Stability and Convergence Analysis of a Class of Continuous Piecewise Polynomial Approximations for Time-Fractional Differential Equations

We propose and study a class of numerical schemes to approximate time-fractional differential equations. The methods are based on the approximations of the Caputo fractional derivative of order α∈(0,1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha \in (0, 1)$$\end{document} by using continuous piecewise polynomials, which are strongly related to the backward differentiation formulae. We investigate their theoretical properties, such as the local truncation error and global error estimates with respect to sufficiently smooth solutions, and the numerical stability in terms of stability region and A(π2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A(\frac{\pi }{2})$$\end{document}-stability. Numerical experiments are given to verify our theoretical investigations.