High-order discretization errors for the Caputo derivative in H\"{o}lder spaces
Abstract
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.
Cite
@article{arxiv.2504.07391,
title = {High-order discretization errors for the Caputo derivative in H\"{o}lder spaces},
author = {Xiangyi Peng and Lisen Ding and Dongling Wang},
journal= {arXiv preprint arXiv:2504.07391},
year = {2025}
}