A novel analytical treatment for the Ambartsumian delay differential equation with a variable coefficient

Rana M. S. Alyoubi; Abdelhalim Ebaid; Essam R. El-Zahar; Mona D. Aljoufi; Rana M. S. Alyoubi; Abdelhalim Ebaid; Essam R. El-Zahar; Mona D. Aljoufi

doi:10.3934/math.20241696

AIMS Mathematics

2024, Volume 9, Issue 12: 35743-35758. doi: 10.3934/math.20241696

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Research article

A novel analytical treatment for the Ambartsumian delay differential equation with a variable coefficient

1.
Department of Mathematics, Faculty of Science, University of Tabuk, P.O. Box 741, Tabuk 71491, Saudi Arabia
2.
Department of Mathematics, College of Science and Humanities in Al-Kharj, Prince Sattam bin Abdulaziz University, P.O. Box 83, Al-Kharj 11942, Saudi Arabia

Received: 17 November 2024 Revised: 06 December 2024 Accepted: 09 December 2024 Published: 24 December 2024
MSC : 34K06, 65L03

The Ambartsumian delay differential equation with a variable coefficient is considered in this paper. An effective transformation is produced to convert the extended Ambartsumian equation to the pantograph model. Two kinds of analytical solutions are determined. The first solution is expressed as an exponential function multiplied by an infinite power series. The second solution is obtained as an infinite series in terms of exponential functions. Several exact solutions are established for different forms of the extended Ambartsumian equation under specific relations. In addition, the convergence analysis is addressed theoretically. Moreover, numeric calculations are conducted to estimate the accuracy. The results reveal that the present analysis is efficient and accurate and can be further applied to similar delay models in a straightforward manner.
- Ambartsumian equation,
- delay,
- exact solution,
- series solution
Citation: Rana M. S. Alyoubi, Abdelhalim Ebaid, Essam R. El-Zahar, Mona D. Aljoufi. A novel analytical treatment for the Ambartsumian delay differential equation with a variable coefficient[J]. AIMS Mathematics, 2024, 9(12): 35743-35758. doi: 10.3934/math.20241696

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Abstract

The Ambartsumian delay differential equation with a variable coefficient is considered in this paper. An effective transformation is produced to convert the extended Ambartsumian equation to the pantograph model. Two kinds of analytical solutions are determined. The first solution is expressed as an exponential function multiplied by an infinite power series. The second solution is obtained as an infinite series in terms of exponential functions. Several exact solutions are established for different forms of the extended Ambartsumian equation under specific relations. In addition, the convergence analysis is addressed theoretically. Moreover, numeric calculations are conducted to estimate the accuracy. The results reveal that the present analysis is efficient and accurate and can be further applied to similar delay models in a straightforward manner.

References

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