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

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

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.

Suggested Citation

  • Abbaszadeh, Mostafa & Zaky, Mahmoud A. & Hendy, Ahmed S. & Dehghan, Mehdi, 2024. "Supervised learning and meshless methods for two-dimensional fractional PDEs on irregular domains," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 216(C), pages 77-103.
    • Handle: RePEc:eee:matcom:v:216:y:2024:i:c:p:77-103
    DOI: 10.1016/j.matcom.2023.08.008
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    References listed on IDEAS

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    1. Pakniyat, A. & Parand, K. & Jani, M., 2021. "Least squares support vector regression for differential equations on unbounded domains," Chaos, Solitons & Fractals, Elsevier, vol. 151(C).
    2. Bu, Weiping & Zhao, Yanmin & Shen, Chen, 2021. "Fast and efficient finite difference/finite element method for the two-dimensional multi-term time-space fractional Bloch-Torrey equation," Applied Mathematics and Computation, Elsevier, vol. 398(C).
    3. Ángel García & Mihaela Negreanu & Francisco Ureña & Antonio M. Vargas, 2021. "A Note on a Meshless Method for Fractional Laplacian at Arbitrary Irregular Meshes," Mathematics, MDPI, vol. 9(22), pages 1-9, November.
    4. Francesco Mainardi, 2018. "Fractional Calculus: Theory and Applications," Mathematics, MDPI, vol. 6(9), pages 1-4, August.
    5. V. G. Pimenov & A. S. Hendy, 2015. "Numerical Studies for Fractional Functional Differential Equations with Delay Based on BDF-Type Shifted Chebyshev Approximations," Abstract and Applied Analysis, Hindawi, vol. 2015, pages 1-12, March.
    6. Mahmoud A. Zaky & Ahmed S. Hendy & Rob H. De Staelen, 2021. "Alikhanov Legendre—Galerkin Spectral Method for the Coupled Nonlinear Time-Space Fractional Ginzburg–Landau Complex System," Mathematics, MDPI, vol. 9(2), pages 1-22, January.
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