Approximations for Fractional Derivatives and Fractional Integrals Using Padé Approximation

This paper tackles the persistent challenge of slow convergence and numerical instability in the fractional calculus when applied to power series–representable functions (Formula presented.), limitations that compromise accuracy in scientific applications. A novel reformulation of fractional derivatives and integrals is achieved by applying Padé approximation to conventional power series solutions, replacing them with optimized rational functions. The modified operators demonstrate significantly improved accuracy and enhanced convergence properties compared to established methods. This approach enables more reliable fractional modeling in physics and engineering domains where traditional operators fail. The work constitutes the first systematic integration (differentiation) of Padé approximation into the foundational definition of fractional operators, overcoming convergence barriers inherent in prior series-based techniques.