In this work, we propose an exponentially convergent numerical method for the Caputo fractional propagator $S_\alpha(t)$ and the associated mild solution of the Cauchy problem with time-independent sectorial operator coefficient $A$ and Caputo fractional derivative of order $\alpha \in (0,2)$ in time. The proposed methods are constructed by generalizing the earlier developed approximation of $S_\alpha(t)$ with help of the subordination principle. Such technique permits us to eliminate the dependence of the main part of error estimate on $\alpha$, while preserving other computationally relevant properties of the original approximation: native support for multilevel parallelism, the ability to handle initial data with minimal spatial smoothness, and stable exponential convergence for all $t \in [0, T]$. Ultimately, the use of subordination leads to a significant improvement of the method's convergence behavior, particularly for small $\alpha < 0.5$, and opens up further opportunities for efficient data reuse. To validate theoretical results, we consider applications of the developed methods to the direct problem of solution approximation, as well as to the inverse problem of fractional order identification.

翻译：本文针对Caputo分数阶传播子$S_\alpha(t)$以及具有时不变扇形算子系数$A$、时间方向$\alpha \in (0,2)$阶Caputo分数阶导数的柯西问题所对应的温和解，提出了一种指数收敛的数值方法。所提出的方法通过推广先前借助从属原理发展的$S_\alpha(t)$近似方案而构建。该技术使我们能够消除误差估计主要部分对$\alpha$的依赖性，同时保留原始近似方案的其他计算相关特性：对多级并行性的原生支持、处理空间光滑性极弱的初始数据的能力，以及对所有$t \in [0, T]$的稳定指数收敛性。最终，从属原理的应用显著改善了方法的收敛行为，特别是在$\alpha < 0.5$的较小值时，并为高效的数据重用开辟了进一步的可能性。为验证理论结果，我们将所发展的方法应用于解近似的正问题以及分数阶辨识的反问题。