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Anomalous diffusion, nonlinear fractional Fokker-Planck equation and solutions. (English) Zbl 1008.82027

Summary: We obtain new exact classes of solutions for the nonlinear fractional Fokker-Planck-like equation \(\partial_t\rho=\partial_x\{D(x)\partial^{\mu-1}_x\rho^{\nu}-F(x)\rho\}\) by considering a diffusion coefficient \(D=D|x|^{-\theta}\) \((\theta\in \mathbb{R}\) and \(D>0)\) and a drift force \(F=-k_1x+\bar k_{\gamma}x|x|^{\gamma-1} (k_1,\bar k_{\gamma},\gamma\in \mathbb{R})\). Connection with nonextensive statistical mechanics based on Tsallis entropy is also discussed.

MSC:

82C70 Transport processes in time-dependent statistical mechanics
82C31 Stochastic methods (Fokker-Planck, Langevin, etc.) applied to problems in time-dependent statistical mechanics
35K55 Nonlinear parabolic equations
35C05 Solutions to PDEs in closed form
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