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Volume 9
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10.3390/math9222843
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Article

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

by
Ángel García
1,†,
Mihaela Negreanu
2,†,
Francisco Ureña
1,*,† and
Antonio M. Vargas
3,†
1
UNED, ETSII, 28040 Madrid, Spain
2
Instituto de Matemática Interdisciplinar, Departamento de Análisis Matemático y Matemática Aplicada, UCM, 28040 Madrid, Spain
3
Departamento de Análisis Matemático y Matemática Aplicada, UCM, 28040 Madrid, Spain
*
Author to whom correspondence should be addressed.
†
These authors contributed equally to this work.
Mathematics 2021, 9(22), 2843; https://doi.org/10.3390/math9222843
Submission received: 28 September 2021 / Revised: 3 November 2021 / Accepted: 8 November 2021 / Published: 10 November 2021
(This article belongs to the Special Issue Applications of Partial Differential Equations in Engineering)
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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.
Keywords: fractional Laplacian; generalized finite difference method; discrete maximum principle; convergence fractional Laplacian; generalized finite difference method; discrete maximum principle; convergence

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MDPI and ACS Style

García, Á.; Negreanu, M.; Ureña, F.; Vargas, A.M. A Note on a Meshless Method for Fractional Laplacian at Arbitrary Irregular Meshes. Mathematics 2021, 9, 2843. https://doi.org/10.3390/math9222843

AMA Style

García Á, Negreanu M, Ureña F, Vargas AM. A Note on a Meshless Method for Fractional Laplacian at Arbitrary Irregular Meshes. Mathematics. 2021; 9(22):2843. https://doi.org/10.3390/math9222843

Chicago/Turabian Style

García, Ángel, Mihaela Negreanu, Francisco Ureña, and Antonio M. Vargas. 2021. "A Note on a Meshless Method for Fractional Laplacian at Arbitrary Irregular Meshes" Mathematics 9, no. 22: 2843. https://doi.org/10.3390/math9222843

APA Style

García, Á., Negreanu, M., Ureña, F., & Vargas, A. M. (2021). A Note on a Meshless Method for Fractional Laplacian at Arbitrary Irregular Meshes. Mathematics, 9(22), 2843. https://doi.org/10.3390/math9222843

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