特慢扩散的一种分数阶结构导数模型

A Fractional Structural Derivative Model for Ultra-Slow Diffusion

自然界和工程中存在很多比幂率慢扩散(sub-diffusion)过程更慢的扩散,即特慢扩散(ultra-slow diffusion).特慢扩散难以用传统的反常扩散建模方法来描述.Sinai(西奈)随机模型描述了一种特殊的对数关系特慢扩散.运用Mittag-Leffler(米塔格-累夫勒)函数的反函数,将Sinai扩散拓展为一般的特慢扩散.此外,该文的模型引入初始状态参量,解决了Sinai对数扩散不适用于初始时刻附近的问题.作为分数阶导数的一般情况,该文也引入了分数阶结构导数的概念,并用来建立特慢扩散的控制微分方程.

The ultra-slowdiffusion is even more slowthan the power-lawsub-diffusion and is widely observed in a variety of natural and engineering fields. The ultra-slowdiffusion cannot be well described with the traditional anomalous diffusion models. The Sinai＇s lawof diffusion depicts a special type of ultra-slowdiffusion which is characterized by a logarithmic stochastic relationship. In this study,the Sinai diffusion was extended to a general ultra-slowdiffusion. In addition,in the proposed model the initial parameters were introduced to remedy the perplexing issue that the Sinai diffusion was not feasible around the initial period of the ultra-slowdiffusion. As a generalized fractional-order derivative,the concept of the fractional structural derivative was also presented to establish the partial differential equation governing the ultra-slowdiffusion.