Mittag–Leffler Memory Kernel in Lévy Flights - Maike A. F. dos Santos (Mathematics, 2019)
In this article, we make a detailed study of some mathematical aspects associated with a generalized Lévy process using fractional diffusion equation with Mittag−Leffler kernel in the context of Atangana−Baleanu operator. The Lévy process has several applications in science, with a particular emphasis on statistical physics and biological systems. Using the continuous time random walk, we constructed a fractional diffusion equation that includes two fractional operators, the Riesz operator to Laplacian term and the Atangana−Baleanu in time derivative, i.e., <semantics> <msubsup> <mspace width="0.277778em"></mspace> <mspace width="0.277778em"></mspace> a A B </msubsup> <msubsup> D t α </msubsup> ρ ( x , t ) = K α , μ <mspace width="4pt"></mspace> <msubsup> ∂ x μ </msubsup> ρ ( x , t ) </semantics> . We present the exact solution to model and discuss how the Mittag−Leffler kernel brings a new point of view to Lévy process. Moreover, we discuss a series of scenarios where the present model can be useful in the description of real systems.