IDEAS home Printed from https://ideas.repec.org/a/eee/matcom/v216y2024icp77-103.html
   My bibliography  Save this article

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

Author

Listed:
  • 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
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0378475423003348
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/j.matcom.2023.08.008?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. 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).
    2. 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).
    3. 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.
    4. 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.
    5. Francesco Mainardi, 2018. "Fractional Calculus: Theory and Applications," Mathematics, MDPI, vol. 6(9), pages 1-4, August.
    6. 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.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Omran, A.K. & Zaky, M.A. & Hendy, A.S. & Pimenov, V.G., 2022. "An easy to implement linearized numerical scheme for fractional reaction–diffusion equations with a prehistorical nonlinear source function," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 200(C), pages 218-239.
    2. Feng, Libo & Liu, Fawang & Anh, Vo V., 2023. "Galerkin finite element method for a two-dimensional tempered time–space fractional diffusion equation with application to a Bloch–Torrey equation retaining Larmor precession," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 206(C), pages 517-537.
    3. Ahadian, P. & Parand, K., 2022. "Support vector regression for the temperature-stimulated drug release," Chaos, Solitons & Fractals, Elsevier, vol. 165(P2).
    4. Francisco Ureña & Ángel García & Antonio M. Vargas, 2022. "Preface to “Applications of Partial Differential Equations in Engineering”," Mathematics, MDPI, vol. 11(1), pages 1-4, December.
    5. Vasily E. Tarasov & Valentina V. Tarasova, 2019. "Dynamic Keynesian Model of Economic Growth with Memory and Lag," Mathematics, MDPI, vol. 7(2), pages 1-17, February.
    6. Nurlana Alimbekova & Abdumauvlen Berdyshev & Muratkan Madiyarov & Yerlan Yergaliyev, 2024. "Finite Element Method for a Fractional-Order Filtration Equation with a Transient Filtration Law," Mathematics, MDPI, vol. 12(16), pages 1-20, August.
    7. Qiang Wang & Pyeoungkee Kim & Wenzhen Qu, 2022. "A Hybrid Localized Meshless Method for the Solution of Transient Groundwater Flow in Two Dimensions," Mathematics, MDPI, vol. 10(3), pages 1-14, February.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:eee:matcom:v:216:y:2024:i:c:p:77-103. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Catherine Liu (email available below). General contact details of provider: http://www.journals.elsevier.com/mathematics-and-computers-in-simulation/ .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.