Numerical Study of Caputo Fractional-Order Differential Equations by Developing New Operational Matrices of Vieta–Lucas Polynomials
In this article, we develop a numerical method based on the operational matrices of shifted Vieta–Lucas polynomials (VLPs) for solving Caputo fractional-order differential equations (FDEs). We derive a new operational matrix of the fractional-order derivatives in the Caputo sense, which is then used...
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| Format: | Article |
| Language: | English |
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MDPI AG
2022-01-01
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| Series: | Fractal and Fractional |
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| Online Access: | https://www.mdpi.com/2504-3110/6/2/79 |
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| author | Zulfiqar Ahmad Noor Imran Talib Thabet Abdeljawad Manar A. Alqudah |
| author_facet | Zulfiqar Ahmad Noor Imran Talib Thabet Abdeljawad Manar A. Alqudah |
| author_sort | Zulfiqar Ahmad Noor |
| collection | DOAJ |
| description | In this article, we develop a numerical method based on the operational matrices of shifted Vieta–Lucas polynomials (VLPs) for solving Caputo fractional-order differential equations (FDEs). We derive a new operational matrix of the fractional-order derivatives in the Caputo sense, which is then used with spectral tau and spectral collocation methods to reduce the FDEs to a system of algebraic equations. Several numerical examples are given to show the accuracy of this method. These examples show that the obtained results have good agreement with the analytical solutions in both linear and non-linear FDEs. In addition to this, the numerical results obtained by using our method are compared with the numerical results obtained otherwise in the literature. |
| first_indexed | 2024-03-09T21:55:27Z |
| format | Article |
| id | doaj.art-1d30db2d6dad408faf210055cf44868e |
| institution | Directory Open Access Journal |
| issn | 2504-3110 |
| language | English |
| last_indexed | 2024-03-09T21:55:27Z |
| publishDate | 2022-01-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Fractal and Fractional |
| spelling | doaj.art-1d30db2d6dad408faf210055cf44868e2023-11-23T19:58:52ZengMDPI AGFractal and Fractional2504-31102022-01-01627910.3390/fractalfract6020079Numerical Study of Caputo Fractional-Order Differential Equations by Developing New Operational Matrices of Vieta–Lucas PolynomialsZulfiqar Ahmad Noor0Imran Talib1Thabet Abdeljawad2Manar A. Alqudah3Nonlinear Analysis Group (NAG), Mathematics Department, Virtual University of Pakistan, Lahore 54000, PakistanNonlinear Analysis Group (NAG), Mathematics Department, Virtual University of Pakistan, Lahore 54000, PakistanDepartment of Mathematics and Sciences, Prince Sultan University, Riyadh 11586, Saudi ArabiaDepartment of Mathematical Sciences, Faculty of Sciences, Princess Nourah Bint Abdulrahman University, Riyadh 11671, Saudi ArabiaIn this article, we develop a numerical method based on the operational matrices of shifted Vieta–Lucas polynomials (VLPs) for solving Caputo fractional-order differential equations (FDEs). We derive a new operational matrix of the fractional-order derivatives in the Caputo sense, which is then used with spectral tau and spectral collocation methods to reduce the FDEs to a system of algebraic equations. Several numerical examples are given to show the accuracy of this method. These examples show that the obtained results have good agreement with the analytical solutions in both linear and non-linear FDEs. In addition to this, the numerical results obtained by using our method are compared with the numerical results obtained otherwise in the literature.https://www.mdpi.com/2504-3110/6/2/79fractional-order differential equationsoperational matricesshifted Vieta–Lucas polynomialsCaputo derivative |
| spellingShingle | Zulfiqar Ahmad Noor Imran Talib Thabet Abdeljawad Manar A. Alqudah Numerical Study of Caputo Fractional-Order Differential Equations by Developing New Operational Matrices of Vieta–Lucas Polynomials Fractal and Fractional fractional-order differential equations operational matrices shifted Vieta–Lucas polynomials Caputo derivative |
| title | Numerical Study of Caputo Fractional-Order Differential Equations by Developing New Operational Matrices of Vieta–Lucas Polynomials |
| title_full | Numerical Study of Caputo Fractional-Order Differential Equations by Developing New Operational Matrices of Vieta–Lucas Polynomials |
| title_fullStr | Numerical Study of Caputo Fractional-Order Differential Equations by Developing New Operational Matrices of Vieta–Lucas Polynomials |
| title_full_unstemmed | Numerical Study of Caputo Fractional-Order Differential Equations by Developing New Operational Matrices of Vieta–Lucas Polynomials |
| title_short | Numerical Study of Caputo Fractional-Order Differential Equations by Developing New Operational Matrices of Vieta–Lucas Polynomials |
| title_sort | numerical study of caputo fractional order differential equations by developing new operational matrices of vieta lucas polynomials |
| topic | fractional-order differential equations operational matrices shifted Vieta–Lucas polynomials Caputo derivative |
| url | https://www.mdpi.com/2504-3110/6/2/79 |
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