On some analytic properties of nabla tempered fractional calculus

TL;DR

This paper introduces a new nabla tempered fractional calculus, exploring its fundamental properties such as equivalence relations, Taylor formula, and Laplace transform, enriching the mathematical framework for practical modeling.

Contribution

It develops the foundational properties of nabla tempered fractional calculus, a novel approach that addresses the modeling challenges of processes with nonlocal but finite memory.

Findings

Established equivalence relations for nabla tempered fractional calculus

Derived the nabla Taylor formula for this calculus

Analyzed the nabla Laplace transform in this context

Abstract

Despite many applications regarding fractional calculus have been reported in literature, it is still unknown how to model some practical process. One major challenge in solving such a problem is that, the nonlocal property is needed while the infinite memory is undesired. Under this context, a new kind nabla fractional calculus accompanied by a tempered function is formulated. However, many properties of such fractional calculus needed to be discovered. From this, this paper gives particular emphasis to the topic. Some remarkable properties like the equivalence relation, the nabla Taylor formula, and the nabla Laplace transform for such nabla fractional calculus are developed and analyzed. It is believed that this work greatly enriches the mathematical theory of nabla tempered fractional calculus and provides high value and huge potential for further applications.