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All Journal YASIN: Jurnal Pendidikan dan Sosial Budaya Asian Journal of Science, Technology, Engineering, and Art Mikailalsys Journal of Mathematics and Statistics
Okai, J. O.
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Arts Computer Science & IT Engineering Mathematics Mechanical Engineering 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 One-Step Modified New Iterative Method for Solving Partial Differential Equation Abdulmalik, Ibrahim; Kwami, A. M.; Okai, J. O.; Barde, A.; Abichele, Ogboche; Jeremiah, Adejoh
YASIN Vol 5 No 3 (2025): JUNI
Publisher : Lembaga Yasin AlSys

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.58578/yasin.v5i3.5498

Abstract

This study introduces a reliable semi-analytical approach for solving partial differential equations (PDEs) using a Modified New Iterative Method (MNIM). The primary aim is to enhance the efficiency of deriving closed-form solutions through an innovative formulation of an integral operator based on n-fold integration. This approach circumvents the conventional necessity of transforming PDEs into systems of multiple integral equations, thereby streamlining the solution process. The effectiveness of the MNIM is assessed through a series of examples, demonstrating its rapid convergence and superior performance in solving an array of evolution and partial differential equations. The results indicate that the MNIM not only simplifies the solution process but also significantly improves computational efficiency compared to traditional methods. This contribution holds substantial implications for both theoretical advancements in numerical analysis and practical applications across various fields where PDEs are prevalent, thereby facilitating more effective problem-solving strategies in complex systems.
Integrated Mahgoub–VIM Hybrid Transform Technique for Solving Linear, Nonlinear, and Fractional Differential Equations Aliyu, Umar Mujahid; Kwami, A. M.; Bello, M. I.; Madaki, A. G.; Okai, J. O.; Hussaini, Abubakar Assidiq
Asian Journal of Science, Technology, Engineering, and Art Vol 4 No 3 (2026): Asian Journal of Science, Technology, Engineering, and Art
Publisher : Darul Yasin Al Sys

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.58578/ajstea.v4i3.9234

Abstract

This study develops an integrated Mahgoub–Variational Iteration Method (VIM) hybrid transform technique for solving linear, nonlinear, and fractional-order ordinary and partial differential equations. The study addresses the limitations of classical integral transforms in handling nonlinearities, fractional derivatives, and memory-dependent effects, while ensuring physically consistent initial conditions through the Caputo fractional derivative. The proposed Mahgoub–VIM framework was applied to higher-order nonlinear ordinary differential equations, fractional ordinary differential equations, time-fractional partial differential equations, and fractional relaxation models. The results demonstrate rapid convergence, high stability, and close agreement with exact solutions. Comparative analysis further indicates that the proposed method consistently outperforms the Sumudu transform in terms of accuracy and error control, particularly for nonlinear and fractional problems. By avoiding linearization and discretization, the technique provides an efficient analytical framework for modeling realistic phenomena, including diffusion, heat transfer, viscoelasticity, and damping. The study contributes to the development of hybrid transform-based methods by offering a robust, accurate, and versatile analytical tool for solving complex differential systems.
Co-Authors Abdulmalik, Ibrahim Abichele, Ogboche Aliyu, Umar Mujahid Barde, A. Bello, M. I. Hussaini, Abubakar Assidiq Jeremiah, Adejoh Kwami, A. M. M. Y. Adamu Madaki, A. G. Martha, Iliya Mujahid, U. A. Sanda, L. N.