$\mathcal{N}$-dimensional fractional diffusion equation and Green function approach: Spatially dependent diffusion coefficient and external force

We investigate an $\mathcal{N}$-dimensional fractional diffusion equation with radial symmetry by using the Green function approach. We consider, in our analysis, the spatial dependence on the diffusion coefficient and the presence of an external force. In particular, we employ boundary conditions in a finite interval and after we extend it to a semi-infinite interval. We also show that a rich class of diffusive processes, including normal and anomalous ones, can be obtained from the solutions found here.

Figure 1

Behavior of $\mathcal{G}(r,\xi ,t)$ versus $r$ for typical values of $\theta$ and $\gamma$, by considering, for simplicity, $t=1.0$, $\xi =2.0$, $a=6.0$, $\mathcal{N}=3.0$, and $\mathcal{D}=1.0$, in arbitrary units.

Figure 2

Behavior of $\rho (r,t)$ versus $r$ for typical values of $t$ by considering, for simplicity, $\gamma =0.5$, $a=6.0$, $\mathcal{N}=3.0$, $\theta =0$, and $\mathcal{D}=1.0$. The initial condition used for this case is the $\mathcal{N}$-dimensional Dirac $\delta$ function, i.e., $\rho (r,0)=\delta \left(r\right)∕r^{\mathcal{N}-1}$, in arbitrary units.

Figure 3

Behavior of $\mathcal{G}(r,\xi ,t)$ versus $r$ for a typical values of $\theta$ by considering, for simplicity, $t=1.0$, $\xi =2.0$, $\mathcal{N}=3.0$, and $\mathcal{D}=1.0$, in arbitrary units.