Derivative with two fractional orders: New door of investigation toward revolution in fractional calculus (1604.05184v1)

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.