Fractional diffusion equation, boundary conditions and surface effects

Abstract

We investigate a system governed by a fractional diffusion equation with an integro-differential boundary condition on the surface. This condition can be connected with several processes such as adsorption and/ or desorption or chemical reactions due to the presence of active sites on the surface. The solutions are obtained by using the Green function approach and show a rich class of behaviors, which can be related to anomalous diffusion.

New Solution of Diffusion–Advection Equation for Cosmic-Ray Transport Using Ultradistributions

“…This equation can be cast as a diffusion one, in which the Laplacian is replaced by a term involving fractional derivatives [18]. See also [19,20,21,22,23,24,25], and references therein. An interesting step towards a more accurate analytical treatment of this problem was recently provided by Litvinenko and Effenberger (LE) in [8].…”

General solution of a fractional diffusion–advection equation for solar cosmic-ray transport

“…It arises as solution a non local diffusive process governed by an integral equation that can be recast under the guise of a diffusion equation where the well-known Laplacian term is replaced by a term involving fractional derivatives [7]. Diffusion equations with fractional derivatives have attracted considerable attention recently (see [8,9,10,11,12] and references therein) and have lots of potential applications [13,14]. In particular, the observed distributions of solar cosmic ray particles are often consistent with power-law tails, suggesting that a superdiffusive process is at work.…”

Features of the fractional diffusion-advection equation

“…It arises as solution a non local diffusive process governed by an integral equation that can be recast under the guise of a diffusion equation where the well-known Laplacian term is replaced by a term involving fractional derivatives [5]. Diffusion equations with fractional derivatives have attracted considerable attention recently (see [9,10,11,12,13] and references therein) and have lots of potential applications [14,15]. In particular, the observed distributions of solar cosmic ray particles are often consistent with power-law tails, suggesting that a superdif-fusive process is at work.…”