标题： 基于移位Gegenbauer伪谱方法的黎曼-刘维尔分数阶积分的超指数逼近 显示英文标题

标题： Super-Exponential Approximation of the Riemann-Liouville Fractional Integral via Shifted Gegenbauer Pseudospectral Methods

摘要： 本文介绍了一种位移Gegenbauer伪谱（SGPS）方法，用于高精度逼近左Riemann-Liouville分数阶积分（RLFI）。通过使用可预先计算的分数阶位移Gegenbauer积分矩阵（FSGIMs），该方法对光滑函数实现了超指数收敛，在计算成本最小的情况下提供了接近机器精度的准确性。可调节的位移Gegenbauer（SG）参数使得在不同问题中可以灵活优化，而严格的误差分析证实了在最优设置下误差迅速衰减。数值实验表明，SGPS方法在精度上比MATLAB的integral、MATHEMATICA的NIntegrate和现有技术高出多达两个数量级，对于不同的分数阶0 <\alpha < 1，其效率也更优。其适应性和精确性使SGPS方法成为分数阶微积分的一种变革性工具，非常适合建模具有记忆性和非局部行为的复杂系统。 显示英文摘要

摘要： This paper introduces a shifted Gegenbauer pseudospectral (SGPS) 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 confirms rapid error decay under optimal settings. Numerical experiments demonstrate that the SGPS 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 SGPS method a transformative tool for fractional calculus, ideal for modeling complex systems with memory and non-local behavior.