Chaos, Solitons and Fractals 127 (2019) 158–164 Contents lists available at ScienceDirect Chaos, Solitons and Fractals Nonlinear Science, and Nonequilibrium and Complex Phenomena journal homepage: www.elsevier.com/locate/chaos Analysis of the fractional diffusion equations described by Atangana-Baleanu-Caputo fractional derivative Ndolane Sene a,∗ , Karima Abdelmalek b a Laboratoire Lmdan, Département de Mathématiques de la Décision, Université Cheikh Anta Diop de Dakar, Faculté des Sciences Economiques et Gestion, Dakar Fann BP 5683, Senegal b Laboratory LAMIS, Department of Mathematics and Informatiques, University Larbi Tebessi, Tebessa 12002, Algeria a r t i c l e i n f o Article history: Received 14 May 2019 Revised 20 June 2019 Accepted 26 June 2019 Keywords: Fractional diffusion equations Mean square displacement Atangana-Baleanu fractional derivative operator a b s t r a c t In this paper, we analyze two types of diffusion processes obtained with the fractional diffusion equa- tions described by the Atangana-Baleanu-Caputo (ABC) fractional derivative. The mean square displace- ment (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 subdif- fusive 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 get- ting the mean square displacement of the fractional diffusion equation. The central limit theorem has been discussed too, and the main results illustrated graphically. © 2019 Published by Elsevier Ltd. 1. Introduction In the literature, many researchers prove the applicability of the Atangana-Baleanu-Caputo derivative in real word problems. No- tably in physics [30,33], in epidemiology, in statistics and probabil- ity [8–10,16], in science and engineering, and in many other fields. Recently, it was introduced in the literature the discrete forms of the fractional derivatives with nonsingular kernels, and many ap- plications were proposed with these new issues. Many investiga- tions related the discrete form of the Atananga-Baleanu fractional derivative can be found in [1–4]. Many applications of the fractional order derivatives in the fractional diffusion equations were recently performed. Nowadays, there exist many types of fractional diffusion equations. We enu- merate some of them. The fractional diffusion reaction equation [29], the fractional subdiffusion equation [12], the fractional dis- persion equation [18] and many others. Many investigations on the fractional diffusion equations concern the analytical solutions [28,31,32], the semi-analytical solutions [29], the approximate so- lutions [11,13] and the numerical solutions [18]. Many methods ex- ∗ Corresponding author. E-mail addresses: ndolanesene@yahoo.fr (N. Sene), abdelmalekmath@gmail.com (K. Abdelmalek). ist in the studies of the solutions of the fractional diffusion equa- tions. We enumerate certain of them. In [20], Kader has proposed the numerical solution of the fractional diffusion equation using the Caputo fractional derivative. In [16], Henry et al. have proposed an introduction of modeling the fractional diffusion equation de- scribed by the Caputo fractional derivative. The statistical proper- ties of the fractional diffusion equations were also provided in this paper [16]. In [21], Tasbozan has proposed the numerical solution fractional diffusion equation for force-free case. In [15], Hashemi has solved the time fractional diffusion equation using a lie group integrator. In [19], Al-Refai et al. have proposed a complete analysis of the fractional diffusion equations with Atangana-Baleanu-Caputo fractional derivative. In [28,30], Sene has introduced the solution of the fractional diffusion equations and the stokes’ equation us- ing the Laplace and Fourier transform methods. For recent inves- tigations on the subdiffusive and superdiffusive processes, see in Owolabi et al. [23–27]. Motivated by the fact to understand, what are the types of diffusion generated by the new fractional order derivatives, in this paper, we propose to study the probability aspect of the fractional diffusion equations. We particularly use the Atananga- Baleanu-Caputo fractional derivative. The objective of the paper is to study the Mean Square Displacement of the fractional diffusion equation to analyze the diffusion processes generated by the https://doi.org/10.1016/j.chaos.2019.06.036 0960-0779/© 2019 Published by Elsevier Ltd.