We propose a fast and stable method for constructing matrix approximations to fractional integral operators applied to series in the Chebyshev fractional polynomials. This method utilizes a recurrence relation satisfied by the fractional integrals of mapped Chebyshev polynomials and significantly outperforms existing methods. Through numerical examples, we highlight the broad applicability of these matrix approximations, including the solution of boundary value problems for fractional integral and differential equations. Additional applications include fractional differential equation initial value problems and fractional eigenvalue problems.

翻译：本文提出了一种快速且稳定的方法，用于构建应用于切比雪夫分数多项式级数的分数积分算子的矩阵逼近。该方法利用了映射切比雪夫多项式的分数积分所满足的递推关系，显著优于现有方法。通过数值算例，我们展示了这些矩阵逼近的广泛适用性，包括求解分数积分与微分方程的边值问题。其他应用还包括分数微分方程的初值问题及分数特征值问题。