Comparative Analysis of Fractional Euler-Type and Runge–Kutta-Type Methods for Newton’s Law of Cooling and Page Models

In this study, the fractional forward Euler method (FFEM), fractional backward Euler method (FBEM), explicit fractional Runge–Kutta (EFORK), and implicit fractional Runge–Kutta (IFORK) are employed to numerically solve the Caputo fractional Newton’s law of cooling and Page models. A new analytical solution of the fractional Page model is derived in terms of the Kilbas–Saigo generalized Mittag–Leffler function. The existence and uniqueness of solutions for both fractional models are established using Banach’s fixed-point theorem. Numerical simulations are performed for various fractional orders, step sizes, and model parameters to investigate the accuracy and convergence behavior of the considered methods. Their performance is evaluated through absolute error analysis, experimental convergence orders, and graphical comparisons with the corresponding analytical solutions. The results show that all four methods converge toward the corresponding analytical solutions as the step size decreases. Overall, this study presents a systematic comparison of Euler-type and Runge–Kutta-type fractional numerical methods for the numerical solution of Caputo fractional kinetic models.