Two difference schemes for solving the one-dimensional time distributed-order fractional wave equations

Two difference schemes are derived for numerically solving the one-dimensional time distributed-order fractional wave equations. It is proved that the schemes are unconditionally stable and convergent in the L∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$L^{\infty }$\end{document} norm with the convergence orders O(τ2 + h2+Δγ2) and O(τ2 + h4+Δγ4), respectively, where τ,h, and Δγ are the step sizes in time, space, and distributed order. A numerical example is implemented to confirm the theoretical results.