Cattaneo--type subdiffusion equation

TL;DR

The paper introduces Cattaneo-type subdiffusion equations (CTSE) by incorporating a random delay in activating the ordinary subdiffusion flux, encoded through a delay-time density $R$ and resulting in an additional memory operator (AO). It derives CTSE within a standard CTRW framework and analyzes two concrete AO realizations: (i) a Caputo fractional derivative of order $1+\kappa$ (via $\hat{\gamma}(s)=s^{\kappa}$) and (ii) an AO generated by a slowly varying function $\omega$, obtaining explicit Green's functions, MSD, and first-passage times. The long-time behavior shows the Cattaneo effect decays as $\tau/t^{\kappa}$ for the Caputo case or as $\tau\omega(t)$ for slowly varying AO, while the short-time behavior can exhibit enhanced or delayed spreading depending on the AO. Although differences in MSD may be small, the CTSE can significantly alter the tails of the Green's function and the first-passage statistics, with potential implications for diffusion-driven processes like epidemic spread or antibiotic transport in complex media.

Abstract

The ordinary subdiffusion equation, with a fractional time derivative of at most first order, describes a process in which the propagation velocity of diffusing molecules is unlimited. To avoid this non-physical property different forms of the Cattaneo subdiffusion equation have been proposed. We define the Cattaneo effect as a delay of the ordinary subdiffusion flux activation by a random time. By incorporating this effect into the flux equation we get a Cattaneo--type subdiffusion equation (CTSE). We study the CTSE that differs from the ordinary subdiffusion equation by an additional integro--differential operator (AO) controlled by a time delay probability distribution. A method for deriving CTSE within the standard continuous time random walk model is also shown. As examples, we consider the CTSE with AO being the Caputo fractional time derivative of the order independent of the subdiffusion exponent and with the AO with a kernel that is a slowly varying function. In the first case the Cattaneo effect disappears over time much faster than in the second one. Based on the Green's functions, the time evolutions of the first passage time distribution and of the mean square displacement of a diffusing molecule, we discuss whether the influence of the Cattaneo effect is significant. In the considered examples, this influence seems to be small. However, a relative probability of finding a molecule at a long distance from the starting point for the CTSE equation with respect to the ordinary subdiffusion equation increases rapidly with distance. Even small changes caused by the Cattaneo effect can lead to different results in modeling processes where a faster appearance of a diffusing object changes the nature of the process. The effect may be important, for example, in modeling the spread of an epidemic when a diffusing object is a source of infection.

Figures (9)

• Figure 1: Plots of the time evolution of the MSD $\sigma^2$ Eq. (\ref{['eq58']}) (panel $a$) and the function $\Delta\sigma^2=\sigma_{\tau=0}^2-\sigma^2$ for the values of parameters in the legend (panel $b$), here $D=5$.
• Figure 2: Time evolution of the relative MSD for the parameters given in the legend, $D=5$.
• Figure 3: Time evolution of the first passage time distribution (panel $a$) and the function $\Delta F=F_{\tau=0}-F$ (panel $b$) for the parameters given in the legend, $x=10$ and $D=5$.
• Figure 4: Time evolution of the relative first passage time for the parameters given in the legend, $x=10$ and $D=5$.
• Figure 5: Plots of Green's functions for $\alpha=0.5$ (panel $a$) and functions $\Delta P=P_{\tau=0}-P$ for parameters given in the legend, $t=20$ and $D=10$.