Abstract This paper investigates the time-local discretization, using Gaussian quadrature, of a class of diffusive operators that includes fractional operators, for application in fractional differential equations and related eigenvalue problems. A discretization based on the Gauss–Legendre quadrature rule is analyzed both theoretically and numerically. Numerical comparisons with both optimization-based and quadrature-based methods highlight its applicability. In addition, it is shown, on the example of a fractional delay differential equation, that quadrature-based discretization methods are spectrally correct, i.e. that they yield an unpolluted and convergent approximation of the essential spectrum linked to the fractional derivative, by contrast with optimization-based methods that can yield polluted spectra whose convergence is difficult to assess.

中文翻译：

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使用高斯正交的分数和相关扩散算子的时间局部离散化与应用

摘要 本文研究了使用高斯求积的一类扩散算子的时间局部离散化，其中包括分数运算符，用于分数微分方程和相关的特征值问题。从理论上和数值上分析了基于 Gauss-Legendre 求积法则的离散化。与基于优化和基于正交的方法的数值比较突出了它的适用性。此外，在分数延迟微分方程的例子中表明，基于正交的离散化方法在谱上是正确的，即它们产生与分数导数相关的基本谱的未污染和收敛的近似值，