Asymptotic expansions and approximations for the Caputo derivative

In this paper we use the asymptotic expansions of the binomial coefficients and the weights of the L1 approximation to obtain approximations of order $2-\alpha$ and second-order approximations of the Caputo derivative by modifying the weights of the shifted Gr"unwald-Letnikov difference approximation and the L1 approximation of the Caputo derivative. A modification of the shifted Gr"unwald-Letnikov approximation is obtained which allows second-order numerical solutions of fractional differential equations with arbitrary values of the solutions and their first derivatives at the initial point.