Numerical methods for nonlocal and fractional models
Numerical methods for nonlocal and fractional models
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DOI：
10.1017/s096249292000001x
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发表时间：
2020-05-01
期刊：
影响因子：
14.2
通讯作者：
Zhou, Zhi
中科院分区：
文献类型：
--
作者：
D'Elia, Marta;Du, Qiang;Zhou, Zhi
偏微分方程（PDE）在所有科学和工程学科中用于对现象进行建模方面取得了巨大成功。然而，在同样宽的范围内，存在偏微分方程无法充分模拟观察到的现象的情况，或者不是最好的可用模型。另一方面，在许多情况下，非局部模型，说明发生在一个距离的相互作用已被证明更忠实和有效地模拟观察到的现象，涉及可能的奇点和其他异常。在这篇文章中，我们考虑一个通用的非局部模型，开始与它的定义，它的解决方案，它的数学分析和具体的具体例子的属性的简短回顾。然后，我们提供了广泛的讨论数值方法，包括有限元，有限差分和谱方法，用于确定所考虑的非局部模型的近似解。在讨论中，我们特别注意一类特殊的非局部模型，这是文献中研究最广泛的，即那些涉及分数阶导数。文章最后简要考虑几个建模和算法的扩展，这有助于显示非局部建模的广泛适用性。
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.