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:我们为封闭间隔的单线分解分解方程式提出了一个光谱方法,在各种方程式,包括非理性顺序、多分顺序、非三边变量系数和初始界限条件下的方程式,实现指数快速趋同。该方法使用我们称为雅各比分数多式的正方位基,这是从加权古典雅各比多式变量的适当变化中获得的。提出并比较了用于构建用于代表分数集操作员的矩阵的新算法。尽管这些算法不稳定,需要使用高精度计算法,但光谱法仍然产生完善的线性系统,因此是稳定而有效的。对于时间折射热和波方程式,我们表明我们的方法(不稀疏,但使用一个正方位基)比稀薄的光谱法(使用一个非正方位的基)要高,因为它的稳定性。