Subdiffusion equation with fractional Caputo time derivative with respect to another function in modeling transition from ordinary subdiffusion to superdiffusion

We use a subdiffusion equation with fractional Caputo time derivative with respect to another function $g$ ($g$-subdiffusion equation) to describe a smooth transition from ordinary subdiffusion to superdiffusion. Ordinary subdiffusion is described by the equation with the “ordinary” fractional Caputo time derivative, superdiffusion is described by the equation with a fractional Riesz-type spatial derivative. We find the function $g$ for which the solution (Green's function, GF) to the $g$-subdiffusion equation takes the form of GF for ordinary subdiffusion in the limit of small time and GF for superdiffusion in the limit of long time. To solve the $g$-subdiffusion equation we use the $g$-Laplace transform method. It is shown that the scaling properties of the GF for $g$-subdiffusion and the GF for superdiffusion are the same in the long time limit. We conclude that for a sufficiently long time the $g$-subdiffusion equation describes superdiffusion well, despite a different stochastic interpretation of the processes. Then, paradoxically, a subdiffusion equation with a fractional time derivative describes superdiffusion. The superdiffusive effect is achieved here not by making anomalously long jumps by a diffusing particle, but by greatly increasing the particle jump frequency which is derived by means of the $g$-continuous-time random walk model. The $g$-subdiffusion equation is shown to be quite general, it can be used in modeling of processes in which a kind of diffusion change continuously over time. In addition, some methods used in modeling of ordinary subdiffusion processes, such as the derivation of local boundary conditions at a thin partially permeable membrane, can be used to model $g$-subdiffusion processes, even if this process is interpreted as superdiffusion.

Figure 1

Plots of Green's functions for fractional superdiffusion Eq. (45) (solid lines with filled symbols) with $\gamma =1.5$ and $D_{\gamma }=4$, for ordinary subdiffusion (solid lines with half-filled symbols) Eq. (20) with $\alpha =0.7$ and $D_{\alpha }=10$, and for $g$-subdiffusion Eq. (36) (dashed lines with open symbols) with $g$ Eq. (59) and $E$ Eq. (56), $A=1, \nu =1.2$, and $\alpha , D_{\alpha }$ given above. The plots are made for times given in the legend. The plot of Green's function for fractional superdiffusion for $t=0.1$ (solid lines with filled squares) is presented in the interval $x\in (-1.6,1.6)$ only due to limited domain in which the series occurring in Eq. (45) is convergent.

Figure 2

Plots of Green's functions for fractional superdiffusion equation (45) (solid lines with filled symbols), and for $g$-subdiffusion equation (36) (dashed lines with open symbols) for times given in the legend, the values of the parameters are the same as in the caption of Fig. 1.

Figure 3

Plots of Green's functions for fractional superdiffusion Eq. (45) (solid lines with filled symbols), for $g$-subdiffusion Eq. (36) (dashed lines with open symbols) for $\nu$ given in the legend, and for ordinary subdiffusion (solid lines with half–filled symbols) Eq. (20), $t=1.5$, the values of the other parameters are the same as in the caption of Fig. 1.

Figure 4

Scaling property of Green's functions for $g$-subdiffusion Eq. (36), $v={(t^{\prime }/t)}^{1/\gamma }, t^{\prime }=1000$, values of $t$ are given in the legend, the other parameters are the same as in the caption of Fig. 1, more detailed description is in the text.

Figure 5

Time evolution of the mean number of particle steps for ordinary subdiffusion (denoted as “subdif” in the legend) and $g$-subdiffusion for different values of $\gamma$ given in the legend, $\tau =1$ and $\nu =1.2$.

Figure 6

Time evolution of the mean number of particle steps done on a logarithmic scale, the description is similar to that in Fig. 5.

Figure 7

Time evolution of the jump frequency $f\left(t\right)=d\langle n\left(t\right)\rangle /dt$ for the cases presented in Fig. 5. For ordinary subdiffusion $f$ is an decreasing function of time, the superdiffusion effect is achieved when $f$ is a function increasing with time.

Figure 8

Time evolution of the mean number of steps for ordinary subdiffusion and $g$-subdiffusion for different values of $\nu$ given in the legend, the values of other parameters are the same as in Figs. 1 and 5.