Integrodifferential diffusion equation for continuous-time random walk

Abstract

In this paper we present an integro-differential diffusion equation for continuous time random walk that is valid for a generic waiting time probability density function. Using this equation we also study diffusion behaviors for a couple of specific waiting time probability density functions such as exponential, and a combination of power law and generalized Mittag-Leffler function. We show that for the case of the exponential waiting time probability density function a normal diffusion is generated and the probability density function is Gaussian distribution. In the case of the combination of a power-law and generalized Mittag-Leffler waiting probability density function we obtain the subdiffusive behavior for all the time regions from small to large times, and probability density function is non-Gaussian distribution.

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

“…There the memory kernel is connected to the cumulative distribution function of waiting times, i.e., one minus the probability to observe no step up to time t. Similar equations in Caputo or Riemann-Liouville form were considered in Refs. [23,104,116]. In Ref.…”

Uncoupled continuous-time random walk: finite jump length probability density function

“…In this work, we have investigated the uncoupled CTRW model with the exponential jump length PDF in a one-dimensional space given by (20). We have presented analytical solutions for the probability density function and second moment with different kinds of waiting-time PDF.…”

“…Recently, an integro-differential diffusion equation was derived from equation ( 8) with a generic waiting-time PDF [20], which is written as…”

From continuous time random walks to the generalized diffusion equation

“…In this generalized diffusion equation the memory kernel appears on the right hand side of the equation, i.e., this equation is of what we call the modified form in comparison to the generalized diffusion equation in normal form (or natural form) where the memory kernel appears on the left side of the equation. Special cases of the generalized equations in normal and modified forms have been extensively investigated in different contexts, for example, in[3,4,6,7,10,22,26,46,47,59].From Eq. (3.4) we derive the general form of the n-th moment (n ∈ N ), by using…”