Robust Integral Transform Methods for the Solution of Nonlinear Fractional Ordinary Differential Equations in Viscoelastic and Biological Systems - Umar Mujahid Aliyu; David Opeoluwa Oyewola; Joel John Taura; Salisu Lukunti; Hassan Muhammad; Abubakar Yahya Adamu; Abdulhalim Isah Ibrahim; Mubarak Muhammad; Imafidor Hassan Ibrahim; Mohammed Abubakar Kolo; Isah Adamu; Wallen Juliet Piapna'an; Mustapha Mohammed Mansur; Ibrahim Abubakar Adamu; Mohammed Yusuf Marafa; Abdulwasiu Umar; Sulaiman Ahmad; Nura Hashim (Darul Yasin Al Sys, 2026)
Nonlinear and fractional-order differential equations frequently arise in viscoelastic and biological systems; however, their solution remains challenging due to the presence of nonlocal operators, memory effects, and complex boundary conditions. Classical integral transforms, including the Laplace and Fourier transforms, often have limitations in addressing these features effectively. This study presents a robust hybrid methodology that combines the Mahgoub Transform with the Variational Iteration Method (VIM) to solve nonlinear and fractional-order ordinary differential equations (ODEs). The proposed approach was systematically applied to linear, nonlinear, and fractional-order ODEs to evaluate its convergence, accuracy, and capacity to handle memory-dependent effects. The findings demonstrate that the Mahgoub–VIM method achieves rapid convergence, high accuracy, and improved performance compared with traditional transforms such as the Sumudu Transform. These results indicate that the proposed method provides a reliable and efficient analytical framework for modeling complex viscoelastic and biological phenomena governed by nonlinear and fractional-order dynamics. This study contributes to the advancement of integral transform-based solution methods and offers practical implications for the mathematical modeling of systems characterized by memory-dependent behavior and nonlinear responses.