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Civil-Comp Proceedings
ISSN 1759-3433 CCP: 84
PROCEEDINGS OF THE FIFTH INTERNATIONAL CONFERENCE ON ENGINEERING COMPUTATIONAL TECHNOLOGY Edited by: B.H.V. Topping, G. Montero and R. Montenegro
Paper 10
Space Fractional Diffusion in a Bound Domain M.C. Néel1, L. Di Pietro2 and N. Krepysheva2
1UMRA Climat Sol Environnement, Faculty of Science, University of Avignon, France
Full Bibliographic Reference for this paper
, "Space Fractional Diffusion in a Bound Domain", in B.H.V. Topping, G. Montero, R. Montenegro, (Editors), "Proceedings of the Fifth International Conference on Engineering Computational Technology", Civil-Comp Press, Stirlingshire, UK, Paper 10, 2006. doi:10.4203/ccp.84.10
Keywords: space-fractional diffusion, Lévy flights, impermeable boundaries, bound domain.
Summary
Field and laboratory experiments indicate that in heterogeneous porous media, the spreading of matter may be described by long-tailed non-Gaussian statistics. Such results would be impossible if the spreading of matter were ruled by the classical diffusion equation. More general, so-called fractional models, were proposed for such situations, but little attention was paid to the influence of boundary conditions. Since in practical situations and experiments we frequently have to consider bounded domains, but fractional derivatives are non-local operators, so introducing boundary conditions may be non-obvious. Here we focus on fractional diffusion in a domain, bounded by reflective walls.
A small scale model for diffusive transport behavior is based on the continuous time random walk approach. Space-fractional diffusion equations were shown to represent the diffusive limit of a wide class of uncoupled continuous time random walks which contains symmetric Lévy flights. In the latter model the spreading of many particles, performing successive independent jumps whose length is a random variable, are distributed according to a stable Lévy law. Jumps are separated by waiting times, which also are independent identically distributed random variables, whose mean The above cited results were stated for infinite domains. Impermeable boundaries were shown to correspond to a non-trivial modification of the macroscopic model, in the case of a semi-infinite domain. In the same spirit, the diffusive limit of Lévy flights in a bounded medium, limited by two reflective walls, takes a slightly different form.
Consider particles, being in ![]() Here ![]() ![]() ![]() ![]() ![]() ![]() ![]() Indeed, a jump from ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]()
When
Adapting a numerical scheme, borrowed from the infinite media case and tested against exact solutions, then allows us to display solutions to (19), obtained for several values of purchase the full-text of this paper (price £20)
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