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Mittag–Leffler Memory Kernel in Lévy Flights

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  • Maike A. F. dos Santos

    (Centro Brasileiro de Pesquisas Físicas and National Institute of Science and Technology for Complex Systems–Rua Dr. Xavier Sigaud 150, Rio de Janeiro 22290-180, Brazil)

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.

Suggested Citation

  • Maike A. F. dos Santos, 2019. "Mittag–Leffler Memory Kernel in Lévy Flights," Mathematics, MDPI, vol. 7(9), pages 1-13, August.
    • Handle: RePEc:gam:jmathe:v:7:y:2019:i:9:p:766-:d:259546
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    References listed on IDEAS

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