On the Atangana–Baleanu Derivative and Its Relation to the Fading Memory Concept: The Diffusion Equation Formulation

Abstract

The constructions of physically adequate forms of the diffusion equation with implementation of the Atangana–Baleanu derivative with Mittag-Leffler exponential kernel have been discussed. The specific form of the corresponding Atangana–Baleanu integral relates it directly to the fading memory concept, following the Boltzmann linear superposition principle with the standard Riemann-Liouville integral as the time-fading term. This approach relates the new fractional operators with non-singular kernel to the classical Riemann-Liouville integral. Using the concept of the fading memory and the specific form of the Atangana–Baleanu integral three forms of the diffusion equation have been investigated. The adequate definition of the flux to gradient relationship has been the main focus of the study resulting in two physically adequate formulations of the diffusion equation. The direct (formalistic) fractionalization of the classical diffusion equation results in physically inadequate relationships.