In this paper, we deal with both the discretization and the efficient solution of initial and initial-boundary value problems with a time derivative of distributed order. A new discretization based on an adaptive Gauss quadrature and product integral formulas is introduced and analyzed. The efficient solution of the resulting sequence of linear systems by Krylov iterative methods and approximate inverse preconditioning is discussed, together with the spectral analysis of the relative matrix sequences. Several numerical examples showing the effectiveness of the approach are included.

中文翻译：

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新的自适应Caputo–Dzherbashyan导数的新的自适应多项离散化的时间分数阶微分方程的有效解

在本文中，我们处理离散问题和具有分布阶数时间导数的初值和初值问题的有效解决方案。介绍并分析了一种基于自适应高斯求积和积积分公式的离散化方法。讨论了通过Krylov迭代方法和近似逆预处理对线性系统所得序列的有效解决方案，以及相关矩阵序列的频谱分析。包括几个数值示例，说明了该方法的有效性。