Sandev Trifce, Chechkin Aleksei V, Korabel Nickolay, Kantz Holger, Sokolov Igor M, Metzler Ralf
Max Planck Institute for the Physics of Complex Systems, Nöthnitzer Strasse 38, 01187 Dresden, Germany.
Radiation Safety Directorate, Partizanski odredi 143, P.O. Box 22, 1020 Skopje, Macedonia.
Phys Rev E Stat Nonlin Soft Matter Phys. 2015 Oct;92(4):042117. doi: 10.1103/PhysRevE.92.042117. Epub 2015 Oct 7.
We study distributed-order time fractional diffusion equations characterized by multifractal memory kernels, in contrast to the simple power-law kernel of common time fractional diffusion equations. Based on the physical approach to anomalous diffusion provided by the seminal Scher-Montroll-Weiss continuous time random walk, we analyze both natural and modified-form distributed-order time fractional diffusion equations and compare the two approaches. The mean squared displacement is obtained and its limiting behavior analyzed. We derive the connection between the Wiener process, described by the conventional Langevin equation and the dynamics encoded by the distributed-order time fractional diffusion equation in terms of a generalized subordination of time. A detailed analysis of the multifractal properties of distributed-order diffusion equations is provided.
我们研究了以多重分形记忆核为特征的分布阶时间分数阶扩散方程，这与普通时间分数阶扩散方程的简单幂律核形成对比。基于开创性的Scher-Montroll-Weiss连续时间随机游走所提供的反常扩散物理方法，我们分析了自然形式和修正形式的分布阶时间分数阶扩散方程，并对这两种方法进行了比较。得到了均方位移并分析了其极限行为。我们根据时间的广义从属关系，推导了由传统朗之万方程描述的维纳过程与分布阶时间分数阶扩散方程所编码的动力学之间的联系。对分布阶扩散方程的多重分形性质进行了详细分析。
