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Fractal Fract 2018, 2(3), 20; https://doi.org/10.3390/fractalfract2030020

Non-Gaussian Distributions to Random Walk in the Context of Memory Kernels

Instituto de Física, Universidade Federal do Rio Grande do Sul, Caixa Postal 15051, CEP 91501-970 Porto Alegre, RS, Brazil
Received: 13 July 2018 / Revised: 24 July 2018 / Accepted: 27 July 2018 / Published: 29 July 2018
(This article belongs to the Special Issue The Craft of Fractional Modelling in Science and Engineering 2018)
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Abstract

The investigation of diffusive process in nature presents a complexity associated with memory effects. Thereby, it is necessary new mathematical models to involve memory concept in diffusion. In the following, I approach the continuous time random walks in the context of generalised diffusion equations. To do this, I investigate the diffusion equation with exponential and Mittag-Leffler memory-kernels in the context of Caputo-Fabrizio and Atangana-Baleanu fractional operators on Caputo sense. Thus, exact expressions for the probability distributions are obtained, in that non-Gaussian distributions emerge. I connect the distribution obtained with a rich class of diffusive behaviour. Moreover, I propose a generalised model to describe the random walk process with resetting on memory kernel context. View Full-Text
Keywords: fractional diffusion equation; memory kernels; random walk; diffusion models; solution techniques; anomalous diffusion fractional diffusion equation; memory kernels; random walk; diffusion models; solution techniques; anomalous diffusion
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dos Santos, M.A.F. Non-Gaussian Distributions to Random Walk in the Context of Memory Kernels. Fractal Fract 2018, 2, 20.

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