Nonlinear diffusion equation and nonlinear external force: Exact solution

Abstract

The solutions of the nonlinear diffusion equation ∂tρ=r1−ND∂r{rN−1−θργ∂r[r−ηρν]}−r1−N∂r[rN−1Fρ] are investigated by considering the presence of an external force F which exhibits an explicit dependence on the distribution. First, the stationary case is considered; after that the dynamical case, i.e., the case dependent on time. The stationary solution is obtained by considering the external force F(r;ρ)=F1(r)+F2(r)[ρ(r)]ν+γ−1 and the result found is related to the distributions which emerge from the Tsallis statistics or the Boltzmann-Gibbs statistics. The dynamical solution is investigated by considering the external force F(r,t;ρ)=−k(t)r+K∕r1+θ+η[ρ(r,t)]γ+ν−1 and related to the Levy distributions in the asymptotic limit. In both cases, the solutions are expressed in terms of the q-exponentials and the q-logarithmics functions which emerge from the Tsallis formalism.