The Exact Solutions of Fractional Differential Systems with <i>n</i> Sinusoidal Terms under Physical Conditions
This paper considers the classes of the first-order fractional differential systems containing a finite number <i>n</i> of sinusoidal terms. The fractional derivative employs the Riemann–Liouville fractional definition. As a method of solution, the Laplace transform is an efficient tool...
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| Format: | Article |
| Language: | English |
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MDPI AG
2022-12-01
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| Series: | Symmetry |
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| Online Access: | https://www.mdpi.com/2073-8994/14/12/2539 |
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| author | Laila F. Seddek Essam R. El-Zahar Abdelhalim Ebaid |
| author_facet | Laila F. Seddek Essam R. El-Zahar Abdelhalim Ebaid |
| author_sort | Laila F. Seddek |
| collection | DOAJ |
| description | This paper considers the classes of the first-order fractional differential systems containing a finite number <i>n</i> of sinusoidal terms. The fractional derivative employs the Riemann–Liouville fractional definition. As a method of solution, the Laplace transform is an efficient tool to solve linear fractional differential equations. However, this method requires to express the initial conditions in certain fractional forms which have no physical meaning currently. This issue formulated a challenge to solve fractional systems under real/physical conditions when applying the Riemann–Liouville fractional definition. The principal incentive of this work is to overcome such difficulties via presenting a simple but effective approach. The proposed approach is successfully applied in this paper to solve linear fractional systems of an oscillatory nature. The exact solutions of the present fractional systems under physical initial conditions are derived in a straightforward manner. In addition, the obtained solutions are given in terms of the entire exponential and periodic functions with arguments of a fractional order. The symmetric/asymmetric behaviors/properties of the obtained solutions are illustrated. Moreover, the exact solutions of the classical/ordinary versions of the undertaken fractional systems are determined smoothly. In addition, the properties and the behaviors of the present solutions are discussed and interpreted. |
| first_indexed | 2024-03-09T03:31:23Z |
| format | Article |
| id | doaj.art-5028669ec65441d290542917a87250de |
| institution | Directory Open Access Journal |
| issn | 2073-8994 |
| language | English |
| last_indexed | 2024-03-09T03:31:23Z |
| publishDate | 2022-12-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Symmetry |
| spelling | doaj.art-5028669ec65441d290542917a87250de2023-12-03T14:55:15ZengMDPI AGSymmetry2073-89942022-12-011412253910.3390/sym14122539The Exact Solutions of Fractional Differential Systems with <i>n</i> Sinusoidal Terms under Physical ConditionsLaila F. Seddek0Essam R. El-Zahar1Abdelhalim Ebaid2Department of Mathematics, College of Science and Humanities in Al-Kharj, Prince Sattam Bin Abdul-Aziz University, Al-Kharj 11942, Saudi ArabiaDepartment of Mathematics, College of Science and Humanities in Al-Kharj, Prince Sattam Bin Abdul-Aziz University, Al-Kharj 11942, Saudi ArabiaDepartment of Mathematics, Faculty of Science, University of Tabuk, P.O. Box 741, Tabuk 71491, Saudi ArabiaThis paper considers the classes of the first-order fractional differential systems containing a finite number <i>n</i> of sinusoidal terms. The fractional derivative employs the Riemann–Liouville fractional definition. As a method of solution, the Laplace transform is an efficient tool to solve linear fractional differential equations. However, this method requires to express the initial conditions in certain fractional forms which have no physical meaning currently. This issue formulated a challenge to solve fractional systems under real/physical conditions when applying the Riemann–Liouville fractional definition. The principal incentive of this work is to overcome such difficulties via presenting a simple but effective approach. The proposed approach is successfully applied in this paper to solve linear fractional systems of an oscillatory nature. The exact solutions of the present fractional systems under physical initial conditions are derived in a straightforward manner. In addition, the obtained solutions are given in terms of the entire exponential and periodic functions with arguments of a fractional order. The symmetric/asymmetric behaviors/properties of the obtained solutions are illustrated. Moreover, the exact solutions of the classical/ordinary versions of the undertaken fractional systems are determined smoothly. In addition, the properties and the behaviors of the present solutions are discussed and interpreted.https://www.mdpi.com/2073-8994/14/12/2539Riemann–Liouville fractional derivativefractional differential equationsinusoidalexact solution |
| spellingShingle | Laila F. Seddek Essam R. El-Zahar Abdelhalim Ebaid The Exact Solutions of Fractional Differential Systems with <i>n</i> Sinusoidal Terms under Physical Conditions Symmetry Riemann–Liouville fractional derivative fractional differential equation sinusoidal exact solution |
| title | The Exact Solutions of Fractional Differential Systems with <i>n</i> Sinusoidal Terms under Physical Conditions |
| title_full | The Exact Solutions of Fractional Differential Systems with <i>n</i> Sinusoidal Terms under Physical Conditions |
| title_fullStr | The Exact Solutions of Fractional Differential Systems with <i>n</i> Sinusoidal Terms under Physical Conditions |
| title_full_unstemmed | The Exact Solutions of Fractional Differential Systems with <i>n</i> Sinusoidal Terms under Physical Conditions |
| title_short | The Exact Solutions of Fractional Differential Systems with <i>n</i> Sinusoidal Terms under Physical Conditions |
| title_sort | exact solutions of fractional differential systems with i n i sinusoidal terms under physical conditions |
| topic | Riemann–Liouville fractional derivative fractional differential equation sinusoidal exact solution |
| url | https://www.mdpi.com/2073-8994/14/12/2539 |
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