An adaptation of the Gear scheme for fractional derivatives

Abstract : The Gear scheme is a three-level step algorithm, backward in time and second-order accurate for the approximation of classical time derivatives. In this contribution, the formal power of this scheme is proposed to approximate fractional derivative operators in the context of finite difference methods. Some numerical examples are presented and analysed in order to show the effectiveness of the present Gear scheme at the power a (G a-scheme) when compared to the classical Gru¨nwald–Letnikov approximation. In particular, for a fractional damped oscillator problem, the combined G a-Newmark scheme is shown to be second-order accurate.