Anomalous diffusion problems are used to describe the evolution of particle's motion in crowded environments with many applications, such as modeling the intracellular transport and disordered media. In the present paper, we develop a Petrov–Galerkin spectral method for the fourth‐order anomalous fractional diffusion equations. For the dimension reduction, we use a discretization in time by the convolution quadrature. We then introduce the basis sets for the trial‐and‐test spaces using modal Bernstein basis functions with a presentation of the method in a weak spectral formulation along with a discussion of the structure of the resulting systems, the convergence, and stability of the proposed method. The theoretical results are supported by illustrating some numerical examples.

中文翻译：

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Petrov-Galerkin谱方法用于分数异常扩散的数值模拟和分析

异常扩散问题用于描述拥挤环境中粒子运动的演变，具有许多应用程序，例如对细胞内运输和无序介质进行建模。在本文中，我们为四阶反常分数阶扩散方程开发了Petrov-Galerkin谱方法。对于降维，我们通过卷积正交使用时间离散化。然后，我们使用模态Bernstein基函数介绍试验空间的基集，并在弱频谱公式中介绍该方法，并讨论所得系统的结构，拟议系统的收敛性和稳定性方法。通过举例说明一些数值示例来支持理论结果。