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, 2026)
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.