Random walk's models for fractionaldiffusion equation

Wafa  Hamrouni; Ali  Abdennadher

doi:10.3934/dcdsb.2016058

Article Contents

2016, Volume 21, Issue 8: 2509-2530. Doi: 10.3934/dcdsb.2016058

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Random walk's models for fractional diffusion equation

Wafa Hamrouni^1, and
Ali Abdennadher^2,

1.
Laboratoire d'Ingénierie Mathématique, Université de Carthage, Ecole Polytechnique de Tunisie, BP 743, 2078 La Marsa
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

Received: April 2015

Revised: May 2016

Published: October 2016

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Abstract

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.

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Mathematics Subject Classification: Primary: 60J60, 45K05, 26A33; Secondary: 60G50.

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