Space-fractional advection-diffusion and reflective boundary condition
- 22 February 2006
- journal article
- research article
- Published by American Physical Society (APS) in Physical Review E
- Vol. 73 (2), 021104
- https://doi.org/10.1103/physreve.73.021104
Abstract
Anomalous diffusive transport arises in a large diversity of disordered media. Stochastic formulations in terms of continuous time random walks (CTRWs) with transition probability densities showing space- and/or time-diverging moments were developed to account for anomalous behaviors. A broad class of CTRWs was shown to correspond, on the macroscopic scale, to advection-diffusion equations involving derivatives of noninteger order. In particular, CTRWs with Lévy distribution of jumps and finite mean waiting time lead to a space-fractional equation that accounts for superdiffusion and involves a nonlocal integral-differential operator. Within this framework, we analyze the evolution of particles performing symmetric Lévy flights with respect to a fluid moving at uniform speed . The particles are restricted to a semi-infinite domain limited by a reflective barrier. We show that the introduction of the boundary condition induces a modification in the kernel of the nonlocal operator. Thus, the macroscopic space-fractional advection-diffusion equation obtained is different from that in an infinite medium.
Keywords
This publication has 25 references indexed in Scilit:
- Uncoupled continuous-time random walks: Solution and limiting behavior of the master equationPhysical Review E, 2004
- Discrete random walk models for space–time fractional diffusionChemical Physics, 2002
- Fractional diffusion: probability distributions and random walk modelsPhysica A: Statistical Mechanics and its Applications, 2002
- The random walk's guide to anomalous diffusion: a fractional dynamics approachPhysics Reports, 2000
- Fractional reaction–diffusionPhysica A: Statistical Mechanics and its Applications, 2000
- Stochastic foundations of fractional dynamicsPhysical Review E, 1996
- Stochastic pathway to anomalous diffusionPhysical Review A, 1987
- Random walks on lattices. IV. Continuous-time walks and influence of absorbing boundariesJournal of Statistical Physics, 1973
- Generalized master equations for continuous-time random walksJournal of Statistical Physics, 1973
- Random Walks on Lattices. IIJournal of Mathematical Physics, 1965