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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
J. O. Okai
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Arts Computer Science & IT Engineering Mathematics Mechanical Engineering Social Sciences Other

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An Overview of Integral Transformation Methods for Solving Physical Problems Umar Mujahid Aliyu; A. G. Madaki; A. M. Kwami; M. I. Bello; J. O. Okai; Abubakar Assidiq Hussaini
Mikailalsys Journal of Mathematics and Statistics Vol 4 No 3 (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.v4i3.9378

Abstract

Integral transformation methods are widely used to solve physical problems formulated through differential equations; however, their effectiveness may vary across linear, nonlinear, and fractional systems. This study provides an analytical overview of major integral transforms, with particular emphasis on the Kamal and Laplace transforms and their integration with decomposition-based techniques, including the Adomian decomposition method. Through a critical discussion of relevant analytical procedures and illustrative examples, the study examines the applicability, efficiency, and accuracy of these methods in solving linear, nonlinear, and fractional differential equations arising in physical systems. The analysis indicates that integral transforms offer efficient and accurate solution procedures, particularly for linear differential equations. Nevertheless, their direct application to nonlinear and fractional problems presents computational and analytical challenges, thereby requiring hybrid approaches that combine transformation techniques with decomposition methods. The study concludes that hybrid integral-transform methods can extend the applicability of conventional analytical techniques to more complex differential equations. It contributes to the literature by synthesizing the strengths and limitations of integral-transform approaches and identifying opportunities to improve their computational efficiency and applicability in modeling physical systems.
Integrated Mahgoub–VIM Hybrid Transform Technique for Solving Linear, Nonlinear, and Fractional Differential Equations Umar Mujahid Aliyu; A. M. Kwami; M. I. Bello; A. G. Madaki; J. O. Okai; Abubakar Assidiq Hussaini
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.
A One-Step Modified New Iterative Method for Solving Partial Differential Equation Ibrahim Abdulmalik; A. M. Kwami; J. O. Okai; A. Barde; Ogboche Abichele; Adejoh Jeremiah
YASIN Vol 5 No 3 (2025): YASIN: Jurnal Pendidikan dan Sosial Budaya
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.
Co-Authors A. Barde A. G. Madaki A. M. Kwami Abubakar Assidiq Hussaini Adejoh Jeremiah Ibrahim Abdulmalik M. I. Bello Ogboche Abichele Umar Mujahid Aliyu