title: 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 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.
authors.affiliation: INRA d'Avignon, UMRA Climat Sol Environnement, Domaine Saint Paul--Site Agroparc, F-84914 Avignon Cedex 9, France. natalia@avignon.inra.fr
authors.forename: Natalia
authors.initials: N
authors.lastname: Krepysheva
authors.forename: Liliana
authors.initials: L
authors.lastname: Di Pietro
authors.forename: Marie-Christine
authors.initials: MC
authors.lastname: Néel