Asymptotic expansions and approximations for the Caputo derivative

Abstract: 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.