Retarding subdiffusion and accelerating superdiffusion governed by distributed-order fractional diffusion equations

Abstract

We propose diffusionlike equations with time and space fractional derivatives of the distributed order for the kinetic description of anomalous diffusion and relaxation phenomena, whose diffusion exponent varies with time and which, correspondingly, cannot be viewed as self-affine random processes possessing a unique Hurst exponent. We prove the positivity of the solutions of the proposed equations and establish their relation to the continuous-time random walk theory. We show that the distributed-order time fractional diffusion equation describes the subdiffusion random process that is subordinated to the Wiener process and whose diffusion exponent decreases in time (retarding subdiffusion). This process may lead to superslow diffusion, with the mean square displacement growing logarithmically in time. We also demonstrate that the distributed-order space fractional diffusion equation describes superdiffusion phenomena with the diffusion exponent increasing in time (accelerating superdiffusion).