Derivative with two fractional orders: New door of investigation toward revolution in fractional calculus

TL;DR

This paper introduces a novel concept of fractional derivatives with two orders based on power law and Mittag-Leffler functions, enabling more accurate modeling of complex layered systems in science and engineering.

Contribution

It presents a new definition of fractional derivatives with two orders, expanding the tools available for modeling layered media with different properties.

Findings

New fractional derivative definitions based on power law and Mittag-Leffler functions.

Application potential in thermal science and groundwater flow modeling.

Opens new avenues for research in fractional calculus and complex system modeling.

Abstract

In order to describe more complex problem using the concept of fractional derivatives, we introduce in this paper the concept of fractional derivatives with orders. The new definitions are based upon the concept of power law together with the generalized Mittag-Leffler function. The first order is included in the power law function and the second is in the generalized Mittag-Leffler function. Each order therefore plays an important rule while modelling for instance problems with two layers with different properties. This is the case for instance in thermal science for a reaction diffusion within a media with two different layers with different properties. Another case is that of groundwater flowing within an aquifer where geological formation is formed with two layers with different properties. The paper presents new useful information within the scope of fractional calculus that will open new doors for research and investigations in modelling real world problems.