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

Abstract

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

3 pages, 2 eps figures

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