Elzaki Transform Approach for Solving Linear Proportional Delay Differential Equations
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Page Numbers:
120-128
Digital Object Identifier:
10.58578/mjaei.v3i1.8105
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Authors:
Sanda L. N.1, Okai J. O.2*, Nasir U.M.3, U. A. Mujahid4, Michael Cornelius5, Ndam G.S.6 
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Abstract
Proportional delay differential equations (PDDEs) arise naturally in physics, economics, population dynamics, epidemiology, and viscoelasticity due to delays that scale proportionally with the independent variable, yet they remain analytically challenging because the delayed argument disrupts the classical structure of ordinary differential equations. This paper presents a human-centered, simplified, and computationally friendly method for solving linear PDDEs using a hybrid approach that combines the Elzaki Transform with established decomposition techniques. Within this framework, the Elzaki Transform is used to convert the original PDDE into an associated functional equation, which is then handled through a systematic decomposition process that avoids excessive algebraic complexity. Two illustrative examples are worked out in detail to demonstrate the step-by-step implementation of the method, showing that the proposed approach yields solutions efficiently while preserving mathematical rigor and interpretability. The analysis highlights that the hybrid Elzaki–decomposition technique offers conceptual transparency, reduces computational overhead, and provides a practical alternative to classical transform-based and purely numerical schemes for linear PDDEs. The study concludes that this approach can serve as an accessible yet robust tool for applied researchers who routinely encounter PDDEs, and it opens pathways for future extensions to more general classes of delay and functional differential equations.
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References
Baker, C. T. H., et al. (2021). Pantograph equations and their applications. Appl. Math. Lett.
Iyiola, O. S. (2023). Recent advances in proportional delay systems. Mathematics.
Ockendon, J., & Tayler, A. B. (2020). The dynamics of the pantograph equation. Phil. Trans. Royal Soc.
Xu, J. (2022). Proportional delays in epidemic modelling. Chaos.
Lin, Q. (2023). Viscoelastic waves with proportional delays. J. Appl. Mech.
Hassan, M. M. (2022). Thermal memory models with PDDEs. Int. J. Heat Mass Transfer.
Kipnis, M. (2020). Population models with proportional delay. Ecological Modelling.
El-Masri, R. (2020). Laplace techniques for delay systems. Math. Methods Appl. Sci.
Ahmadi, F. (2023). Sumudu transform for delay problems. AIMS Math.
Elzaki, T. M. (2020). On the Elzaki transform. Global J. Pure Appl. Math.
Ameen, I. (2023). New applications of Elzaki transform in differential systems. Alexandria Eng. J.
Batiha, B. (2022). Extended properties of the Elzaki transform. Math. Model. Appl.
Sharma, R. (2024). Scaled-argument behavior of the Elzaki transform. J. Math. Anal.
Derfel, G. (2021). Dynamics of pantograph-type equations. J. Differential Equations.
Atangana, A. (2020). Analytic approaches to proportional delay equations. Chaos Solitons Fractals.
Jafari, H. (2024). Decomposition strategies for delay systems. Commun. Nonlinear Sci.
Khan, S. (2022). Hybrid transforms for multi-delay problems. Results Math.
Suleiman, I. (2023). Transform-based solutions for delay operators. Appl. Math. Comput.
Ibrahim, M. D., Okai, J. O., & Cornelius, M. (2024). A robust numerical method for solving linear delay differential equations.
Quitaiba, W. I. (2014). Numerical solution for solving fractional-order delay differential equations.