Distributed-order diffusion equations and multifractality: Models and solutions

We study distributed-order time fractional diffusion equations characterized by multifractal memory kernels, in contrast to the simple power-law kernel of common time fractional diffusion equations. Based on the physical approach to anomalous diffusion provided by the seminal Scher-Montroll-Weiss continuous time random walk, we analyze both natural and modified-form distributed-order time fractional diffusion equations and compare the two approaches. The mean squared displacement is obtained and its limiting behavior analyzed. We derive the connection between the Wiener process, described by the conventional Langevin equation and the dynamics encoded by the distributed-order time fractional diffusion equation in terms of a generalized subordination of time. A detailed analysis of the multifractal properties of distributed-order diffusion equations is provided.

Figure 1

PDF (14) of the fractional diffusion equation (2) for $\alpha =1/2, K_{\alpha }=1$, and $t=0.1$ (blue solid line), $t=1$ (red dashed line), and $t=10$ (green dot-dashed line).

Figure 2

Double-logarithmic plot of the waiting-time PDF (21) for $\tau =1$ and ${\mathcal{D}}_{\alpha }=1$, with $\alpha =1/2$ (blue solid line), $\alpha =5/8$ (red dashed line), and $\alpha =3/4$ (green dot-dashed line). Inset: $q\mathrm{th}$-order moments (7) for the same parameter values and color codes.

Figure 3

Double-logarithmic plot of the MSD (40) for $2\mathcal{K}\tau =1, B_1=B_2=1/2, {\lambda }_1=1/4$, and ${\lambda }_2=1$ (blue solid line); ${\lambda }_2=3/4$ (red dashed line); ${\lambda }_2=1/2$ (green dot-dashed line). Inset: Fourth moments (44) for the same parameters.

Figure 4

MSD of a CTRW process with mixed waiting-time PDFs (blue line). The MSD for the corresponding CTRW process with a single-power waiting-time PDF with the larger exponent and its asymptotic for the short-time limit is given by the green lines, and correspondingly with the smaller exponent and its asymptotic in the long-time limit is shown by the red lines. The fractional exponents are ${\lambda }_1=0.35$ and ${\lambda }_2=0.75, 2\mathcal{K}\tau =1$, and the weights are set to $B_1=0.01$ and $B_2=0.99$.

Figure 5

Second derivative with respect to $q$ of $c_1\left(q\right)$ (blue line) and $c_2\left(q\right)$ (red line) for ${\lambda }_1=1/2$ and ${\lambda }_2=7/8$.

Figure 6

Double-logarithmic representation of the MSD (84) for $\mathcal{K}\tau =1/2, B_1=B_2=1/2, {\lambda }_1=1/4$, and ${\lambda }_2=1$ (blue solid line); ${\lambda }_2=3/4$ (red dashed line); and ${\lambda }_2=1/2$ (green dot-dashed line). Inset: Fourth-order moment (86) for the same values of parameters.

Figure 7

Sample trajectories of two CTRWs $x_1$ (red), $x_2$ (blue), and the composite process (black) which is the sum of $x_1$ and $x_2$. Parameters are ${\lambda }_1=0.3, K_{{\lambda }_1}=3.06, {\lambda }_2=0.75, K_{{\lambda }_2}=0.138 [B_1=K_{{\lambda }_1}/\left(K_{{\lambda }_1}+K_{{\lambda }_2}\right)$ and $B_2=K_{{\lambda }_2}/\left(K_{{\lambda }_1}+K_{{\lambda }_2}\right)]$. Inset: MSD of the composite process $x_1+x_2$ (solid line) with ${\lambda }_1=0.3, K_{{\lambda }_1}=3.06$ and ${\lambda }_2=0.75, K_{{\lambda }_2}=0.08$. Dashed lines represent the two limits $t^{{\lambda }_1}$ and $t^{{\lambda }_2}$.

Figure 8

PDF of the sum of two CTRWs with ${\lambda }_1=0.3, K_{{\lambda }_1}=3.06$ and ${\lambda }_2=0.75, K_{{\lambda }_2}=0.138$ calculated for $t=10, t={10}^2, t={10}^3, t={10}^4$, and $t={10}^5$. Dashed lines represent the PDF of the single CTRW process with ${\lambda }_1=0.3, K_{{\lambda }_1}=3.06$ calculated for $t=10, t={10}^2$; dashed-dotted lines represent the PDF of the single CTRW with ${\lambda }_2=0.75, K_{{\lambda }_2}=0.138$ calculated for $t={10}^4$ and $t={10}^5 [B_1=K_{{\lambda }_1}/\left(K_{{\lambda }_1}+K_{{\lambda }_2}\right), B_2=K_{{\lambda }_2}/\left(K_{{\lambda }_1}+K_{{\lambda }_2}\right)]$. (b) Same as in (a) in double-logarithmic scale.

Figure 9

Second derivative with respect to $q$ of $c_3\left(q\right)$ (blue line) and $c_4\left(q\right)$ (red line), for ${\lambda }_1=1/2$ and ${\lambda }_2=7/8$.

Figure 10

Double-logarithmic representation of the three-parameter Mittag-Leffler function (A13) for $\alpha =3/4, \beta =1$, and $\delta =1/2$ (blue solid line). The red dashed line is the stretched exponential function (A15) and the green dot-dashed line is the power-law function (A16) for the same values of parameters.