Mittag–Leffler Memory Kernel in Lévy Flights
oleh: Maike A. F. dos Santos
| Format: | Article |
|---|---|
| Diterbitkan: | MDPI AG 2019-08-01 |
Deskripsi
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., <inline-formula> <math display="inline"> <semantics> <mrow> <msubsup> <mrow></mrow> <mrow> <mspace width="0.277778em"></mspace> <mspace width="0.277778em"></mspace> <mi>a</mi> </mrow> <mrow> <mi>A</mi> <mi>B</mi> </mrow> </msubsup> <msubsup> <mi mathvariant="script">D</mi> <mi>t</mi> <mi>α</mi> </msubsup> <mi>ρ</mi> <mrow> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>=</mo> <msub> <mi mathvariant="script">K</mi> <mrow> <mi>α</mi> <mo>,</mo> <mi>μ</mi> </mrow> </msub> <mspace width="4pt"></mspace> <msubsup> <mi>∂</mi> <mi>x</mi> <mi>μ</mi> </msubsup> <mi>ρ</mi> <mrow> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </semantics> </math> </inline-formula>. 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.