A Comparative Study of Adomian–Kamal Decomposition and Euler-Based Methods for Solving the Fractional Abel Differential Equation

This study presents the development and application of two semi-analytical methods—namely the Adomian Laplace Theorem and the Adomian–Kamal Theorem—for solving the Fractional Abel Differential Equation (FADE). Both approaches integrate the Adomian Decomposition Method (ADM) with distinct integral transforms to enhance accuracy and computational efficiency. The Adomian–Laplace method combines ADM with the Laplace Transform (LT), while the Adomian–Kamal method incorporates the Kamal Integral Transform (KIT), enabling improved handling of the non-local and long-memory characteristics inherent in fractional-order systems. Additionally, a fractional extension of the classical Euler method is implemented for comparative purposes. The methods are evaluated through two case studies, where approximate solutions are compared to exact solutions for various fractional orders α. Graphical analyses demonstrate that both semi-analytical methods yield results that perfectly overlap with exact solutions, indicating high accuracy and convergence. In contrast, the fractional Euler method exhibits reduced accuracy at lower fractional orders due to its limited capability of capturing memory effects. The findings highlight the superior performance and reliability of the Adomian–Kamal and Adomian–Laplace approaches for solving nonlinear FADEs, offering a robust framework for analytical and semi-analytical modeling in physics, engineering, and applied sciences.