Optimizing Fractional Differential Equation Solutions with Novel Müntz Space Basis Functions

This paper introduces innovative basis functions derived from Müntz spaces, aimed at addressing the computational challenges of Fractional Differential Equations (FDEs). Our primary focus is the creation of these functions using singular indices linked to the solutions of FDEs. We thoroughly investigate the properties of these fundamental functions to understand their operational potential. These functions are particularly adept at capturing initial singular indices, making them highly suitable for solving FDEs. The proposed numerical method is distinguished by its rapid convergence rates, showcasing its efficiency in computational evaluations. We validate our approach by presenting numerical examples that highlight its accuracy and reliability. These examples confirm the effectiveness and efficiency of the new basis functions from Müntz spaces in accurately solving FDEs. This research advances numerical methods for FDEs and serves as a valuable resource for researchers seeking robust and reliable techniques