Supervised learning and meshless methods for two-dimensional fractional PDEs on irregular domains

Mostafa Abbaszadeh, Mahmoud A. Zaky, Ahmed S. Hendy and Mehdi Dehghan

Mathematics and Computers in Simulation (MATCOM), 2024, vol. 216, issue C, 77-103

Abstract: Recently, several numerical methods have been developed for solving time-fractional differential equations not only on rectangular computational domains but also on convex and non-convex non-rectangular computational geometries. On the other hand, due to the existence of integrals in the definition of space-fractional operators, there are few numerical schemes for solving space-fractional differential equations on irregular regions. In this paper, we develop a novel numerical solution based on the machine learning technique and a generalized moving least squares approximation for two-dimensional fractional PDEs on irregular domains. The scheme is constructed on the monomials, and this is the strength of this technique. Moreover, it will be used to approximate the space derivatives on convex and non-convex non-rectangular computational domains. The numerical results are extended to solve the fractional Bloch–Torrey equation, fractional Gray–Scott equation, and fractional Fitzhugh–Nagumo equation.

Keywords: Supervised learning algorithm; Least squares support vector regression machines; Fractional Bloch–Torrey equation; Generalized moving least squares approximation; Convex and non-convex computational domains (search for similar items in EconPapers)
Date: 2024
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Persistent link: https://EconPapers.repec.org/RePEc:eee:matcom:v:216:y:2024:i:c:p:77-103

DOI: 10.1016/j.matcom.2023.08.008

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