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All Journal Journal of Multidisciplinary Science: MIKAILALSYS Asian Journal of Science, Technology, Engineering, and Art Mikailalsys Journal of Mathematics and Statistics
A. G. Madaki
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Agriculture, Biological Sciences & Forestry Arts Chemical Engineering, Chemistry & Bioengineering Computer Science & IT Engineering Environmental Science Mathematics Mechanical Engineering Physics Social Sciences Other

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 1 
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.
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.
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.
Co-Authors A. M. Kwami Abubakar Assidiq Hussaini Adejoh Jeremiah Araga Hassan I. B. Mshelia J. O. Okai M. I. Bello M. Y. Adamu Michael Cornelius Ogboche Abichele Okai J. O U. M. Nasir Umar Mujahid Aliyu Yohanna Nehemiah