Ali Abdennadher, Marie-Christine Neel
Abstract:
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 | |
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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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