Abstract Models involving stochastic diffusion equations are utilized for describing the evolution of a number of natural phenomena and are widely discussed in the open literature. In recent years, these models have been revisited in light of experimental observations in which “anomalous” diffusion processes were identified, such as in the propagation of acoustic waves in random media. In this context, a critical characteristic of the theoretical models is the introduction of fractional derivative operators in the associated governing equations. Specifically, anomalous diffusion involves a fractional Laplacian operator replacing the classical Laplacian. Currently, solutions to equations with fractional Laplacians are available for a quite limited numbers of cases. Further, to the authors’ knowledge, no solutions are available for nonlinear equations involving fractional Laplacians. This fact creates the need of developing adequate numerical methods for estimating the response of this kind of systems. This paper proposes a Boundary Element Method (BEM)-based approach to determine the response of a system governed by a nonlinear fractional diffusion equation involving a random excitation. The approach is constructed by utilizing the integral representation of the classical Poisson equation solution, in which unknown constants are determined by the BEM. Then, based on a recently proposed representation of the fractional Laplacian operator, the value of the fractional Laplacian of the response is updated progressively by matrix transformation of these constants. Numerical results pertaining to a system exposed to white noise are presented to elucidate the mechanization of the approach. Further, parameter studies are done for examining the influence of the fractional Laplacian order on the system response.

中文翻译：

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用分数拉普拉斯项求解随机非线性问题的边界元方法

摘要 涉及随机扩散方程的模型被用来描述许多自然现象的演变，并在公开文献中被广泛讨论。近年来，根据实验观察重新审视了这些模型，其中识别出“异常”扩散过程，例如声波在随机介质中的传播。在这种情况下，理论模型的一个关键特征是在相关的控制方程中引入了分数阶导数算子。具体来说，异常扩散涉及替代经典拉普拉斯算子的分数拉普拉斯算子。目前，可用分数拉普拉斯算子方程的解在数量非常有限的情况下可用。此外，据作者所知，对于包含分数拉普拉斯算子的非线性方程，没有可用的解。这一事实导致需要开发适当的数值方法来估计此类系统的响应。本文提出了一种基于边界元方法 (BEM) 的方法来确定由涉及随机激励的非线性分数扩散方程控制的系统的响应。该方法是通过利用经典泊松方程解的积分表示构建的，其中未知常数由边界元法确定。然后，基于最近提出的分数拉普拉斯算子的表示，响应的分数拉普拉斯算子的值通过这些常数的矩阵变换逐步更新。提供了与暴露于白噪声的系统有关的数值结果，以阐明该方法的机械化。此外，还进行了参数研究，以检查分数拉普拉斯阶数对系统响应的影响。