Approximations for the Caputo Derivative (I)

In this paper we construct approximations for the Caputo derivative of order $1-\alpha,2-\alpha,2$ and $3-\alpha$. The approximations have weights $0.5\left((k+1)^{-\alpha}-(k-1)^{-\alpha}\right)/\Gamma(1-\alpha)$ and $k^{-1-\alpha}/\Gamma(-\alpha)$, and the higher accuracy is achieved by modifying the initial and last weights using the expansion formulas for the left and right endpoints. The approximations are applied for computing the numerical solution of ordinary fractional differential equations. The properties of the weights of the approximations of order $2-\alpha$ are similar to the properties of the $L1$ approximation. In all experiments presented in the paper the accuracy of the numerical solutions using the approximation of order $2-\alpha$ which has weights $k^{-1-\alpha}/\Gamma(-\alpha)$ is higher than the accuracy of the numerical solutions using the $L1$ approximation for the Caputo derivative.