Solution of a modified fractional diffusion equation

Recently, a modified fractional diffusion equation has been proposed [I. Sokolov, J. Klafter, From diffusion to anomalous diffusion: a century after Einstein’s brownian motion, Chaos 15 (2005) 026103; A.V. Chechkin, R. Gorenflo, I.M. Sokolov, V.Yu. Gonchar, Distributed order time fractional diffusion equation, Frac. Calc. Appl. Anal. 6 (3) (2003) 259–279; I.M. Sokolov, A.V. Chechkin, J. Klafter, Distributed-order time fractional kinetics, Acta. Phys. Pol. B 35 (2004) 1323.] for describing processes that become less anomalous as time progresses by the inclusion of a second fractional time derivative acting on the diffusion term. In this letter we give the solution of the modified equation on an infinite domain. In contrast to the solution of the traditional fractional diffusion equation, the solution of the modified equation requires a summation of Fox functions instead of a single Fox function

Dynamical continuous time random walk

© 2015, EDP Sciences, SIF, Springer-Verlag Berlin Heidelberg. We consider a continuous time random walk model in which each jump is considered to be dynamical process. Dissipative launch velocity and hopping time in each jump is the key factor in this model. Within the model, normal diffusion and anomalous diffusion is realized theoretically and numerically in the force free potential. Besides, external potential can be introduced naturally, so the random walker’s behavior in the linear potential and quartic potential is discussed, especially the walker with Lévy velocity in the quartic potential, bimodal behavior of the spatial distribution is observed, it is shown that due to the inertial effect induced by damping term, there exists transition from unimodality to bimodality for the walker’s spatial stationary distribution