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All Journal Journal of Multidisciplinary Science: MIKAILALSYS Mikailalsys Journal of Advanced Engineering International
L. N., Sanda
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A Hybrid Elzaki Transform-Daftardar-Jafari Method for Solving Nonlinear Proportional Delay Differential Equations L. N., Sanda; J. O., Okai; U.M., Nasir; Mujahid, U. A.; Cornelius, Michael; G.S., Ndam
Journal of Multidisciplinary Science: MIKAILALSYS Vol 4 No 1 (2026): Journal of Multidisciplinary Science: MIKAILALSYS
Publisher : Darul Yasin Al Sys

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.58578/mikailalsys.v4i1.8106

Abstract

Proportional delay differential equations (PDDEs) arise naturally in viscoelasticity, control theory, biology, population dynamics, and fractional-order physical models in which the future state depends on the value of the solution at a proportion of the current time, but their nonlinear nature and delay terms make analytic treatment challenging. This study develops a hybrid computational scheme that combines the Elzaki Transform (ET) and the Daftardar–Jafari Method (DJM) to obtain accurate analytical–approximate solutions for linear and nonlinear PDDEs. In the proposed approach, the Elzaki transform converts the PDDE into an algebraic functional equation, which is subsequently decomposed using DJM without the need for Adomian polynomials. The method is straightforward, computationally efficient, and capable of handling strong nonlinearities. Several illustrative examples are presented to demonstrate its efficiency, and the results confirm that the ET–DJM hybrid provides a powerful alternative to classical methods such as the Laplace transform, Adomian Decomposition Method (ADM), Variational Iteration Method (VIM), Homotopy Perturbation Method (HPM), and homotopy analysis methods.
Elzaki Transform Approach for Solving Linear Proportional Delay Differential Equations L. N., Sanda; J. O., Okai; U.M., Nasir; Mujahid, U. A.; Cornelius, Michael; G.S., Ndam
Mikailalsys Journal of Advanced Engineering International Vol 3 No 1 (2026): Mikailalsys Journal of Advanced Engineering International
Publisher : Darul Yasin Al Sys

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.58578/mjaei.v3i1.8105

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.
Co-Authors Cornelius, Michael G.S., Ndam J. O., Okai Mujahid, U. A. U.M., Nasir