A Hybrid Elzaki Transform-Daftardar-Jafari Method for Solving Nonlinear Proportional Delay Differential Equations

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Page Numbers: 13-25
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Published: Jan 28, 2026
Digital Object Identifier: 10.58578/mikailalsys.v4i1.8106
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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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Sanda L. N.
Abubakar Tafawa Balewa University Bauchi State, Nigeria
Okai J. O.
Abubakar Tafawa Balewa University Bauchi State, Nigeria
Nasir U.M.
Abubakar Tafawa Balewa University Bauchi State, Nigeria
U. A. Mujahid
Federal Polytechnic Kaltungo, Gombe State, Nigeria
Michael Cornelius
Gombe State University, Gombe State, Nigeria
Ndam G.S.
Abubakar Tafawa Balewa University Bauchi State,

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.

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Article Details

How to Cite
L. N., S., J. O., O., U.M., N., Mujahid, U. A., Cornelius, M., & G.S., N. (2026). A Hybrid Elzaki Transform-Daftardar-Jafari Method for Solving Nonlinear Proportional Delay Differential Equations. Journal of Multidisciplinary Science: MIKAILALSYS, 4(1), 13-25. https://doi.org/10.58578/mikailalsys.v4i1.8106

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Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License that allows others to share the work with an acknowledgement of the work's authorship and initial publication in this journal.

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