Abstract Various natural phenomena are described by anomalous diffusion processes. Notable examples relate to the study of diffusion of tracers in particles in turbulent flows, or in the propagation of acoustic waves. In this context, the governing equations involve a fractional Laplacian operator, which replaces the classical Laplacian, and may involve nonlinear terms. This leads to problems described by fractional nonlinear diffusion equations. In general, no analytical solutions are available for determining the response of these systems. Thus, the development of approximate approaches circumventing the use of computationally demanding numerical techniques is desirable. This paper proposes a statistical linearization based approach, which allows calculating approximately, albeit iteratively, the response statistics. The method is developed using a recently proposed representation of the fractional Laplacian in conjunction with a mode expansion of the system response. It is implemented by introducing non-orthogonal eigenfunctions of the fractional Laplacian of the response, which are obtained from the linear modes of the classical diffusion equation. Such a representation allows deriving a system of nonlinear ordinary differential equations, which is linearized in a stochastic mean square sense. Then, the response statistics and power spectral density are determined by an iterative procedure. Numerical results pertaining to a system with white noise excitation demonstrate the efficiency of the proposed approximate approach. Further, comparisons with data from relevant Monte Carlo results assess the reliability of the estimated response.

中文翻译：

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二维分数扩散系统响应统计的高效计算

摘要 异常扩散过程描述了各种自然现象。值得注意的例子与湍流中粒子中示踪剂扩散或声波传播的研究有关。在这种情况下，控制方程涉及分数拉普拉斯算子，它取代了经典拉普拉斯算子，并且可能涉及非线性项。这导致由分数非线性扩散方程描述的问题。通常，没有可用的分析解决方案来确定这些系统的响应。因此，开发近似方法来规避计算要求高的数值技术的使用是可取的。本文提出了一种基于统计线性化的方法，该方法允许近似地（尽管是迭代地）计算响应统计量。该方法是使用最近提出的分数拉普拉斯算子的表示结合系统响应的模式展开来开发的。它是通过引入响应的分数拉普拉斯算子的非正交特征函数来实现的，这些特征函数是从经典扩散方程的线性模式中获得的。这种表示允许推导非线性常微分方程系统，该系统在随机均方意义上线性化。然后，响应统计和功率谱密度由迭代过程确定。与具有白噪声激励的系统有关的数值结果证明了所提出的近似方法的效率。此外，与来自相关蒙特卡罗结果的数据进行比较可评估估计响应的可靠性。