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 $\frac{\partial }{\partial t}P(x,t)=D\frac{{\partial }^{\gamma }}{\partial x^{\gamma }}[P(x,t){]}^{\nu }$. Exact time-dependent solutions are found for $\nu =\frac{2-\gamma }{1+\gamma }$ ($-\infty <\gamma \le 2$). By considering the long-distance {\it asymptotic} behavior of these solutions, a connection is established, namely $q=\frac{\gamma +3}{\gamma +1}$ ($0<\gamma \le 2$), 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., $\nu =1$ and $0<\gamma \le 2$). Finally, for $(\gamma ,\nu )=(2,0)$ we obtain $q=5/3$ which differs from the value $q=2$ corresponding to the $\gamma =2$ solutions available in the literature ($\nu <1$ porous medium equation), thus exhibiting nonuniform convergence.

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References in corpus (2)

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