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A Note on a Meshless Method for Fractional Laplacian at Arbitrary Irregular Meshes

Author

Listed:
  • Ángel García

    (UNED, ETSII, 28040 Madrid, Spain
    These authors contributed equally to this work.)

  • Mihaela Negreanu

    (Instituto de Matemática Interdisciplinar, Departamento de Análisis Matemático y Matemática Aplicada, UCM, 28040 Madrid, Spain
    These authors contributed equally to this work.)

  • Francisco Ureña

    (UNED, ETSII, 28040 Madrid, Spain
    These authors contributed equally to this work.)

  • Antonio M. Vargas

    (Departamento de Análisis Matemático y Matemática Aplicada, UCM, 28040 Madrid, Spain
    These authors contributed equally to this work.)

Abstract

The existence and uniqueness of the discrete solutions of a porous medium equation with diffusion are demonstrated. The Cauchy problem contains a fractional Laplacian and it is equivalent to the extension formulation in the sense of trace and harmonic extension operators. By using the generalized finite difference method, we obtain the convergence of the numerical solution to the classical/theoretical solution of the equation for nonnegative initial data sufficiently smooth and bounded. This procedure allows us to use meshes with complicated geometry (more realistic) or with an irregular distribution of nodes (providing more accurate solutions where needed). Some numerical results are presented in arbitrary irregular meshes to illustrate the potential of the method.

Suggested Citation

  • 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.
    • Handle: RePEc:gam:jmathe:v:9:y:2021:i:22:p:2843-:d:675619
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    References listed on IDEAS

    as
    1. Hu, Ye & Li, Changpin & Li, Hefeng, 2017. "The finite difference method for Caputo-type parabolic equation with fractional Laplacian: One-dimension case," Chaos, Solitons & Fractals, Elsevier, vol. 102(C), pages 319-326.
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    Cited by:

    1. 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.
    2. 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.
    3. 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.

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