Two Alternating Direction Implicit Difference Schemes for Solving the Two-Dimensional Time Distributed-Order Wave Equations

Two alternating direction implicit difference schemes are established for solving a class of two-dimensional time distributed-order wave equations. The schemes are proved to be unconditionally stable and convergent in the maximum norm with the convergence orders O(τ2+h12+h22+Δγ2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$O(\tau ^2+h_1^2+h_2^2+\Delta \gamma ^2)$$\end{document} and O(τ2+h14+h24+Δγ4),\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$O(\tau ^2+h_1^4+h_2^4+\Delta \gamma ^4),$$\end{document} respectively, where τ,hi(i=1,2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau , h_i\; (i=1,2)$$\end{document} and Δγ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta \gamma $$\end{document} are the step sizes in time, space and distributed order. Also, several numerical experiments are carried out to validate the theoretical results.