On the Exact Solution of a Scalar Differential Equation via a Simple Analytical Approach
The existence of the advance parameter in a scalar differential equation prevents the application of the well-known standard methods used for solving classical ordinary differential equations. A simple procedure is introduced in this paper to remove the advance parameter from a special kind of first...
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MDPI AG
2024-02-01
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| Online Access: | https://www.mdpi.com/2075-1680/13/2/129 |
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| author | Nada A. M. Alshomrani Abdelhalim Ebaid Faten Aldosari Mona D. Aljoufi |
| author_facet | Nada A. M. Alshomrani Abdelhalim Ebaid Faten Aldosari Mona D. Aljoufi |
| author_sort | Nada A. M. Alshomrani |
| collection | DOAJ |
| description | The existence of the advance parameter in a scalar differential equation prevents the application of the well-known standard methods used for solving classical ordinary differential equations. A simple procedure is introduced in this paper to remove the advance parameter from a special kind of first-order scalar differential equation. The suggested approach transforms the given first-order scalar differential equation to an equivalent second-order ordinary differential equation (ODE) without the advance parameter. Using this method, we are able to construct the exact solution of both the transformed model and the given original model. The exact solution is obtained in a wave form with specified amplitude and phase. Furthermore, several special cases are investigated at certain values/relationships of the involved parameters. It is shown that the exact solution in the absence of the advance parameter reduces to the corresponding solution in the literature. In addition, it is declared that the current model enjoys various kinds of solutions, such as constant solutions, polynomial solutions, and periodic solutions under certain constraints of the included parameters. |
| first_indexed | 2024-03-07T22:43:00Z |
| format | Article |
| id | doaj.art-cf52d2de87ed47e1a2e83020102fdf7a |
| institution | Directory Open Access Journal |
| issn | 2075-1680 |
| language | English |
| last_indexed | 2024-03-07T22:43:00Z |
| publishDate | 2024-02-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Axioms |
| spelling | doaj.art-cf52d2de87ed47e1a2e83020102fdf7a2024-02-23T15:07:30ZengMDPI AGAxioms2075-16802024-02-0113212910.3390/axioms13020129On the Exact Solution of a Scalar Differential Equation via a Simple Analytical ApproachNada A. M. Alshomrani0Abdelhalim Ebaid1Faten Aldosari2Mona D. Aljoufi3Department of Mathematics, Faculty of Science, University of Tabuk, P.O. Box 741, Tabuk 71491, Saudi ArabiaDepartment of Mathematics, Faculty of Science, University of Tabuk, P.O. Box 741, Tabuk 71491, Saudi ArabiaDepartment of Mathematics, College of Science, Shaqra University, P.O. Box 15572, Shaqra 11961, Saudi ArabiaDepartment of Mathematics, Faculty of Science, University of Tabuk, P.O. Box 741, Tabuk 71491, Saudi ArabiaThe existence of the advance parameter in a scalar differential equation prevents the application of the well-known standard methods used for solving classical ordinary differential equations. A simple procedure is introduced in this paper to remove the advance parameter from a special kind of first-order scalar differential equation. The suggested approach transforms the given first-order scalar differential equation to an equivalent second-order ordinary differential equation (ODE) without the advance parameter. Using this method, we are able to construct the exact solution of both the transformed model and the given original model. The exact solution is obtained in a wave form with specified amplitude and phase. Furthermore, several special cases are investigated at certain values/relationships of the involved parameters. It is shown that the exact solution in the absence of the advance parameter reduces to the corresponding solution in the literature. In addition, it is declared that the current model enjoys various kinds of solutions, such as constant solutions, polynomial solutions, and periodic solutions under certain constraints of the included parameters.https://www.mdpi.com/2075-1680/13/2/129ordinary differential equationdelayexact solutioninitial value problem |
| spellingShingle | Nada A. M. Alshomrani Abdelhalim Ebaid Faten Aldosari Mona D. Aljoufi On the Exact Solution of a Scalar Differential Equation via a Simple Analytical Approach Axioms ordinary differential equation delay exact solution initial value problem |
| title | On the Exact Solution of a Scalar Differential Equation via a Simple Analytical Approach |
| title_full | On the Exact Solution of a Scalar Differential Equation via a Simple Analytical Approach |
| title_fullStr | On the Exact Solution of a Scalar Differential Equation via a Simple Analytical Approach |
| title_full_unstemmed | On the Exact Solution of a Scalar Differential Equation via a Simple Analytical Approach |
| title_short | On the Exact Solution of a Scalar Differential Equation via a Simple Analytical Approach |
| title_sort | on the exact solution of a scalar differential equation via a simple analytical approach |
| topic | ordinary differential equation delay exact solution initial value problem |
| url | https://www.mdpi.com/2075-1680/13/2/129 |
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