Analysis of the fractional diffusion equations described by Atangana-Baleanu-Caputo fractional derivative

Abstract

• The mean square displacement of the fractional diffusion equation described by the Atangana-Baleanu-Caputo fractional derivative has been presented. • The fractional diffusion equation has been classified in four classes : the fractional diffusion equations with subdiffusive, superdiffusive, ballistic and hyper diffusive processes. • The Laplace transform of the Atangana-Baleanu-Caputo fractional derivative has been used for getting the mean square displacement. • The graphical representations of the MSD have been proposed to illustrate the main results of the paper. In this paper, we analyze two types of diffusion processes obtained with the fractional diffusion equations described by the Atangana-Baleanu-Caputo ( ABC ) fractional derivative. The mean square displacement ( MSD ) concept has been used to discuss the types of diffusion processes obtained when the order of the fractional derivative take certain values. Many types of diffusion processes exist and depend to the value of the order of the used fractional derivatives: the fractional diffusion equation with the subdiffusive process, the fractional diffusion equation with the superdiffusive process, the fractional diffusion equation with the ballistic diffusive process and the fractional diffusion equation with the hyper diffusive process. Here we use the Atangana-Baleanu fractional derivative and analyze the subdiffusion process obtained when the order of ABC α is into (0,1) and the normal diffusion obtained in the limiting case α = 1 . The Laplace transform of the Atangana-Baleanu-Caputo fractional derivative has been used for getting the mean square displacement of the fractional diffusion equation. The central limit theorem has been discussed too, and the main results illustrated graphically.