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Space-fractional advection-diffusion and reflective boundary condition

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 Levy 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 Levy flights with respect to a fluid moving at uniform speed v. 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
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Natalia Krepysheva, Liliana Di Pietro, Marie-Christine Neel. Space-fractional advection-diffusion and reflective boundary condition. Physical Review E : Statistical, Nonlinear, and Soft Matter Physics, American Physical Society, 2006, 73 (2), pp.021104. ⟨10.1103/PhysRevE.73.021104⟩. ⟨hal-02667236⟩

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