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All Journal Journal of Multidisciplinary Science: MIKAILALSYS Mikailalsys Journal of Advanced Engineering International
Michael Cornelius
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Agriculture, Biological Sciences & Forestry Chemical Engineering, Chemistry & Bioengineering Computer Science & IT Electrical & Electronics Engineering Engineering Environmental Science Mechanical Engineering Physics Social Sciences Other

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Elzaki Transform Approach for Solving Linear Proportional Delay Differential Equations Sanda L. N.; Okai J. O.; Nasir U.M.; U. A. Mujahid; Michael Cornelius; Ndam G.S.
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.
An Enhanced Variational Iteration Method for Solving Ordinary and Partial Differential Equations Araga Hassan; M. Y. Adamu; A. G. Madaki; Yohanna Nehemiah; Michael Cornelius; U. M. Nasir
Journal of Multidisciplinary Science: MIKAILALSYS Vol 3 No 2 (2025): Journal of Multidisciplinary Science: MIKAILALSYS
Publisher : Darul Yasin Al Sys

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.58578/mikailalsys.v3i2.5317

Abstract

The Variational Iteration Method (VIM) has proven to be a powerful technique for solving both ordinary and partial differential equations. However, its reliance on Lagrange multipliers for each type of equation has posed significant limitations, complicating its application and reducing its efficiency. This study introduces a Modified Variational Iteration Method (MVIM) that eliminates the need for Lagrange multipliers, addressing these challenges. The MVIM reformulates the correctional functional, simplifying the solution process and enhancing computational efficiency. The method is applied to both linear and nonlinear ordinary and partial differential equations, demonstrating its ability to provide accurate and fast-converging solutions. Numerical examples show that the MVIM outperforms traditional VIM in terms of computational time and convergence speed, and compares favourably with other methods such as the Adomian Decomposition Method (ADM) and New Iteration Method (NIM). The results highlight the potential of MVIM as a versatile and efficient tool for solving complex differential equations in a variety of scientific and engineering applications.
A Semi-Analytical Method for Nonlinear System of Delay Differential Equation via Modified He’s Polynomial Yohanna Nehemiah; Aminu Barde; A. Madaki; Michael Cornelius
Journal of Multidisciplinary Science: MIKAILALSYS Vol 3 No 2 (2025): Journal of Multidisciplinary Science: MIKAILALSYS
Publisher : Darul Yasin Al Sys

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.58578/mikailalsys.v3i2.5319

Abstract

In this work, a simple technique based on the combination of sumudu transform and variational iteration method via Modified He’s polynomial is introduced to solve systems of non-linear delay differential equations. The introduced technique is simpler and shorter in its computational procedures and time than the other methods. In addition, the modified He’s polynomial takes care of the nonlinear terms and hence, making the method less stressful in terms of computations. Also, this technique reduces the complexity of calculating Lagrange’s multiplier values which need more computational procedures and time. These advantages make it reliable and its efficiency is demonstrated with numerical examples.
Application of the Kamal-He’s Iterative Method to Klein-Gordons Equations Adejoh Jeremiah; M. Y. Adamu; A. G. Madaki; Okai J. O; Michael Cornelius
Journal of Multidisciplinary Science: MIKAILALSYS Vol 3 No 2 (2025): Journal of Multidisciplinary Science: MIKAILALSYS
Publisher : Darul Yasin Al Sys

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.58578/mikailalsys.v3i2.5320

Abstract

This study demonstrates the effectiveness and accuracy of the KHM for solving both linear and nonlinear Klein-Gordon equations. Through graphical comparisons with other methods such as VIM, TAM, and NIM, and error analysis, the results confirm the high precision and reliability of KHM. The approach is shown to be straightforward, easy to implement, and highly efficient for solving linear PDEs. Additionally, KHM provides the exact solution for nonlinear Klein-Gordon equations in a single iteration, highlighting its computational efficiency. Overall, the KHM is proven to be a powerful and reliable tool for solving a wide range of equations in mathematical physics.
Enhancement of the Kamal Transform Method with the He’s Polynomial for Solving Partial Differential Equations (Telegraph Equation) Ogboche Abichele; I. B. Mshelia; A. G. Madaki; Adejoh Jeremiah; Okai J. O; Michael Cornelius
Journal of Multidisciplinary Science: MIKAILALSYS Vol 3 No 2 (2025): Journal of Multidisciplinary Science: MIKAILALSYS
Publisher : Darul Yasin Al Sys

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.58578/mikailalsys.v3i2.5321

Abstract

This study proposes a hybrid solution methodology that integrates the Kamal Transform Method (KTM) with He’s Polynomial Method (HPM) for solving nonlinear partial differential equations (PDEs), with a focus on the telegraph equation. The telegraph equation, which models wave propagation and diffusive behaviors, presents significant challenges in terms of nonlinearity, complex boundary conditions, and slow convergence in traditional methods. By combining the transformation power of the Kamal method with the iterative, rapidly converging He’s polynomial method, this research aims to enhance the accuracy, convergence, and computational efficiency of existing solution techniques for PDEs. The proposed hybrid approach is applied to both linear and nonlinear forms of the telegraph equation, demonstrating excellent agreement with exact solutions and offering significant improvements in accuracy, especially in the presence of nonlinearities. Comparative analyses with traditional methods, including Elzaki's transform, show that the Kamal-He’s polynomial method outperforms existing techniques in terms of error reduction. The results highlight the method's potential for broader application in various fields of engineering, physics, and applied sciences, where complex, nonlinear PDEs are commonly encountered.
A Hybrid Elzaki Transform-Daftardar-Jafari Method for Solving Nonlinear Proportional Delay Differential Equations Sanda L. N.; Okai J. O.; Nasir U.M.; U. A. Mujahid; Michael Cornelius; Ndam G.S.
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.
Co-Authors A. G. Madaki A. Madaki Adejoh Jeremiah Aminu Barde Araga Hassan I. B. Mshelia M. Y. Adamu Nasir U.M. Ndam G.S. Ogboche Abichele Okai J. O Okai J. O. Sanda L. N. U. A. Mujahid U. M. Nasir Yohanna Nehemiah