Stability and convergence of a class of continuous piecewise polynomial approximations for time-fractional differential equations
Fractional calculus, as a generalization of ordinary calculus, has been an interesting topic since the 17th century. Recently, there are some attractive applications in the fields of physics, chemistry and engineering by applying fractional integrals and derivatives to construct mathematical models that describes anomalous diffusion process. In this report, we propose and study a class of numerical schemes to approximate time-fractional differential equations. The methods are based on the approximations to the Caputo fractional derivative of order from 0 to 1, which are strongly related to the backward differentiation formulae. We investigate their theoretical properties, such as the truncation errors with respect to sufficiently smooth solutions, and the numerical stability. Numerical experiments are given to verify our theoretical investigations.