Vogl Christopher J, Miksis Michael J, Davis Stephen H
Department of Engineering Sciences and Applied Mathematics , Northwestern University , 2145 Sheridan Road, Evanston, IL 60208-3125, USA.
Proc Math Phys Eng Sci. 2012 Nov 8;468(2147):3348-3369. doi: 10.1098/rspa.2012.0170. Epub 2012 Jun 20.
Anomalous diffusion can be characterized by a mean-squared displacement 〈x(2)(t)〉 that is proportional to t(α) where α≠1. A class of one-dimensional moving boundary problems is investigated that involves one or more regions governed by anomalous diffusion, specifically subdiffusion (α<1). A novel numerical method is developed to handle the moving interface as well as the singular history kernel of subdiffusion. Two moving boundary problems are solved: the first involves a subdiffusion region to the one side of an interface and a classical diffusion region to the other. The interface will display non-monotone behaviour. The subdiffusion region will always initially advance until a given time, after which it will always recede. The second problem involves subdiffusion regions to both sides of an interface. The interface here also reverses direction after a given time, with the more subdiffusive region initially advancing and then receding.
反常扩散可以通过均方位移〈x²(t)〉来表征，其与t^α成正比，其中α≠1。研究了一类一维移动边界问题，该问题涉及一个或多个由反常扩散（特别是亚扩散，α<1）控制的区域。开发了一种新颖的数值方法来处理移动界面以及亚扩散的奇异历史核。求解了两个移动边界问题：第一个问题涉及界面一侧的亚扩散区域和另一侧的经典扩散区域。界面将呈现非单调行为。亚扩散区域最初总是会前进，直到给定时间，之后它总是会后退。第二个问题涉及界面两侧的亚扩散区域。这里的界面在给定时间后也会改变方向，亚扩散性更强的区域最初前进然后后退。
