We present a spectral method for one-sided linear fractional integral equations on a closed interval that achieves exponentially fast convergence for a variety of equations, including ones with irrational order, multiple fractional orders, non-trivial variable coefficients, and initial-boundary conditions. The method uses an orthogonal basis that we refer to as Jacobi fractional polynomials, which are obtained from an appropriate change of variable in weighted classical Jacobi polynomials. New algorithms for building the matrices used to represent fractional integration operators are presented and compared. Even though these algorithms are unstable and require the use of high-precision computations, the spectral method nonetheless yields well-conditioned linear systems and is therefore stable and efficient. For time-fractional heat and wave equations, we show that our method (which is not sparse but uses an orthogonal basis) outperforms a sparse spectral method (which uses a basis that is not orthogonal) due to its superior stability.

翻译：本文提出一种闭区间上一侧线性分数阶积分方程的谱方法，该方法对包括无理阶数、多分数阶、非平凡变系数及初边值条件在内的多种方程均能实现指数级快速收敛。该方法采用一种我们称为雅可比分数阶多项式的正交基，该基函数通过加权经典雅可比多项式的适当变量变换得到。本文提出并比较了构造分数阶积分算子矩阵的新算法。尽管这些算法不稳定且需要采用高精度计算，但谱方法仍能生成良态线性系统，因此具有稳定性和高效性。对于时间分数阶热传导方程与波动方程，我们证明该方法（虽非稀疏但使用正交基）因其优越的稳定性而优于稀疏谱方法（使用非正交基）。