Fourth-Order Difference Scheme and a Matrix Transform Approach for Solving Fractional PDEs
Proposing a matrix transform method to solve a fractional partial differential equation is the main aim of this paper. The main model can be transferred to a partial-integro differential equation (PIDE) with a weakly singular kernel. The spatial direction is approximated by a fourth-order difference...
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| Format: | Article |
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MDPI AG
2023-09-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/11/17/3786 |
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| author | Zahrah I. Salman Majid Tavassoli Kajani Mohammed Sahib Mechee Masoud Allame |
| author_facet | Zahrah I. Salman Majid Tavassoli Kajani Mohammed Sahib Mechee Masoud Allame |
| author_sort | Zahrah I. Salman |
| collection | DOAJ |
| description | Proposing a matrix transform method to solve a fractional partial differential equation is the main aim of this paper. The main model can be transferred to a partial-integro differential equation (PIDE) with a weakly singular kernel. The spatial direction is approximated by a fourth-order difference scheme. Also, the temporal derivative is discretized via a second-order numerical procedure. First, the spatial derivatives are approximated by a fourth-order operator to compute the second-order derivatives. This process produces a system of differential equations related to the time variable. Then, the Crank–Nicolson idea is utilized to achieve a full-discrete scheme. The kernel of the integral term is discretized by using the Lagrange polynomials to overcome its singularity. Subsequently, we prove the convergence and stability of the new difference scheme by utilizing the Rayleigh–Ritz theorem. Finally, some numerical examples in one-dimensional and two-dimensional cases are presented to verify the theoretical results. |
| first_indexed | 2024-03-10T23:17:07Z |
| format | Article |
| id | doaj.art-23f3b6bc399e424fb37355022944658e |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-10T23:17:07Z |
| publishDate | 2023-09-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj.art-23f3b6bc399e424fb37355022944658e2023-11-19T08:32:07ZengMDPI AGMathematics2227-73902023-09-011117378610.3390/math11173786Fourth-Order Difference Scheme and a Matrix Transform Approach for Solving Fractional PDEsZahrah I. Salman0Majid Tavassoli Kajani1Mohammed Sahib Mechee2Masoud Allame3Department of Mathematics, Isfahan (Khorasgan) Branch, Islamic Azad University, Isfahan P.O. Box 158-81595, IranDepartment of Mathematics, Isfahan (Khorasgan) Branch, Islamic Azad University, Isfahan P.O. Box 158-81595, IranInformation Technology Research and Development Center, University of Kufa, Najaf 540011, IraqDepartment of Mathematics, Isfahan (Khorasgan) Branch, Islamic Azad University, Isfahan P.O. Box 158-81595, IranProposing a matrix transform method to solve a fractional partial differential equation is the main aim of this paper. The main model can be transferred to a partial-integro differential equation (PIDE) with a weakly singular kernel. The spatial direction is approximated by a fourth-order difference scheme. Also, the temporal derivative is discretized via a second-order numerical procedure. First, the spatial derivatives are approximated by a fourth-order operator to compute the second-order derivatives. This process produces a system of differential equations related to the time variable. Then, the Crank–Nicolson idea is utilized to achieve a full-discrete scheme. The kernel of the integral term is discretized by using the Lagrange polynomials to overcome its singularity. Subsequently, we prove the convergence and stability of the new difference scheme by utilizing the Rayleigh–Ritz theorem. Finally, some numerical examples in one-dimensional and two-dimensional cases are presented to verify the theoretical results.https://www.mdpi.com/2227-7390/11/17/3786matrix transform methodfourth-order difference schemepartial-integro differential equationRayleigh–Ritz theoremerror estimate |
| spellingShingle | Zahrah I. Salman Majid Tavassoli Kajani Mohammed Sahib Mechee Masoud Allame Fourth-Order Difference Scheme and a Matrix Transform Approach for Solving Fractional PDEs Mathematics matrix transform method fourth-order difference scheme partial-integro differential equation Rayleigh–Ritz theorem error estimate |
| title | Fourth-Order Difference Scheme and a Matrix Transform Approach for Solving Fractional PDEs |
| title_full | Fourth-Order Difference Scheme and a Matrix Transform Approach for Solving Fractional PDEs |
| title_fullStr | Fourth-Order Difference Scheme and a Matrix Transform Approach for Solving Fractional PDEs |
| title_full_unstemmed | Fourth-Order Difference Scheme and a Matrix Transform Approach for Solving Fractional PDEs |
| title_short | Fourth-Order Difference Scheme and a Matrix Transform Approach for Solving Fractional PDEs |
| title_sort | fourth order difference scheme and a matrix transform approach for solving fractional pdes |
| topic | matrix transform method fourth-order difference scheme partial-integro differential equation Rayleigh–Ritz theorem error estimate |
| url | https://www.mdpi.com/2227-7390/11/17/3786 |
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