Nonlinear equation for anomalous diffusion: unified power-law and stretched exponential exact solution

Abstract

The nonlinear diffusion equation $\frac{\partial ρ}{\partial t}=D \tildeΔ ρ^ν$ is analyzed here, where $\tildeΔ\equiv \frac{1}{r^{d-1}}\frac{\partial}{\partial r} r^{d-1-θ} \frac{\partial}{\partial r}$, and $d$, $θ$ and $ν$ are real parameters. This equation unifies the anomalous diffusion equation on fractals ($ν=1$) and the spherical anomalous diffusion for porous media ($θ=0$). Exact point-source solution is obtained, enabling us to describe a large class of subdiffusion ($θ> (1-ν)d$), normal diffusion ($θ= (1-ν)d$) and superdiffusion ($θ< (1-ν)d$). Furthermore, a thermostatistical basis for this solution is given from the maximum entropic principle applied to the Tsallis entropy.