Application of Nonlinear Time-Fractional Partial Differential Equations to Image Processing via Hybrid Laplace Transform Method
This work considers a hybrid solution method for the time-fractional diffusion model with a cubic nonlinear source term in one and two dimensions. Both Dirichlet and Neumann boundary conditions are considered for each dimensional case. The hybrid method involves a Laplace transformation in the tempo...
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| Format: | Article |
| Language: | English |
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Wiley
2018-01-01
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| Series: | Journal of Mathematics |
| Online Access: | http://dx.doi.org/10.1155/2018/8924547 |
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| _version_ | 1832552073678290944 |
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| author | B. A. Jacobs C. Harley |
| author_facet | B. A. Jacobs C. Harley |
| author_sort | B. A. Jacobs |
| collection | DOAJ |
| description | This work considers a hybrid solution method for the time-fractional diffusion model with a cubic nonlinear source term in one and two dimensions. Both Dirichlet and Neumann boundary conditions are considered for each dimensional case. The hybrid method involves a Laplace transformation in the temporal domain which is numerically inverted, and Chebyshev collocation is employed in the spatial domain due to its increased accuracy over a standard finite-difference discretization. Due to the fractional-order derivative we are only able to compare the accuracy of this method with Mathematica’s NDSolve in the case of integer derivatives; however, a detailed discussion of the merits and shortcomings of the proposed hybridization is presented. An application to image processing via a finite-difference discretization is included in order to substantiate the application of this method. |
| format | Article |
| id | doaj-art-3f3f30fccc18474186cf3bb1c124f237 |
| institution | Kabale University |
| issn | 2314-4629 2314-4785 |
| language | English |
| publishDate | 2018-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Journal of Mathematics |
| spelling | doaj-art-3f3f30fccc18474186cf3bb1c124f2372025-02-03T05:59:41ZengWileyJournal of Mathematics2314-46292314-47852018-01-01201810.1155/2018/89245478924547Application of Nonlinear Time-Fractional Partial Differential Equations to Image Processing via Hybrid Laplace Transform MethodB. A. Jacobs0C. Harley1School of Computer Science and Applied Mathematics, University of the Witwatersrand, Johannesburg, Private Bag 3, Wits 2050, South AfricaSchool of Computer Science and Applied Mathematics, University of the Witwatersrand, Johannesburg, Private Bag 3, Wits 2050, South AfricaThis work considers a hybrid solution method for the time-fractional diffusion model with a cubic nonlinear source term in one and two dimensions. Both Dirichlet and Neumann boundary conditions are considered for each dimensional case. The hybrid method involves a Laplace transformation in the temporal domain which is numerically inverted, and Chebyshev collocation is employed in the spatial domain due to its increased accuracy over a standard finite-difference discretization. Due to the fractional-order derivative we are only able to compare the accuracy of this method with Mathematica’s NDSolve in the case of integer derivatives; however, a detailed discussion of the merits and shortcomings of the proposed hybridization is presented. An application to image processing via a finite-difference discretization is included in order to substantiate the application of this method.http://dx.doi.org/10.1155/2018/8924547 |
| spellingShingle | B. A. Jacobs C. Harley Application of Nonlinear Time-Fractional Partial Differential Equations to Image Processing via Hybrid Laplace Transform Method Journal of Mathematics |
| title | Application of Nonlinear Time-Fractional Partial Differential Equations to Image Processing via Hybrid Laplace Transform Method |
| title_full | Application of Nonlinear Time-Fractional Partial Differential Equations to Image Processing via Hybrid Laplace Transform Method |
| title_fullStr | Application of Nonlinear Time-Fractional Partial Differential Equations to Image Processing via Hybrid Laplace Transform Method |
| title_full_unstemmed | Application of Nonlinear Time-Fractional Partial Differential Equations to Image Processing via Hybrid Laplace Transform Method |
| title_short | Application of Nonlinear Time-Fractional Partial Differential Equations to Image Processing via Hybrid Laplace Transform Method |
| title_sort | application of nonlinear time fractional partial differential equations to image processing via hybrid laplace transform method |
| url | http://dx.doi.org/10.1155/2018/8924547 |
| work_keys_str_mv | AT bajacobs applicationofnonlineartimefractionalpartialdifferentialequationstoimageprocessingviahybridlaplacetransformmethod AT charley applicationofnonlineartimefractionalpartialdifferentialequationstoimageprocessingviahybridlaplacetransformmethod |