  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.

Numerical Analysis