Anomalous diffusion associated with nonlinear fractional derivative fokker-planck-like equation: exact time-dependent solutions
We consider the d=1 nonlinear Fokker-Planck-like equation with fractional derivatives ( partial differential/ partial differentialt)P(x,t)=D( partial differential(gamma)/ partial differentialx(gamma))[P(x,t)](nu). Exact time-dependent solutions are found for nu=(2-gamma)/(1+gamma)(-infinity<gamma</=2). By considering the long-distance asymptotic behavior of these solutions, a connection is established, namely, q=(gamma+3)/(gamma+1)(0<gamma</=2), with the solutions optimizing the nonextensive entropy characterized by index q. Interestingly enough, this relation coincides with the one already known for Levy-like superdiffusion (i.e., nu=1 and 0<gamma</=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.
Bologna
M
Department of Physics, University of North Texas, P.O. Box 311427, Denton, Texas 76203, USA.
Tsallis
C
Grigolini
P