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

Abstract

We propose diffusion-like 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, can not 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 the relation to the Continuous Time Random Walk theory. We show that the distributed order time fractional diffusion equation describes the sub-diffusion random process which is subordinated to the Wiener process and whose diffusion exponent diminishes in time (retarding sub-diffusion) leading to superslow diffusion, for which the square displacement grows logarithmically in time. We also demonstrate that the distributed order space fractional diffusion equation describes super-diffusion phenomena when the diffusion exponent grows in time (accelerating super-diffusion).

Time distributed-order diffusion-wave equation. II. Applications of Laplace and Fourier transformations

“…Similarly, as in Chechkin et al (2002) and Kochubei (2008), rewrite (3.12) in the formĝ ðu; sÞ Z ð N 0 e KpðB u ðsÞCu 2 Þ dp; uR 0; ReB u ðsÞO 0; s 2 C C :…”

“…Another paper dealing with the similar equation is that of Langlands (2006). Furthermore, we mention that the time and space distributed-order diffusion equation was studied by Chechkin et al (2002), through the analysis of second moments of certain probability density functions. A similar approach can be found in the paper of Sokolov et al (2004).…”

A series of high-order quasi-compact schemes for space fractional diffusion equations based on the superconvergent approximations for fractional derivatives

“…Fractional derivatives have been widely applied to model the problems in physics [2,4,5,9,15,18,28], finance [26,27], and hydrology [3,30], especially to the anomalous diffusion or dispersion, where a particle plume spreads at a rate inconsistent with the classical Brownian motion [10,16,21]. The suitable mathematical models are the generalization to the classical diffusion equations formally replacing the classical first order derivative in time by the Caputo fractional derivative of order α ∈ (0, 1), and the second order derivative in space by the Riemann-Liouville fractional derivative of order α ∈ (1, 2].…”

Time distributed-order diffusion-wave equation. I. Volterra-type equation

“…For example, when the time derivative in the classical diffusion equation is replaced by the fractional time derivative (we call it fractionalization) of order a, 0!a!2, then a diffusion-wave equation is obtained, (see Mainardi 1996Mainardi , 1997Gorenflo et al 2000;Hanyga 2002a,b;Atanackovic et al 2007;Mainardi et al 2008;Chechkin et al 2002;Atanackovic et al 2005). The fractionalization of differential equations of mathematical physics leads to the analysis of parameter a, which has to be determined through experimental results.…”