A truncated $\mathcal{V}$-fractional derivative in $\mathbb{R}^n$

TL;DR

This paper introduces a new truncated $ ext{V}$-fractional derivative using Mittag-Leffler functions, extends it to vector functions and Jacobian matrices, and explores properties like commutativity and Green's theorem in fractional calculus.

Contribution

It proposes a novel truncated $ ext{V}$-fractional derivative, extends it to multivariable functions, and establishes foundational properties and theorems for this new fractional calculus framework.

Findings

The truncated $ ext{V}$-fractional derivative generalizes previous derivatives.

The derivative can be extended to vector functions and Jacobian matrices.

Properties like commutativity and Green's theorem are established for the new derivative.

Abstract

Using the six parameters truncated Mittag-Leffler function, we introduce a convenient truncated function to define the so-called truncated $\mathcal{V}$-fractional derivative type. After a discussion involving some properties associated with this derivative, we propose the derivative of a vector valued function and define the $\mathcal{V}$-fractional Jacobian matrix whose properties allow us to say that: the multivariable truncated $\mathcal{V}$-fractional derivative type, as proposed here, generalizes the truncated $\mathcal{V}$-fractional derivative type and can bee extended to obtain a truncated $\mathcal{V}$-fractional partial derivative type. As applications we discuss and prove the change of order associated with two index i.e., the commutativity of two truncated $\mathcal{V}$-fractional partial derivative type and propose the truncated $\mathcal{V}$-fractional Green's theorem.