The Global Diffusion Limit for the Space Dependent Variable-Order Time-Fractional Diffusion Equation

TL;DR

This work addresses diffusion in spatially inhomogeneous media by deriving a globally valid space-dependent variable-order fractional diffusion equation from stochastic models. It demonstrates that naive local diffusion limits fail when α varies in space and provides two robust pathways—the CTRW with space-dependent Mittag-Leffler waiting times (with a global scale τ and a characteristic time t0) and the DTRW with self-jumping—to realize a global diffusion limit with κ_{α(x)}=D t_0^{−α(x)}. The authors further develop numerical methods and Monte Carlo schemes that align with the VO-FDE, enabling calibration and validation for systems with traps and obstacles. The framework yields insights into symmetry-breaking localization at minima of α(x) and offers a physically grounded basis for modeling diffusion in spatially heterogeneous environments.

Abstract

The diffusion equation and its time-fractional counterpart can be obtained via the diffusion limit of continuous-time random walks with exponential and heavy-tailed waiting time distributions. The space dependent variable-order time-fractional diffusion equation is a generalization of the time-fractional diffusion equation with a fractional exponent that varies over space, modelling systems with spatial heterogeneity. However, there has been limited work on defining a global diffusion limit and an underlying random walk for this macroscopic governing equation, which is needed to make meaningful interpretations of the parameters for applications. Here, we introduce continuous time and discrete time random walk models that limit to the variable-order fractional diffusion equation via a global diffusion limit and space- and time- continuum limits. From this, we show how the master equation of the discrete time random walk can be used to provide a numerical method for solving the variable-order fractional diffusion equation. The results in this work provide underlying random walks and an improved understanding of the diffusion limit for the variable-order fractional diffusion equation, which is critical for the development, calibration and validation of models for diffusion in spatially inhomogeneous media with traps and obstacles.

Figures (4)

• Figure 1: Plot of the PDFs, $p(x,t)$, calculated via numerical integration of the DTRW (solid lines) and Monte Carlo simulation (dashed lines). Different colors show different times. The space dependent exponent was $\alpha(x) = 0.8+0.1x$ with $x\in[0,1]$ and $11$ discrete spatial lattice points such that $\Delta x=0.1$. The initial condition was $p(x,t) = \delta(x-0.5)$.
• Figure 2: Plot of the PDFs, $p(x,t)$, with the same parameters as Figure \ref{['fig:dtrw_mc_1']} except the initial condition was $p(x,t) = \delta(x-0.2)$.
• Figure 3: Plot of the PDFs, $p(x,t)$, with the same parameters as Figure \ref{['fig:dtrw_mc_1']} except the fractional exponent is $\alpha(x) = 0.7-0.1\sin(2\pi x)$.
• Figure 4: Plot of the PDFs, $p(x,t)$, with the same parameters as Figure \ref{['fig:dtrw_mc_1']} except the fractional exponent is $\alpha(x) = 0.7+0.1\sin(2\pi x)$.