Solving Some Physics Problems Involving Fractional-Order Differential Equations with the Morgan-Voyce Polynomials
In this research, we present a new computational technique for solving some physics problems involving fractional-order differential equations including the famous Bagley–Torvik method. The model is considered one of the important models to simulate the coupled oscillator and various other applicati...
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| Format: | Article |
| Language: | English |
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MDPI AG
2023-03-01
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| Series: | Fractal and Fractional |
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| Online Access: | https://www.mdpi.com/2504-3110/7/4/301 |
| _version_ | 1797605368463884288 |
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| author | Hari Mohan Srivastava Waleed Adel Mohammad Izadi Adel A. El-Sayed |
| author_facet | Hari Mohan Srivastava Waleed Adel Mohammad Izadi Adel A. El-Sayed |
| author_sort | Hari Mohan Srivastava |
| collection | DOAJ |
| description | In this research, we present a new computational technique for solving some physics problems involving fractional-order differential equations including the famous Bagley–Torvik method. The model is considered one of the important models to simulate the coupled oscillator and various other applications in science and engineering. We adapt a collocation technique involving a new operational matrix that utilizes the Liouville–Caputo operator of differentiation and Morgan–Voyce polynomials, in combination with the Tau spectral method. We first present the differentiation matrix of fractional order that is used to convert the problem and its conditions into an algebraic system of equations with unknown coefficients, which are then used to find the solutions to the proposed models. An error analysis for the method is proved to verify the convergence of the acquired solutions. To test the effectiveness of the proposed technique, several examples are simulated using the presented technique and these results are compared with other techniques from the literature. In addition, the computational time is computed and tabulated to ensure the efficacy and robustness of the method. The outcomes of the numerical examples support the theoretical results and show the accuracy and applicability of the presented approach. The method is shown to give better results than the other methods using a lower number of bases and with less spent time, and helped in highlighting some of the important features of the model. The technique proves to be a valuable approach that can be extended in the future for other fractional models having real applications such as the fractional partial differential equations and fractional integro-differential equations. |
| first_indexed | 2024-03-11T05:00:09Z |
| format | Article |
| id | doaj.art-7886807384434ab787570319f3c7843e |
| institution | Directory Open Access Journal |
| issn | 2504-3110 |
| language | English |
| last_indexed | 2024-03-11T05:00:09Z |
| publishDate | 2023-03-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Fractal and Fractional |
| spelling | doaj.art-7886807384434ab787570319f3c7843e2023-11-17T19:19:15ZengMDPI AGFractal and Fractional2504-31102023-03-017430110.3390/fractalfract7040301Solving Some Physics Problems Involving Fractional-Order Differential Equations with the Morgan-Voyce PolynomialsHari Mohan Srivastava0Waleed Adel1Mohammad Izadi2Adel A. El-Sayed3Department of Mathematics and Statistics, University of Victoria, Victoria, BC V8W 3R4, CanadaDepartment of Technology of Informatics and Communications, Université Française D’Egypte, Ismailia Desert Road, Cairo 11837, EgyptDepartment of Applied Mathematics, Faculty of Mathematics and Computer, Shahid Bahonar University of Kerman, Kerman 76169-14111, IranDepartment of Mathematics, Faculty of Science, Fayoum University, Fayoum 63514, EgyptIn this research, we present a new computational technique for solving some physics problems involving fractional-order differential equations including the famous Bagley–Torvik method. The model is considered one of the important models to simulate the coupled oscillator and various other applications in science and engineering. We adapt a collocation technique involving a new operational matrix that utilizes the Liouville–Caputo operator of differentiation and Morgan–Voyce polynomials, in combination with the Tau spectral method. We first present the differentiation matrix of fractional order that is used to convert the problem and its conditions into an algebraic system of equations with unknown coefficients, which are then used to find the solutions to the proposed models. An error analysis for the method is proved to verify the convergence of the acquired solutions. To test the effectiveness of the proposed technique, several examples are simulated using the presented technique and these results are compared with other techniques from the literature. In addition, the computational time is computed and tabulated to ensure the efficacy and robustness of the method. The outcomes of the numerical examples support the theoretical results and show the accuracy and applicability of the presented approach. The method is shown to give better results than the other methods using a lower number of bases and with less spent time, and helped in highlighting some of the important features of the model. The technique proves to be a valuable approach that can be extended in the future for other fractional models having real applications such as the fractional partial differential equations and fractional integro-differential equations.https://www.mdpi.com/2504-3110/7/4/301fractional-order equationscollocation methodLiouville–Caputo’s fractional derivative operatorerror analysisTau method |
| spellingShingle | Hari Mohan Srivastava Waleed Adel Mohammad Izadi Adel A. El-Sayed Solving Some Physics Problems Involving Fractional-Order Differential Equations with the Morgan-Voyce Polynomials Fractal and Fractional fractional-order equations collocation method Liouville–Caputo’s fractional derivative operator error analysis Tau method |
| title | Solving Some Physics Problems Involving Fractional-Order Differential Equations with the Morgan-Voyce Polynomials |
| title_full | Solving Some Physics Problems Involving Fractional-Order Differential Equations with the Morgan-Voyce Polynomials |
| title_fullStr | Solving Some Physics Problems Involving Fractional-Order Differential Equations with the Morgan-Voyce Polynomials |
| title_full_unstemmed | Solving Some Physics Problems Involving Fractional-Order Differential Equations with the Morgan-Voyce Polynomials |
| title_short | Solving Some Physics Problems Involving Fractional-Order Differential Equations with the Morgan-Voyce Polynomials |
| title_sort | solving some physics problems involving fractional order differential equations with the morgan voyce polynomials |
| topic | fractional-order equations collocation method Liouville–Caputo’s fractional derivative operator error analysis Tau method |
| url | https://www.mdpi.com/2504-3110/7/4/301 |
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