Stability and convergence of a local discontinuous Galerkin method for the fractional diffusion equation with distributed order

In this paper, a numerical method is proposed for solving distributed order diffusion equation, which arises in the mathematical modeling of ultra-slow diffusion processes observed in some physical problems, whose solution decays logarithmically as the time t tends to infinity. Based on local discontinuous Galerkin method in space, we develop a fully discrete scheme and prove that the scheme is unconditionally stable and convergent with the order O(hk+1+Δt+Δα2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$O(h^{k+1}+\Delta t+\Delta \alpha ^2)$$\end{document}, where h,Δt\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$h, \Delta t$$\end{document},Δα\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta \alpha $$\end{document} and k are the step size in space, time, distributed order and the degree of piecewise polynomials, respectively. Extensive numerical examples are carried out to illustrate the effectiveness of the numerical schemes.