标题： 基于Gegenbauer的分数阶逼近方法对黎曼-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 Gegenbauer-Based Fractional Approximation Methods

摘要： 本文介绍了一种基于Gegenbauer的分数阶逼近（GBFA）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移位Gegenbauer积分矩阵（FSGIMs），该方法对光滑函数实现了超指数收敛，以最小的计算成本提供了接近机器精度的准确性。 可调节的移位Gegenbauer（SG）参数使得在不同问题中能够灵活优化，而严格的误差分析验证了当选择适当的参数时，逼近误差可以快速减少。 数值实验表明，GBFA方法在准确性上比MATLAB的integral、MATHEMATICA的NIntegrate和现有技术高出多达两个数量级，并且在不同分数阶0 <{\alpha }< 1的情况下表现出更高的效率。 其适应性和精度使GBFA方法成为分数阶微积分的一种变革性工具，非常适合用于建模具有记忆和非局部行为的复杂系统，因为在这些系统中，理解底层结构通常受益于识别内在对称性或模式。 显示英文摘要

摘要： 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.