标题： 高阶离散化误差在Hölder空间中的Caputo导数 显示英文标题

标题： High-order discretization errors for the Caputo derivative in Hölder spaces

摘要： 基于Teso和Plociniczak（2025）关于Hölder空间中Caputo导数的L1离散化误差的最新工作，本研究将分析扩展到同一函数框架内的高阶离散化误差。 我们首先研究了L2和L1-2方法的截断误差，这些方法通过分段二次插值近似Caputo导数。 然后我们将结果推广到任意高阶离散化。 理论分析揭示了所有方案中的统一误差结构：收敛阶等于函数空间的光滑度与分数阶导数阶数之间的差，即误差阶 = 光滑度阶 - 导数阶。 数值实验验证了这些理论结果。 显示英文摘要

摘要： Building upon the recent work of Teso and Plociniczak (2025) regarding L1 discretization errors for the Caputo derivative in H\"{o}lder spaces, this study extends the analysis to higher-order discretization errors within the same functional framework. We first investigate truncation errors for the L2 and L1-2 methods, which approximate the Caputo derivative via piecewise quadratic interpolation. Then we generalize the results to arbitrary high-order discretization. Theoretical analyses reveal a unified error structure across all schemes: the convergence order equals the difference between the smoothness degree of the function space and the fractional derivative order, i.e., order of error = degree of smoothness - order of the derivative. Numerical experiments validate these theoretical findings.