Distributed-order diffusion equations and multifractality: Models and solutions

Abstract

We study distributed-order time fractional diffusion equations characterized by multifractal memory kernels, in contrast to the simple power-law kernel of common time fractional diffusion equations. Based on the physical approach to anomalous diffusion provided by the seminal Scher-Montroll-Weiss continuous time random walk, we analyze both natural and modified-form distributed-order time fractional diffusion equations and compare the two approaches. The mean squared displacement is obtained and its limiting behavior analyzed. We derive the connection between the Wiener process, described by the conventional Langevin equation and the dynamics encoded by the distributed-order time fractional diffusion equation in terms of a generalized subordination of time. A detailed analysis of the multifractal properties of distributed-order diffusion equations is provided.

Diffusion and Fokker-Planck-Smoluchowski Equations with Generalized Memory Kernel

“…This case corresponds to the distributed order diffusion equation with two fractional exponents [89] (compare also Refs. [13,105]), which can be obtained if we substitute γ(t) in the generalized diffusion equation (2.11).…”

“…Ref. [89] shows that distributed order diffusion equations yield more complex forms of the q-th moment (4.3). In the context of strong anomalous diffusion and Lévy walks multi-scaling behaviour is connected to the theory of infinite invariant densities [75,85].…”

“…with 0 < α 1 < α 2 < 1 the waiting time PDF is an infinite series in the three parameter M-L functions (8.3) [89],…”

Numerical solution for diffusion equations with distributed order in time using a Chebyshev collocation method

“…As a second example [52,15,60,58], let us consider a finite variance of the jump length and the wait time assumed to be given by a power-law distribution ϕ (t) = τ α /t 1+α (0 < α < 1) that leads to a divergent mean waiting time. A process modelled in this way leads to subdiffusion with a mean squared displacement given by < x 2 >= (2K α /Γ (1 + α)) t α .…”

“…A close look into the physics of some complex diffusive processes, suggests that an even more general theory for fractional derivatives should be devised, and, this will no doubt be developed in the near future ( [58,50,52,39]). …”