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All Journal Journal of Multidisciplinary Science: MIKAILALSYS Mikailalsys Journal of Mathematics and Statistics
Mujahid, U. A.
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Agriculture, Biological Sciences & Forestry Chemical Engineering, Chemistry & Bioengineering Engineering Environmental Science Mathematics Mechanical Engineering Physics Social Sciences Other

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On the Numerical Solutions of Linear and Nonlinear Differential Equations by the Modified Laplace–Adomian Polynomials Method Okai, J. O.; Martha, Iliya; Adamu, M. Y.; Mujahid, U. A.; Sanda, L. N.
Mikailalsys Journal of Mathematics and Statistics Vol 4 No 1 (2026): Mikailalsys Journal of Mathematics and Statistics
Publisher : Darul Yasin Al Sys

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.58578/mjms.v4i1.7493

Abstract

This study employs the Laplace–Adomian Polynomial Method (LAPM) to obtain approximate solutions for both linear and nonlinear ordinary differential equations. LAPM integrates the Laplace transform with Adomian polynomials to manage nonlinear terms effectively, avoiding the need for linearization or perturbation techniques. To evaluate the method’s accuracy and computational efficiency, three representative examples were solved, with the results benchmarked against corresponding exact solutions. The numerical outcomes, presented through tables and graphical comparisons, demonstrate that LAPM provides highly accurate approximations with minimal error using only a few series terms. The findings affirm that the method is not only straightforward and computationally efficient but also broadly applicable to various nonlinear problems. Given its robustness and simplicity, LAPM holds promise for extension to more complex systems, including partial differential equations and multi-dimensional models in applied sciences.
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.
Co-Authors Cornelius, Michael G.S., Ndam J. O., Okai L. N., Sanda M. Y. Adamu Martha, Iliya Okai, J. O. Sanda, L. N. U.M., Nasir