非线性反常扩散方程与分形维数：精确广义高斯解
Nonlinear anomalous diffusion equation and fractal dimension: exact generalized Gaussian solution.
作者信息
Pedron I T, Mendes R S, Malacarne L C, Lenzi E K
机构信息
Departamento de Física, Universidade Estadual de Maringá, Avenida Colombo 5790, 87020-900 Maringá, Paraná, Brazil.
出版信息
Phys Rev E Stat Nonlin Soft Matter Phys. 2002 Apr;65(4 Pt 1):041108. doi: 10.1103/PhysRevE.65.041108. Epub 2002 Apr 10.
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.
在这项工作中，我们通过考虑N维非线性扩散方程(\frac{\partial\rho}{\partial t}=\nabla\cdot(K\nabla\rho^{\nu})-\nabla\cdot(\mu F\rho)-\alpha\rho)的径向依赖性，以统一的方式纳入了两种反常行为，即幂律行为和拉伸指数行为，其中(K = D r^{-\theta})，(\nu)、(\theta)、(\mu)和(D)是实参数，(F)是外力，(\alpha)是与时间有关的源项。该方程统一了分形上的奥肖内西 - 普罗卡恰反常扩散方程（(\nu = 1)）和多孔介质的球形反常扩散（(\theta = 0)）。我们得到了这个非线性福克 - 普朗克方程的一个精确球对称解，从而得到了一大类反常行为。通过引入有效势，还讨论了这个类福克 - 普朗克方程的定态解。
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