The Novel Analytical–Numerical Method for Multi-Dimensional Multi-Term Time-Fractional Equations with General Boundary Conditions
This article presents a simple but effective two-step analytical–numerical algorithm for solving multi-dimensional multi-term time-fractional equations. The first step is to derive an analytic representation that satisfies boundary requirements for 1D, 2D, and 3D problems, respectively. The second s...
| Main Authors: | , , , , |
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| Format: | Article |
| Language: | English |
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MDPI AG
2023-02-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/11/4/929 |
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| author | Ji Lin Sergiy Reutskiy Yuhui Zhang Yu Sun Jun Lu |
| author_facet | Ji Lin Sergiy Reutskiy Yuhui Zhang Yu Sun Jun Lu |
| author_sort | Ji Lin |
| collection | DOAJ |
| description | This article presents a simple but effective two-step analytical–numerical algorithm for solving multi-dimensional multi-term time-fractional equations. The first step is to derive an analytic representation that satisfies boundary requirements for 1D, 2D, and 3D problems, respectively. The second step is the meshless approximation where the Müntz polynomials are used to form the approximate solution and the unknown parameters are obtained by imposing the approximation for the governing equations. We illustrate first the detailed derivation of the analytic approximation and then the numerical implementation of the solution procedure. Several numerical examples are provided to verify the accuracy, efficiency, and adaptability to problems with general boundary conditions. The numerical results are compared with exact solutions and numerical methods reported in the literature, showing that the algorithm has great potential for multi-dimensional multi-term time-fractional equations with various boundary conditions. |
| first_indexed | 2024-03-11T08:28:40Z |
| format | Article |
| id | doaj.art-bd1165def8e540bc8cc5f35e4c31a4b1 |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-11T08:28:40Z |
| publishDate | 2023-02-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj.art-bd1165def8e540bc8cc5f35e4c31a4b12023-11-16T21:55:58ZengMDPI AGMathematics2227-73902023-02-0111492910.3390/math11040929The Novel Analytical–Numerical Method for Multi-Dimensional Multi-Term Time-Fractional Equations with General Boundary ConditionsJi Lin0Sergiy Reutskiy1Yuhui Zhang2Yu Sun3Jun Lu4College of Mechanics and Materials, Hohai University, Nanjing 210098, ChinaA. Pidhornyi Institute of Mechanical Engineering Problems of NAS of Ukraine, 2/10 Pozharsky Street, 61046 Kharkiv, UkraineCollege of Mechanics and Materials, Hohai University, Nanjing 210098, ChinaNanjing Hydraulic Research Institute, Nanjing 210029, ChinaNanjing Hydraulic Research Institute, Nanjing 210029, ChinaThis article presents a simple but effective two-step analytical–numerical algorithm for solving multi-dimensional multi-term time-fractional equations. The first step is to derive an analytic representation that satisfies boundary requirements for 1D, 2D, and 3D problems, respectively. The second step is the meshless approximation where the Müntz polynomials are used to form the approximate solution and the unknown parameters are obtained by imposing the approximation for the governing equations. We illustrate first the detailed derivation of the analytic approximation and then the numerical implementation of the solution procedure. Several numerical examples are provided to verify the accuracy, efficiency, and adaptability to problems with general boundary conditions. The numerical results are compared with exact solutions and numerical methods reported in the literature, showing that the algorithm has great potential for multi-dimensional multi-term time-fractional equations with various boundary conditions.https://www.mdpi.com/2227-7390/11/4/929multi-dimensional fractional equationsmulti-term fractional equationsmeshless methodcollocation methodanalytic representation |
| spellingShingle | Ji Lin Sergiy Reutskiy Yuhui Zhang Yu Sun Jun Lu The Novel Analytical–Numerical Method for Multi-Dimensional Multi-Term Time-Fractional Equations with General Boundary Conditions Mathematics multi-dimensional fractional equations multi-term fractional equations meshless method collocation method analytic representation |
| title | The Novel Analytical–Numerical Method for Multi-Dimensional Multi-Term Time-Fractional Equations with General Boundary Conditions |
| title_full | The Novel Analytical–Numerical Method for Multi-Dimensional Multi-Term Time-Fractional Equations with General Boundary Conditions |
| title_fullStr | The Novel Analytical–Numerical Method for Multi-Dimensional Multi-Term Time-Fractional Equations with General Boundary Conditions |
| title_full_unstemmed | The Novel Analytical–Numerical Method for Multi-Dimensional Multi-Term Time-Fractional Equations with General Boundary Conditions |
| title_short | The Novel Analytical–Numerical Method for Multi-Dimensional Multi-Term Time-Fractional Equations with General Boundary Conditions |
| title_sort | novel analytical numerical method for multi dimensional multi term time fractional equations with general boundary conditions |
| topic | multi-dimensional fractional equations multi-term fractional equations meshless method collocation method analytic representation |
| url | https://www.mdpi.com/2227-7390/11/4/929 |
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