Error estimates of a high order numerical method for solving linear fractional differential equations

In this paper, we first introduce an alternative proof of the error estimates of the numerical methods for solving linear fractional differential equations proposed in Diethelm [6] where a first-degree compound quadrature formula was used to approximate the Hadamard finite-part integral and the convergence order of the proposed numerical method is O(Δt2−α),0<α<1, where α is the order of the fractional derivative and Δt is the step size. We then use a similar idea to prove the error estimates of the high order numerical method for solving linear fractional differential equations proposed in Yan et al. [37], where a second-degree compound quadrature formula was used to approximate the Hadamard finite-part integral and we show that the convergence order of the numerical method is O(Δt3−α),0<α<1. Numerical examples are given to show that the numerical results are consistent with the theoretical results.