A finite-volume scheme for fractional diffusion on bounded domains

IF 2.3 4区 数学 Q1 MATHEMATICS, APPLIED European Journal of Applied Mathematics Pub Date : 2024-04-16 DOI:10.1017/s0956792524000172
Rafael Bailo, José A. Carrillo, Stefano Fronzoni, David Gómez-Castro
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Abstract

We propose a new fractional Laplacian for bounded domains, expressed as a conservation law and thus particularly suited to finite-volume schemes. Our approach permits the direct prescription of no-flux boundary conditions. We first show the well-posedness theory for the fractional heat equation. We also develop a numerical scheme, which correctly captures the action of the fractional Laplacian and its anomalous diffusion effect. We benchmark numerical solutions for the Lévy–Fokker–Planck equation against known analytical solutions. We conclude by numerically exploring properties of these equations with respect to their stationary states and long-time asymptotics.

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有界域上分数扩散的有限体积方案
我们为有界域提出了一种新的分数拉普拉斯,它以守恒定律的形式表示,因此特别适用于有限体积方案。我们的方法允许直接规定无流动边界条件。我们首先展示了分数热方程的拟合理论。我们还开发了一种数值方案,它能正确捕捉分数拉普拉斯的作用及其反常扩散效应。我们将 Lévy-Fokker-Planck 方程的数值解与已知的分析解进行比较。最后,我们用数值方法探讨了这些方程的静止状态和长时间渐近特性。
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来源期刊
CiteScore
4.70
自引率
0.00%
发文量
31
审稿时长
>12 weeks
期刊介绍: Since 2008 EJAM surveys have been expanded to cover Applied and Industrial Mathematics. Coverage of the journal has been strengthened in probabilistic applications, while still focusing on those areas of applied mathematics inspired by real-world applications, and at the same time fostering the development of theoretical methods with a broad range of applicability. Survey papers contain reviews of emerging areas of mathematics, either in core areas or with relevance to users in industry and other disciplines. Research papers may be in any area of applied mathematics, with special emphasis on new mathematical ideas, relevant to modelling and analysis in modern science and technology, and the development of interesting mathematical methods of wide applicability.
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