Analysis of a First-Order Delay Model under a History Function with Discontinuity
This paper analyzes the first-order delay equation <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>y</mi><mo>′</mo></msup><mrow><mo>(</mo><...
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2024-08-01
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| author | Weam G. Alharbi |
| author_facet | Weam G. Alharbi |
| author_sort | Weam G. Alharbi |
| collection | DOAJ |
| description | This paper analyzes the first-order delay equation <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>y</mi><mo>′</mo></msup><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow><mo>=</mo><mi>α</mi><mi>y</mi><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow><mo>+</mo><mi>β</mi><mi>y</mi><mrow><mo>(</mo><mi>t</mi><mo>−</mo><mi>τ</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> subject to a history function in addition to an initial condition that assumes discontinuity at <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>t</mi><mo>=</mo><mn>0</mn></mrow></semantics></math></inline-formula>. The method of steps is successfully applied to derive the exact solution in an explicit form. In addition, a unified formula is provided to describe the solution in any finite sub-interval of the problem’s domain. The characteristics and properties of the solution are theoretically investigated and then confirmed through several plots. The behavior of the solution and its derivative are examined and interpreted. The results show that the method of steps is an effective method of solution to treat the current delay model. The present successful analysis can be used to investigate other delay models with complex initial conditions. Furthermore, the present approach can be generalized to include the inhomogeneous version of the current model without using numerical methods. |
| format | Article |
| id | doaj-art-99f23409f2a241ef8230cb3f6655e8ba |
| institution | OA Journals |
| issn | 1300-686X 2297-8747 |
| language | English |
| publishDate | 2024-08-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematical and Computational Applications |
| spelling | doaj-art-99f23409f2a241ef8230cb3f6655e8ba2025-08-20T02:10:56ZengMDPI AGMathematical and Computational Applications1300-686X2297-87472024-08-012957210.3390/mca29050072Analysis of a First-Order Delay Model under a History Function with DiscontinuityWeam G. Alharbi0Department of Mathematics, Faculty of Science, University of Tabuk, P.O. Box 741, Tabuk 71491, Saudi ArabiaThis paper analyzes the first-order delay equation <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>y</mi><mo>′</mo></msup><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow><mo>=</mo><mi>α</mi><mi>y</mi><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow><mo>+</mo><mi>β</mi><mi>y</mi><mrow><mo>(</mo><mi>t</mi><mo>−</mo><mi>τ</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> subject to a history function in addition to an initial condition that assumes discontinuity at <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>t</mi><mo>=</mo><mn>0</mn></mrow></semantics></math></inline-formula>. The method of steps is successfully applied to derive the exact solution in an explicit form. In addition, a unified formula is provided to describe the solution in any finite sub-interval of the problem’s domain. The characteristics and properties of the solution are theoretically investigated and then confirmed through several plots. The behavior of the solution and its derivative are examined and interpreted. The results show that the method of steps is an effective method of solution to treat the current delay model. The present successful analysis can be used to investigate other delay models with complex initial conditions. Furthermore, the present approach can be generalized to include the inhomogeneous version of the current model without using numerical methods.https://www.mdpi.com/2297-8747/29/5/72delay differential equationsinitial value problemsexact solutionmethod of step |
| spellingShingle | Weam G. Alharbi Analysis of a First-Order Delay Model under a History Function with Discontinuity Mathematical and Computational Applications delay differential equations initial value problems exact solution method of step |
| title | Analysis of a First-Order Delay Model under a History Function with Discontinuity |
| title_full | Analysis of a First-Order Delay Model under a History Function with Discontinuity |
| title_fullStr | Analysis of a First-Order Delay Model under a History Function with Discontinuity |
| title_full_unstemmed | Analysis of a First-Order Delay Model under a History Function with Discontinuity |
| title_short | Analysis of a First-Order Delay Model under a History Function with Discontinuity |
| title_sort | analysis of a first order delay model under a history function with discontinuity |
| topic | delay differential equations initial value problems exact solution method of step |
| url | https://www.mdpi.com/2297-8747/29/5/72 |
| work_keys_str_mv | AT weamgalharbi analysisofafirstorderdelaymodelunderahistoryfunctionwithdiscontinuity |