Finite Difference Scheme and Finite Volume Scheme for Fractional Laplacian Operator and Some Applications
The fractional Laplacian operator is a very important fractional operator that is often used to describe several anomalous diffusion phenomena. In this paper, we develop some numerical schemes, including a finite difference scheme and finite volume scheme for the fractional Laplacian operator, and a...
| Main Authors: | , , |
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| Format: | Article |
| Language: | English |
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MDPI AG
2023-12-01
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| Series: | Fractal and Fractional |
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| Online Access: | https://www.mdpi.com/2504-3110/7/12/868 |
| _version_ | 1797380984820203520 |
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| author | Junjie Wang Shoucheng Yuan Xiao Liu |
| author_facet | Junjie Wang Shoucheng Yuan Xiao Liu |
| author_sort | Junjie Wang |
| collection | DOAJ |
| description | The fractional Laplacian operator is a very important fractional operator that is often used to describe several anomalous diffusion phenomena. In this paper, we develop some numerical schemes, including a finite difference scheme and finite volume scheme for the fractional Laplacian operator, and apply the resulting numerical schemes to solve some fractional diffusion equations. First, the fractional Laplacian operator can be characterized as the weak singular integral by an integral operator with zero boundary condition. Second, because the solutions of fractional diffusion equations are usually singular near the boundary, we use a fractional interpolation function in the region near the boundary and use a classical interpolation function in other intervals. Then, we apply a finite difference scheme to the discrete fractional Laplacian operator and fractional diffusion equation with the above fractional interpolation function and classical interpolation function. Moreover, it is found that the differential matrix of the above scheme is a symmetric matrix and strictly row-wise diagonally dominant in special fractional interpolation functions. Third, we show a finite volume scheme for a discrete fractional diffusion equation by fractional interpolation function and classical interpolation function and analyze the properties of the differential matrix. Finally, the numerical experiments are given, and we verify the correctness of the theoretical results and the efficiency of the schemes. |
| first_indexed | 2024-03-08T20:45:47Z |
| format | Article |
| id | doaj.art-7ffdf10a9b7f47dfb418218ba00b77a5 |
| institution | Directory Open Access Journal |
| issn | 2504-3110 |
| language | English |
| last_indexed | 2024-03-08T20:45:47Z |
| publishDate | 2023-12-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Fractal and Fractional |
| spelling | doaj.art-7ffdf10a9b7f47dfb418218ba00b77a52023-12-22T14:10:01ZengMDPI AGFractal and Fractional2504-31102023-12-0171286810.3390/fractalfract7120868Finite Difference Scheme and Finite Volume Scheme for Fractional Laplacian Operator and Some ApplicationsJunjie Wang0Shoucheng Yuan1Xiao Liu2School of Mathematics and Statistical, Pu’er University, Pu’er 665000, ChinaSchool of Mathematics and Statistical, Pu’er University, Pu’er 665000, ChinaSchool of Mathematics and Statistical, Pu’er University, Pu’er 665000, ChinaThe fractional Laplacian operator is a very important fractional operator that is often used to describe several anomalous diffusion phenomena. In this paper, we develop some numerical schemes, including a finite difference scheme and finite volume scheme for the fractional Laplacian operator, and apply the resulting numerical schemes to solve some fractional diffusion equations. First, the fractional Laplacian operator can be characterized as the weak singular integral by an integral operator with zero boundary condition. Second, because the solutions of fractional diffusion equations are usually singular near the boundary, we use a fractional interpolation function in the region near the boundary and use a classical interpolation function in other intervals. Then, we apply a finite difference scheme to the discrete fractional Laplacian operator and fractional diffusion equation with the above fractional interpolation function and classical interpolation function. Moreover, it is found that the differential matrix of the above scheme is a symmetric matrix and strictly row-wise diagonally dominant in special fractional interpolation functions. Third, we show a finite volume scheme for a discrete fractional diffusion equation by fractional interpolation function and classical interpolation function and analyze the properties of the differential matrix. Finally, the numerical experiments are given, and we verify the correctness of the theoretical results and the efficiency of the schemes.https://www.mdpi.com/2504-3110/7/12/868fractional Laplacian operatorfractional diffusion equationfinite difference schemefinite volume schemeweak singular integral |
| spellingShingle | Junjie Wang Shoucheng Yuan Xiao Liu Finite Difference Scheme and Finite Volume Scheme for Fractional Laplacian Operator and Some Applications Fractal and Fractional fractional Laplacian operator fractional diffusion equation finite difference scheme finite volume scheme weak singular integral |
| title | Finite Difference Scheme and Finite Volume Scheme for Fractional Laplacian Operator and Some Applications |
| title_full | Finite Difference Scheme and Finite Volume Scheme for Fractional Laplacian Operator and Some Applications |
| title_fullStr | Finite Difference Scheme and Finite Volume Scheme for Fractional Laplacian Operator and Some Applications |
| title_full_unstemmed | Finite Difference Scheme and Finite Volume Scheme for Fractional Laplacian Operator and Some Applications |
| title_short | Finite Difference Scheme and Finite Volume Scheme for Fractional Laplacian Operator and Some Applications |
| title_sort | finite difference scheme and finite volume scheme for fractional laplacian operator and some applications |
| topic | fractional Laplacian operator fractional diffusion equation finite difference scheme finite volume scheme weak singular integral |
| url | https://www.mdpi.com/2504-3110/7/12/868 |
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