# American Institute of Mathematical Sciences

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October  2016, 21(8): 2509-2530. doi: 10.3934/dcdsb.2016058

## Random walk's models for fractional diffusion equation

 1 Laboratoire d'Ingénierie Mathématique, Université de Carthage, Ecole Polytechnique de Tunisie, BP 743, 2078 La Marsa, Tunisia 2 Laboratoire d'Ingénierie Mathématique, Université de Carthage, Ecole Polytechnique de Tunisie-Institut National des Sciences Appliquées et de Technologie, Centre Urbain Nord, BP 676 Cedex 1080 Charguia Tunis, Tunisia

Received  April 2015 Revised  May 2016 Published  September 2016

Fractional diffusion equations are used for mass spreading in inhomogeneous media. They are applied to model anomalous diffusion, where a cloud of particles spreads in a different manner than the classical diffusion equation predicts. Thus, they involve fractional derivatives. Here we present a continuous variant of Grünwald-Letnikov's formula, which is useful to compute the flux of particles performing random walks, allowing for heavy-tailed jump distributions. In fact, we set a definition of fractional derivatives yielding the operators which enable us to retrieve the space fractional variant of Fick's law, for enhanced diffusion in disordered media, without passing through any partial differential equation for the space and time evolution of the concentration.
Citation: Wafa Hamrouni, Ali Abdennadher. Random walk's models for fractional diffusion equation. Discrete & Continuous Dynamical Systems - B, 2016, 21 (8) : 2509-2530. doi: 10.3934/dcdsb.2016058
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