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Finite Difference Scheme and Finite Volume Scheme for Fractional Laplacian Operator and Some Applications
Version 1
: Received: 19 October 2023 / Approved: 20 October 2023 / Online: 20 October 2023 (06:32:00 CEST)
A peer-reviewed article of this Preprint also exists.
Wang, J.; Yuan, S.; Liu, X. Finite Difference Scheme and Finite Volume Scheme for Fractional Laplacian Operator and Some Applications. Fractal Fract. 2023, 7, 868. Wang, J.; Yuan, S.; Liu, X. Finite Difference Scheme and Finite Volume Scheme for Fractional Laplacian Operator and Some Applications. Fractal Fract. 2023, 7, 868.
Abstract
In the paper, we develop some numerical schemes including finite difference scheme and finite volume scheme for the fractional Laplacian operator, and apply the resulted numerical schemes
to solve some fractional diffusion equations.
First, the fractional Laplacian operator can be characterized as the weak singular integral by a integral operator with zero boundary
condition.
Second, because of the solution of fractional diffusion equation is usually singular near the boundary, we apply finite difference scheme to discrete fractional Laplacian operator and fractional diffusion equation by fractional interpolation function.
Moreover, it is find that the differential matrix of the above scheme is symmetric matrix, and strictly row-wise diagonally dominant in special fractional interpolation function.
Third, we show finite volume scheme to discrete fractional diffusion equation by fractional interpolation function,
and analyze the properties of differential matrix.
Finally, the numerical experiments are given, and verify the correctness of theoretical results and the efficiency of the schemes.
Keywords
fractional laplacian; fractional diffusion equation; finite difference scheme; finite volume scheme; weak singular integral
Subject
Computer Science and Mathematics, Mathematics
Copyright: This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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