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 modelsthat 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.

中文翻译：

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非局部和分数模型的数值方法

偏微分方程 (PDE) 用于对所有科学和工程学科的现象进行建模取得了巨大成功。然而，在同样广泛的范围内，存在偏微分方程无法充分模拟观察到的现象的情况，或者不是用于该目的的最佳可用模型。另一方面，在很多情况下，非本地模型已经证明，解释发生在远处的相互作用可以更忠实和有效地模拟观察到的现象，这些现象涉及可能的奇点和其他异常。在本文中，我们考虑一个通用的非局部模型，首先简要回顾它的定义、解的性质、数学分析和具体的例子。然后，我们对数值方法进行了广泛的讨论，包括有限元、有限差分和谱方法，以确定所考虑的非局部模型的近似解。在那次讨论中，我们特别关注了文献中研究最广泛的一类特殊的非局部模型，即那些涉及分数导数的模型。文章最后简要考虑了几个建模和算法扩展，