Reliable analytical approach to solving delay differential equations with combined proportional and constant delays

Essam R. El-Zahar; Abdelhalim Ebaid; Laila F. Seddek; Mona D. Aljoufi; Essam R. El-Zahar; Abdelhalim Ebaid; Laila F. Seddek; Mona D. Aljoufi

doi:10.3934/math.20251277

AIMS Mathematics

2025, Volume 10, Issue 12: 29037-29053. doi: 10.3934/math.20251277

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Research article

Reliable analytical approach to solving delay differential equations with combined proportional and constant delays

1.
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
2.
Department of Mathematics, Faculty of Science, University of Tabuk, P.O. Box 741, Tabuk 71491, Saudi Arabia

Received: 22 October 2025 Revised: 19 November 2025 Accepted: 09 December 2025 Published: 11 December 2025
MSC : 34K06, 34K07, 65L03

The investigation of delay differential equations (DDEs) spans a diverse array of practical applications. In the realm of applied sciences, DDEs typically involve either constant/pure delays or proportional delays, each posing significant analytical challenges. The task of deriving exact solutions becomes increasingly intricate when both types of delays are integrated within a single model. This paper derives an explicit unified analytical solution for a DDE with combined delays using the method of steps (MoS). A unified solution formula is presented, applicable across any sub-interval of the problem's domain. Additionally, the theoretical properties of the solution and its derivative—such as continuity and the presence of discontinuities at specific points—are meticulously examined. The proposed methodology also encompasses existing findings in the literature, with applications extending to fields including astronomy and railway electrification.
- ordinary differential equation,
- delay differential equation,
- initial value problem
Citation: Essam R. El-Zahar, Abdelhalim Ebaid, Laila F. Seddek, Mona D. Aljoufi. Reliable analytical approach to solving delay differential equations with combined proportional and constant delays[J]. AIMS Mathematics, 2025, 10(12): 29037-29053. doi: 10.3934/math.20251277

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Abstract

The investigation of delay differential equations (DDEs) spans a diverse array of practical applications. In the realm of applied sciences, DDEs typically involve either constant/pure delays or proportional delays, each posing significant analytical challenges. The task of deriving exact solutions becomes increasingly intricate when both types of delays are integrated within a single model. This paper derives an explicit unified analytical solution for a DDE with combined delays using the method of steps (MoS). A unified solution formula is presented, applicable across any sub-interval of the problem's domain. Additionally, the theoretical properties of the solution and its derivative—such as continuity and the presence of discontinuities at specific points—are meticulously examined. The proposed methodology also encompasses existing findings in the literature, with applications extending to fields including astronomy and railway electrification.

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