In this paper, we mainly study the solution and properties of the multiterm time-fractional diffusion equation. First, we obtained the stochastic representation for this equation, which turns to be a subordinated process. Based on the stochastic representation, we calculated the mean square displacement (MSD) and time average mean square displacement, then proved some properties of this model, including subdiffusion, generalized Einstein relationship, and nonergodicity. Finally, a stochastic simulation algorithm was developed for the visualization of sample path of the abnormal diffusion process. The Monte Carlo method was also employed to show the behavior of the solution of this fractional equation.

中文翻译：

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多项式时间分数阶扩散方程的随机表示和蒙特卡罗模拟

在本文中，我们主要研究多项式时间分数阶扩散方程的解和性质。首先，我们获得了该方程的随机表示，这变成了从属过程。基于随机表示，我们计算了均方位移（MSD）和时间均方位移，然后证明了该模型的一些特性，包括子扩散，广义爱因斯坦关系和非遍历性。最后，开发了一种随机仿真算法，用于可视化异常扩散过程的样本路径。蒙特卡罗方法也被用来显示该分数方程解的行为。