The global attractor of the 2d Boussinesq equations with fractional Laplacian in subcritical case
Stabilization of the wave equation with interior input delay and mixed Neumann-Dirichlet boundary
October 2016, 21(8): 2509-2530. doi: 10.3934/dcdsb.2016058
Random walk's models for fractional diffusion equation
Wafa Hamrouni 1, and Ali Abdennadher 2,
Laboratoire d'Ingénierie Mathématique, Université de Carthage, Ecole Polytechnique de Tunisie, BP 743, 2078 La Marsa, Tunisia
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.
Keywords: transport processes, Integrodifferential equations, random walks, fractional derivatives..
Mathematics Subject Classification: Primary: 60J60, 45K05, 26A33; Secondary: 60G5.
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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Wafa Hamrouni Ali Abdennadher