A fourth-order compact difference method for the nonlinear time-fractional fourth-order reaction–diffusion equation

In this paper, a high-order compact scheme is proposed for solving two-dimensional nonlinear time-fractional fourth-order reaction-diffusion equations. The fractional derivative is the Caputo fractional derivative. A scheme with the second-order accuracy is applied to deal with the time derivative, while the spatial derivatives are discretized by the fourth-order compact numerical differential formulas. The unique solvability of the numerical method is proved in detail. Then by using the energy method, it is proved that the proposed algorithm is convergent with order O(τ2+h14+h24)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$O({\tau ^2} + h_1^4 + h_2^4)$$\end{document}, where τ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau$$\end{document} is the temporal step size and h1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$h_1$$\end{document}, h2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$h_2$$\end{document} are the spatial step sizes. Finally, some numerical examples are given to verify the theoretical analysis and efficiency of the developed scheme.