Exact and Approximate Solutions for Some Classes of the Inhomogeneous Pantograph Equation
The standard pantograph delay equation (SPDDE) is one of the famous delay models. This standard model is basically homogeneous in nature and it has been extensively studied in the literature. However, the studies on the general inhomogeneous form of such a model seem rare. This paper considers the i...
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MDPI AG
2023-12-01
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| author | A. A. Al Qarni |
| author_facet | A. A. Al Qarni |
| author_sort | A. A. Al Qarni |
| collection | DOAJ |
| description | The standard pantograph delay equation (SPDDE) is one of the famous delay models. This standard model is basically homogeneous in nature and it has been extensively studied in the literature. However, the studies on the general inhomogeneous form of such a model seem rare. This paper considers the inhomogeneous pantograph delay equation (IPDDE) with a kind of arbitrary inhomogeneous term. This arbitrary inhomogeneous term is used in different forms to generate various classes of IPDDEs. The solutions of the present classes are obtained in closed series forms which satisfy the criteria of convergence. Also, the exact solutions are determined for these classes under a certain relation between the given initial condition of the model and the initial value of the inhomogeneous term. Several classes are generated and solved when the inhomogeneous term takes the form of trigonometric, exponential, and hyperbolic functions. Some existing results in the literature are recovered as special cases of the present ones. Moreover, the behaviors of the obtained solutions are demonstrated through graphs for various kinds of IPDDEs. |
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| issn | 2075-1680 |
| language | English |
| last_indexed | 2024-03-08T11:06:38Z |
| publishDate | 2023-12-01 |
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| spelling | doaj.art-ba3deff4a25c4fb0b3ffd28bd1c2477d2024-01-26T15:02:43ZengMDPI AGAxioms2075-16802023-12-01131110.3390/axioms13010001Exact and Approximate Solutions for Some Classes of the Inhomogeneous Pantograph EquationA. A. Al Qarni0Department of Mathematics, College of Science, University of Bisha, P.O. Box 344, Bisha 61922, Saudi ArabiaThe standard pantograph delay equation (SPDDE) is one of the famous delay models. This standard model is basically homogeneous in nature and it has been extensively studied in the literature. However, the studies on the general inhomogeneous form of such a model seem rare. This paper considers the inhomogeneous pantograph delay equation (IPDDE) with a kind of arbitrary inhomogeneous term. This arbitrary inhomogeneous term is used in different forms to generate various classes of IPDDEs. The solutions of the present classes are obtained in closed series forms which satisfy the criteria of convergence. Also, the exact solutions are determined for these classes under a certain relation between the given initial condition of the model and the initial value of the inhomogeneous term. Several classes are generated and solved when the inhomogeneous term takes the form of trigonometric, exponential, and hyperbolic functions. Some existing results in the literature are recovered as special cases of the present ones. Moreover, the behaviors of the obtained solutions are demonstrated through graphs for various kinds of IPDDEs.https://www.mdpi.com/2075-1680/13/1/1pantographdelayinhomogeneousexact solutionseries solution |
| spellingShingle | A. A. Al Qarni Exact and Approximate Solutions for Some Classes of the Inhomogeneous Pantograph Equation Axioms pantograph delay inhomogeneous exact solution series solution |
| title | Exact and Approximate Solutions for Some Classes of the Inhomogeneous Pantograph Equation |
| title_full | Exact and Approximate Solutions for Some Classes of the Inhomogeneous Pantograph Equation |
| title_fullStr | Exact and Approximate Solutions for Some Classes of the Inhomogeneous Pantograph Equation |
| title_full_unstemmed | Exact and Approximate Solutions for Some Classes of the Inhomogeneous Pantograph Equation |
| title_short | Exact and Approximate Solutions for Some Classes of the Inhomogeneous Pantograph Equation |
| title_sort | exact and approximate solutions for some classes of the inhomogeneous pantograph equation |
| topic | pantograph delay inhomogeneous exact solution series solution |
| url | https://www.mdpi.com/2075-1680/13/1/1 |
| work_keys_str_mv | AT aaalqarni exactandapproximatesolutionsforsomeclassesoftheinhomogeneouspantographequation |