Numerical methods for nonlocal and fractional models

IF 16.3 1区 数学 Q1 MATHEMATICS
Marta D’Elia, Qiang Du, Christian Glusa, Max Gunzburger, Xiaochuan Tian, Zhi Zhou
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引用次数: 0

Abstract

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.
非局部和分数阶模型的数值方法
偏微分方程(PDEs)在所有科学和工程学科中都被成功地用于建模现象。然而,在同样广泛的范围内,存在着一些情况,其中偏微分方程不能充分模拟观察到的现象,或者不是用于这一目的的最佳可用模型。另一方面,在许多情况下,考虑到发生在远处的相互作用的非局部模型已被证明更忠实和有效地模拟涉及可能的奇点和其他异常的观测现象。在本文中,我们考虑一个一般的非局部模型,从它的定义、解的性质、数学分析和具体例子的简短回顾开始。然后,我们提供了关于数值方法的广泛讨论,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,我们特别关注文献中研究最广泛的一类特殊的非局部模型,即那些涉及分数阶导数的模型。文章最后简要考虑了几种建模和算法扩展,这表明了非局部建模的广泛适用性。
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来源期刊
Acta Numerica
Acta Numerica MATHEMATICS-
CiteScore
26.00
自引率
0.70%
发文量
7
期刊介绍: Acta Numerica stands as the preeminent mathematics journal, ranking highest in both Impact Factor and MCQ metrics. This annual journal features a collection of review articles that showcase survey papers authored by prominent researchers in numerical analysis, scientific computing, and computational mathematics. These papers deliver comprehensive overviews of recent advances, offering state-of-the-art techniques and analyses. Encompassing the entirety of numerical analysis, the articles are crafted in an accessible style, catering to researchers at all levels and serving as valuable teaching aids for advanced instruction. The broad subject areas covered include computational methods in linear algebra, optimization, ordinary and partial differential equations, approximation theory, stochastic analysis, nonlinear dynamical systems, as well as the application of computational techniques in science and engineering. Acta Numerica also delves into the mathematical theory underpinning numerical methods, making it a versatile and authoritative resource in the field of mathematics.
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