Asymptotic Behavior of the Solution of the Space Dependent Variable Order Fractional Diffusion Equation: Ultraslow Anomalous Aggregation

We find the asymptotic representation of the solution of the variable-order fractional diffusion equation, which remains unsolved since it was proposed by Chechkin, Gorenflo, and Sokolov [J. Phys. A, 38, L679 (2005)]. We identify a new advection term that causes ultraslow spatial aggregation of subdiffusive particles due to dominance over the standard advection and diffusion terms in the long-time limit. This uncovers the anomalous mechanism by which nonuniform distributions can occur. We perform Monte Carlo simulations of the underlying anomalous random walk and find excellent agreement with the asymptotic solution.

Figure 1

Asymptotic density (3) (dashed line) and normalized histograms corresponding to simulation of $N={10}^4$ particles jumping between $k=50$ bins in the domain $0\le x\le 1$ with fractional exponent $\mu (x)=0.4+0.5x$ and ${\tau }_0={10}^{-3}$. In this simulation, $r_i=0.5$. The legend shows simulation times, $t$, at which snapshots of the distribution of particles were produced. Inset: Time series of number of particles $N(x,t)$ at specific $x$ positions for the same simulation in the main figure. The plot at $x=0$ clearly shows logarithmic growth as predicted by solution (3).

Figure 2

Asymptotic density (3) (dashed line) and normalized histograms corresponding to simulation of $N={10}^4$ particles jumping between $k=50$ bins in the domain $0\le x\le 1$ with fractional exponent $\mu (x)=0.4+0.5x$ and ${\tau }_0={10}^{-3}$. In this simulation, $r_i=\frac{1}{2}+\frac{0.5}{k}(0.5-(i/k))$. The legend shows simulation times, $t$, at which snapshots of the distribution of particles was produced. Inset: The steady state solution of the fractional Fokker-Planck equation for constant $\mu$ and $v(x)=1–2x$: $p(x)=C{e}^{-(x^2-x)}$ (solid line) and normalized histogram of particles at $t={10}^4$ with the same parameters and initial conditions as the main histograms except with $N=5\times {10}^4$ and mean fractional exponent $\overline{\mu }={\int }_0^1\mu (x)dx=0.65$.