A Finite Difference Scheme for the Fractional Laplacian on Non-uniform Grids

doi:10.1007/s42967-023-00323-4

Communications on Applied Mathematics and Computation ›› 2025, Vol. 7 ›› Issue (4): 1364-1377.doi: 10.1007/s42967-023-00323-4

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A Finite Difference Scheme for the Fractional Laplacian on Non-uniform Grids

A. M. Vargas

Departamento de Matemáticas Fundamentales, UNED, Madrid, Spain

收稿日期:2023-06-13 修回日期:2023-08-16 接受日期:2023-09-17 出版日期:2023-12-16 发布日期:2023-12-16
通讯作者: A.M.Vargas,E-mail:avargas@mat.uned.es E-mail:avargas@mat.uned.es
基金资助:
The author is supported by the Spanish MINECO through Juan de la Cierva fellow-ship FJC2021-046953-I.

A Finite Difference Scheme for the Fractional Laplacian on Non-uniform Grids

A. M. Vargas

Departamento de Matemáticas Fundamentales, UNED, Madrid, Spain

Received:2023-06-13 Revised:2023-08-16 Accepted:2023-09-17 Online:2023-12-16 Published:2023-12-16
Supported by:
The author is supported by the Spanish MINECO through Juan de la Cierva fellow-ship FJC2021-046953-I.

摘要/Abstract

摘要： In this study, we analyze the convergence of the finite difference method on non-uniform grids and provide examples to demonstrate its effectiveness in approximating fractional differential equations involving the fractional Laplacian. By utilizing non-uniform grids, it becomes possible to achieve higher accuracy and improved resolution in specific regions of interest. Overall, our findings indicate that finite difference approximation on non-uniform grids can serve as a dependable and efficient tool for approximating fractional Laplacians across a diverse array of applications.

关键词: Fractional differential equations, Caputo fractional derivative, Fractional Laplacian, Finite difference method, Meshless method

Abstract: In this study, we analyze the convergence of the finite difference method on non-uniform grids and provide examples to demonstrate its effectiveness in approximating fractional differential equations involving the fractional Laplacian. By utilizing non-uniform grids, it becomes possible to achieve higher accuracy and improved resolution in specific regions of interest. Overall, our findings indicate that finite difference approximation on non-uniform grids can serve as a dependable and efficient tool for approximating fractional Laplacians across a diverse array of applications.

Key words: Fractional differential equations, Caputo fractional derivative, Fractional Laplacian, Finite difference method, Meshless method

中图分类号:

65L12
65D25

A. M. Vargas. A Finite Difference Scheme for the Fractional Laplacian on Non-uniform Grids[J]. Communications on Applied Mathematics and Computation, 2025, 7(4): 1364-1377.

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A Finite Difference Scheme for the Fractional Laplacian on Non-uniform Grids

A Finite Difference Scheme for the Fractional Laplacian on Non-uniform Grids

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