This paper tackles the persistent challenge of slow convergence and numerical instability in the fractional calculus when applied to
power series–representable functions f(x)=∑_(i=0)^(∞ )▒〖 c_i x〗^i , 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. |
