Two unconditionally stable difference schemes for time distributed-order differential equation based on Caputo–Fabrizio fractional derivative

We consider distributed-order partial differential equations with time fractional derivative proposed by Caputo and Fabrizio in a one-dimensional space. Two finite difference schemes are established via Grünwald formula. We show that these two schemes are unconditionally stable with convergence rates O(τ2+h2+Δα2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$O(\tau ^{2}+h^{2}+ \Delta \alpha ^{2})$\end{document} and O(τ2+h4+Δα4)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$O(\tau ^{2}+h^{4}+\Delta \alpha ^{4})$\end{document} in discrete L2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$L^{2}$\end{document}, respectively, where Δα, h, and τ are step sizes for distributed-order, space, and time variables, respectively. Finally, the performance of difference schemes is illustrated via numerical examples.