Numerical methods for nonlocal and fractional models
Numerical methods for nonlocal and fractional models
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DOI：
10.2172/1598758
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发表时间：
2020-02
期刊：
影响因子：
14.2
通讯作者：
M. D'Elia;Q. Du;Christian A. Glusa;M. Gunzburger;Xiaochuan Tian;Zhi Zhou
中科院分区：
文献类型：
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作者：
M. D'Elia;Q. Du;Christian A. Glusa;M. Gunzburger;Xiaochuan Tian;Zhi Zhou
偏微分方程（PDE）的成功范围很大，以模拟所有科学和工程学科的现象。但是，在同样宽的片段中，存在PDE无法充分模拟观察到的现象的情况，或者不是该目的的最佳可用模型。另一方面，在许多情况下，出现在距离处发生相互作用的非本地模型已被证明更忠实，有效地模拟了涉及可能的奇异性和其他异常的现象。在本文中，我们考虑了一个通用的非本地模型，首先要简要审查其定义，其解决方案的属性，其数学分析和特定的具体示例。然后，我们提供有关数值方法的广泛讨论，包括有限元，有限差和光谱方法，以确定所考虑的非局部模型的近似解。在该讨论中，我们特别关注特殊的非本地模型，这些模型是文献中最广泛研究的，即涉及分数衍生物的模型。本文以几种建模和算法扩展的简要考虑结尾，这些扩展旨在显示非本地建模的广泛适用性。
Partial differential equations (PDEs) are used with huge success to model phenomena across all scientific and engineering disciplines. However, across an equally wide swath, there exist situations in which PDEs fail to adequately model observed phenomena, or are not the best available model for that purpose. On the other hand, in many situations, nonlocal models that account for interaction occurring at a distance have been shown to more faithfully and effectively model observed phenomena that involve possible singularities and other anomalies. In this article we consider a generic nonlocal model, beginning with a short review of its definition, the properties of its solution, its mathematical analysis and of specific concrete examples. We then provide extensive discussions about numerical methods, including finite element, finite difference and spectral methods, for determining approximate solutions of the nonlocal models considered. In that discussion, we pay particular attention to a special class of nonlocal models that are the most widely studied in the literature, namely those involving fractional derivatives. The article ends with brief considerations of several modelling and algorithmic extensions, which serve to show the wide applicability of nonlocal modelling.