Anomalous diffusion associated with nonlinear fractional derivative Fokker-Planck-like equation: Exact time-dependent solutions

Abstract

We consider the $d=1$ nonlinear Fokker-Planck-like equation with fractional derivatives $\frac{\partial}{\partial t}P(x,t)=D \frac{\partial^γ}{\partial x^γ}[P(x,t) ]^ν$. Exact time-dependent solutions are found for $ ν= \frac{2-γ}{1+ γ}$ ($-\infty<γ\leq 2$). By considering the long-distance {\it asymptotic} behavior of these solutions, a connection is established, namely $q=\frac{γ+3}{γ+1}$ ($0<γ\le 2$), with the solutions optimizing the nonextensive entropy characterized by index $q$ . Interestingly enough, this relation coincides with the one already known for Lévy-like superdiffusion (i.e., $ν=1$ and $0<γ\le 2$). Finally, for $(γ,ν)=(2, 0)$ we obtain $q=5/3$ which differs from the value $q=2$ corresponding to the $γ=2$ solutions available in the literature ($ν<1$ porous medium equation), thus exhibiting nonuniform convergence.