A Fast O(N log N) Finite Difference Method for the One-Dimensional Space-Fractional Diffusion Equation
This paper proposes an approach for the space-fractional diffusion equation in one dimension. Since fractional differential operators are non-local, two main difficulties arise after discretization and solving using Gaussian elimination: how to handle the memory requirement of O(N2) for storing the...
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| Format: | Article |
| Language: | English |
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MDPI AG
2015-10-01
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| Series: | Mathematics |
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| Online Access: | http://www.mdpi.com/2227-7390/3/4/1032 |
| _version_ | 1818515722231349248 |
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| author | Treena Basu |
| author_facet | Treena Basu |
| author_sort | Treena Basu |
| collection | DOAJ |
| description | This paper proposes an approach for the space-fractional diffusion equation in one dimension. Since fractional differential operators are non-local, two main difficulties arise after discretization and solving using Gaussian elimination: how to handle the memory requirement of O(N2) for storing the dense or even full matrices that arise from application of numerical methods and how to manage the significant computational work count of O(N3) per time step, where N is the number of spatial grid points. In this paper, a fast iterative finite difference method is developed, which has a memory requirement of O(N) and a computational cost of O(N logN) per iteration. Finally, some numerical results are shown to verify the accuracy and efficiency of the new method. |
| first_indexed | 2024-12-11T00:32:36Z |
| format | Article |
| id | doaj.art-956f6a962a254ef88f67bc83c04256c1 |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-12-11T00:32:36Z |
| publishDate | 2015-10-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj.art-956f6a962a254ef88f67bc83c04256c12022-12-22T01:27:18ZengMDPI AGMathematics2227-73902015-10-01341032104410.3390/math3041032math3041032A Fast O(N log N) Finite Difference Method for the One-Dimensional Space-Fractional Diffusion EquationTreena Basu0Department of Mathematics, Occidental College, Los Angeles, CA 90041, USAThis paper proposes an approach for the space-fractional diffusion equation in one dimension. Since fractional differential operators are non-local, two main difficulties arise after discretization and solving using Gaussian elimination: how to handle the memory requirement of O(N2) for storing the dense or even full matrices that arise from application of numerical methods and how to manage the significant computational work count of O(N3) per time step, where N is the number of spatial grid points. In this paper, a fast iterative finite difference method is developed, which has a memory requirement of O(N) and a computational cost of O(N logN) per iteration. Finally, some numerical results are shown to verify the accuracy and efficiency of the new method.http://www.mdpi.com/2227-7390/3/4/1032circulant and toeplitz matricesfast finite difference methodsfast fourier transformfractional diffusion equations |
| spellingShingle | Treena Basu A Fast O(N log N) Finite Difference Method for the One-Dimensional Space-Fractional Diffusion Equation Mathematics circulant and toeplitz matrices fast finite difference methods fast fourier transform fractional diffusion equations |
| title | A Fast O(N log N) Finite Difference Method for the One-Dimensional Space-Fractional Diffusion Equation |
| title_full | A Fast O(N log N) Finite Difference Method for the One-Dimensional Space-Fractional Diffusion Equation |
| title_fullStr | A Fast O(N log N) Finite Difference Method for the One-Dimensional Space-Fractional Diffusion Equation |
| title_full_unstemmed | A Fast O(N log N) Finite Difference Method for the One-Dimensional Space-Fractional Diffusion Equation |
| title_short | A Fast O(N log N) Finite Difference Method for the One-Dimensional Space-Fractional Diffusion Equation |
| title_sort | fast o n log n finite difference method for the one dimensional space fractional diffusion equation |
| topic | circulant and toeplitz matrices fast finite difference methods fast fourier transform fractional diffusion equations |
| url | http://www.mdpi.com/2227-7390/3/4/1032 |
| work_keys_str_mv | AT treenabasu afastonlognfinitedifferencemethodfortheonedimensionalspacefractionaldiffusionequation AT treenabasu fastonlognfinitedifferencemethodfortheonedimensionalspacefractionaldiffusionequation |