The main aim of this paper is to present the derivation, stability analysis, and implementation of a 3-stage fractional Runge-Kutta method (3sFRKM). This method is useful as it belongs to the well-known Runge-Kutta family, which was developed to find a better approximate solution to fractional initial value problems (FIVPs.) in the Caputo sense. We have provided Some illustrative examples to demonstrate the efficiency and accuracy of the method. Our results compare them with the exact solution, using figures and tables, and all calculations were done using MATLAB.

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