Fractional Modeling in Action: a Survey of Nonlocal Models for Subsurface Transport, Turbulent Flows, and Anomalous Materials
Fractional Modeling in Action: a Survey of Nonlocal Models for Subsurface Transport, Turbulent Flows, and Anomalous Materials
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DOI：
10.1007/s42102-022-00085-2
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发表时间：
2021-09
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通讯作者：
J. Suzuki;Mamikon A. Gulian;Mohsen Zayernouri;M. D'Elia
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作者：
J. Suzuki;Mamikon A. Gulian;Mohsen Zayernouri;M. D'Elia
通过分数阶微分方程对异常输运等现象进行建模已经成为偏微分方程的有效替代，因为它具有比完全解析的经典模型更有效地描述大规模行为的固有能力。在这篇综述文章中，我们首先提供了分数阶导数的广泛概述，并明确强调了其使用背后的随机过程。然后，我们调查了三个典型的应用领域-地下输送，湍流和异常材料-其中分数阶微分方程提供了准确的预测模型。对于每个领域，我们报告了反常行为的证据，证明了分数阶模型的使用是合理的，并调查了基础模型以及更具表现力的最先进的模型。我们还提出了未来研究的途径，包括更先进和物理健全的模型，以及用于校准和发现分数阶模型的工具。
Modeling of phenomena such as anomalous transport via fractional-order differential equations has been established as an effective alternative to partial differential equations, due to the inherent ability to describe large-scale behavior with greater efficiency than fully resolved classical models. In this review article, we first provide a broad overview of fractional-order derivatives with a clear emphasis on the stochastic processes that underlie their use. We then survey three exemplary application areas — subsurface transport, turbulence, and anomalous materials — in which fractional-order differential equations provide accurate and predictive models. For each area, we report on the evidence of anomalous behavior that justifies the use of fractional-order models, and survey both foundational models as well as more expressive state-of-the-art models. We also propose avenues for future research, including more advanced and physically sound models, as well as tools for calibration and discovery of fractional-order models.