Mittag–Leffler Memory Kernel in Lévy Flights

Abstract

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., a A B D t α ρ ( x , t ) = K α , μ ∂ x μ ρ ( x , t ) . 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.

Monitoring Lévy-Process Crossovers

“…Within this context, one can think of the possibility for a given walker to be governed by two different Lévy processes in distinct time regimes. Similar situations, exhibiting crossovers between anomalous to normal diffusion processes, have been observed in telomeres in the nucleus of mammalian cells [46], diffusion in biological cells [47], as well as in various complex systems [48,49,50,51,52,53].…”

A Multivariate Hybrid Stochastic Differential Equation Model for Whole-Stand Dynamics

“…The trivariate dynamics of the tree density, diameter, and height denominated by the trivariate probability density function (see Equations ( 9) and (10) or, for stationary case, Equations ( 20) and ( 21)) yield a unified system of forest stand development. The focus of this paper is the methodology of growth and yield modeling using a hybrid trivariate SDE.…”

“…Typically, the SDE is considered an ordinary differential equation with a white noise variable, which incorporates an influence that seems random. The possibility of combining SDEs, sophisticated mathematical techniques of parameter estimates, and increased computing power have produced an advanced research methodology for understanding how the stochastic phenomenon affects the predictions in practical applications [8][9][10].…”

A fast implicit difference scheme for solving the generalized time–space fractional diffusion equations with variable coefficients

“…However, for some practical physical processes, it is necessary to make the first moment of the waiting time measure finite. This leads to the generalized time fractional diffusion equation corresponding to the CTRWs model with some more complicated WTDs (beyond the power-law limit) [1,48,49], for example, the tempered [2,15,16,46,59] and the scale-weight [12,57] power law WTDs. In one word, the generalization of time-space fractional diffusion equations where the subdiffusion in time and the super-diffusion in space simultaneously [3] will be meaningful to model the anomalous diffusion with complicated physical processes.…”