Super-Exponential Approximation of the Riemann-Liouville Fractional Integral via Gegenbauer-Based Fractional Approximation Methods

Abstract: This paper introduces a Gegenbauer-based fractional approximation (GBFA) method for high-precision approximation of the left Riemann-Liouville fractional integral (RLFI). By using precomputable fractional-order shifted Gegenbauer integration matrices (FSGIMs), the method achieves super-exponential convergence for smooth functions, delivering near machine-precision accuracy with minimal computational cost. Tunable shifted Gegenbauer (SG) parameters enable flexible optimization across diverse problems, while rigorous error analysis verifies a fast reduction in approximation error when appropriate parameter choices are applied. Numerical experiments demonstrate that the GBFA method outperforms MATLAB's integral, MATHEMATICA's NIntegrate, and existing techniques by up to two orders of magnitude in accuracy, with superior efficiency for varying fractional orders 0 < {\alpha} < 1. Its adaptability and precision make the GBFA method a transformative tool for fractional calculus, ideal for modeling complex systems with memory and non-local behavior, where understanding underlying structures often benefits from recognizing inherent symmetries or patterns.