2006 International Conference in honor of Jacqueline Fleckinger. Electron. J. Diff. Eqns., Conference 16 (2007), pp. 1-13.

Fractional flux and non-normal diffusion

Ali Abdennadher, Marie-Christine Neel

Fractional diffusion equations are widely used for mass spreading in heterogeneous media. The correspondence between fractional equations and random walks based upon stable Levy laws, keeps in analogy with that between heat equation and Brownian motion. Several definitions of fractional derivatives yield operators, which coincide on a wide domain and can be used in fractional partial differential equations. Then, the various definitions are useful in different purposes: they may be very close to some physics, or to numerical schemes, or be based upon important mathematical properties. Here we present a definition, which enables us to describe the flux of particles, performing a random walk. We show that it is a left inverse to fractional integrals. Hence it coincides with Riemann-Liouville and Marchaud's derivatives when applied to functions, belonging to suitable domains.

Published May 15, 2007.
Math Subject Classifications: 26A33, 45K05, 60E07, 60J60, 60J75, 83C40.
Key Words: Diffusion equations; fractional calculus; random walks; stable probability distributions.

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  Ali Abdennadher
Department of mathematics,
Institut National des Sciences Appliquées et de Technologie
Centre Urbain Nord
BP 676 Cedex 1080 Charguia Tunis, Tunisie
email: ali_abdennadher@yahoo.fr
Marie-Christine Neel
UMRA 1114 Climat Sol Environnement
University of Avignon
33, rue Pasteur, 84000 Avignon, France
email: marie-christine.neel@univ-avignon.fr

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