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 (Darul Yasin Al Sys, 2026)
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.