Nonsymmetric and Symmetric Fractional Calculi on Arbitrary Nonempty Closed Sets

TL;DR

This paper develops new fractional calculus operators on arbitrary closed sets, unifying discrete, continuous, and hybrid calculus through nabla, delta, and symmetric approaches, and explores their fundamental properties.

Contribution

It introduces a novel fractional calculus framework on arbitrary closed sets, bridging discrete and continuous calculus in a unified manner.

Findings

Established fundamental properties of the new fractional operators

Demonstrated the interplay between discrete and continuous behaviors

Provided a unified approach to fractional calculus on arbitrary sets

Abstract

We introduce a nabla, a delta, and a symmetric fractional calculus on arbitrary nonempty closed subsets of the real numbers. These fractional calculi provide a study of differentiation and integration of noninteger order on discrete, continuous, and hybrid settings. Main properties of the new fractional operators are investigated, and some fundamental results presented, illustrating the interplay between discrete and continuous behaviors.