A Study on Fractional Operators and Their Modelling via Generalized Integral Transform

This study presents a generalized integral transform (GIT) approach for analyzing and modeling a wide class of fractional-order mathematical systems arising in physical, engineering, and biological sciences. The investigated models include Newton’s law of cooling, the logistic population growth equation, and the blood alcohol concentration model, each formulated using distinct fractional derivatives such as the Caputo, Caputo–Fabrizio (CF), modified Atangana–Baleanu–Caputo (mABC), and constant proportional Caputo (CPC) derivatives. These fractional operators effectively describe memory-dependent and non-local characteristics inherent in many natural and engineered processes. Analytical solutions of the proposed models are derived through the generalized transform method, and graphical illustrations are provided to demonstrate the influence of various fractional orders on system dynamics. The results reveal that the GIT technique offers a unified, powerful, and efficient framework for solving a broad spectrum of fractional differential equations with diverse kernels. Furthermore, it integrates several classical and modern transforms—including the Laplace, Sumudu, Elzaki, and Formable transforms—as special cases, thereby simplifying computation and enhancing both generality and adaptability. This unified formulation provides researchers with a flexible analytical tool capable of addressing diverse problems without redefining operators for each model. The comparative analysis validates the stability, accuracy, and consistency of the proposed technique. Overall, this work highlights the efficacy, robustness, and broad applicability of the generalized integral transform, establishing a firm foundation for future explorations of hybrid fractional models and their interdisciplinary applications.