Nonlinear anomalous diffusion equation and fractal dimension: exact generalized Gaussian solution.

Research paper by I T IT Pedron, R S RS Mendes, L C LC Malacarne, E K EK Lenzi

Indexed on: 15 May '02Published on: 15 May '02Published in: Physical review. E, Statistical, nonlinear, and soft matter physics


In this work we incorporate, in a unified way, two anomalous behaviors, the power law and stretched exponential ones, by considering the radial dependence of the N-dimensional nonlinear diffusion equation partial differential rho/ partial differential t=nabla.(Knablarho(nu))-nabla.(muFrho)-alpharho, where K=Dr(-theta), nu, theta, mu, and D are real parameters, F is the external force, and alpha is a time-dependent source. This equation unifies the O'Shaughnessy-Procaccia anomalous diffusion equation on fractals (nu=1) and the spherical anomalous diffusion for porous media (theta=0). An exact spherical symmetric solution of this nonlinear Fokker-Planck equation is obtained, leading to a large class of anomalous behaviors. Stationary solutions for this Fokker-Planck-like equation are also discussed by introducing an effective potential.