Correction of a High-Order Numerical Method for Approximating Time-Fractional Wave Equation

Abstract

A high-order time discretization scheme to approximate the time-fractional wave equation with the Caputo fractional derivative of order \(\alpha \in (1, 2)\) is studied. We establish a high-order formula for approximating the Caputo fractional derivative of order \(\alpha \in (1, 2)\). Based on this approximation, we propose a novel numerical method to solve the time-fractional wave equation. Remarkably, this method corrects only one starting step and demonstrates second-order convergence in both homogeneous and inhomogeneous cases, regardless of whether the data is smooth or nonsmooth. We also analyze the stability region associated with the proposed numerical method. Some numerical examples are given to elucidate the convergence analysis.