Inhomogeneous parametric scaling and variable-order fractional diffusion equations

We discuss the derivation and the solutions of integrodifferential equations (variable-order time-fractional diffusion equations) following as continuous limits for lattice continuous time random walk schemes with power-law waiting-time probability density functions whose parameters are position-dependent. We concentrate on subdiffusive cases and discuss two situations as examples: A system consisting of two parts with different exponents of subdiffusion, and a system in which the subdiffusion exponent changes linearly from one end of the interval to another one. In both cases we compare the numerical solutions of generalized master equations describing the process on the lattice to the corresponding solutions of the continuous equations, which follow by exact solution of the corresponding equations in the Laplace domain with subsequent numerical inversion using the Gaver-Stehfest algorithm.

Figure 1

Enumeration of sites at the boundary of two media as applied in Eq. (27).

Figure 2

The PDF $p(x,t)$ in a system with ${\alpha }_{-}=0.8$ in its left and ${\alpha }_{+}=0.9$ in its right part for $t={10}^{-2}$ (upper panel) and for $t={10}^3$ (lower panel). The system consists of 100 sites, and the border is placed between sites 50 and 51. The initial condition is $p_i(t=0)=1/100$. The characteristic times are chosen with ${\tau }_{\pm }=\tau ={10}^{-5}$. The graph shows the full numerical solution (black crosses), the (semi)analytical solution as obtained by numerical Laplace inversion of Eq. (36) (red dotted line), and the approximation as given by Eq. (39) (blue dashed line).

Figure 3

The PDF $p(x,t)$ in a system with linearly changing $\alpha \left(x\right)$ for time $t=0.1$ (upper panel) and $t={10}^5$ (lower panel). We used $\alpha \left(x\right)=0.4+0.005x$ for $x=[1,100]$. Choosing $\tau ={10}^{-5}$ led to $K_{\alpha \left(0\right)}=100$ on the left boundary and $K_{\alpha \left(L\right)}\approx 3\times {10}^4$ on the right boundary of the interval. The initial condition is $p(x,t=0)=1/100$. The results of the numerical solution of the GME are shown as crosses, the (semi)analytical solution obtained by a numerical Laplace inversion of the analytical solution, Eq. (45) are shown as red dotted lines, and the results of approximate analytical inversion, Eq. (39), by the blue dashed ones.