduced. The basic tools for fractional differentiation and fractional integration are then developed. The Hilger time-scale calculus is obtained as a particular case, by choosing $\alpha = 1$.