An adaptation of the Gear scheme for fractional derivatives

Ana Cristina Galucio (1) , Jean-François Deü (1) , Stéphanie Mengué (2) , François Dubois (3)

1 LMSSC - Laboratoire de Mécanique des Structures et des Systèmes Couplés
2 ESYCOM - Electronique, Systèmes de communication et Microsystèmes
3 UP11 UFR Sciences - Université Paris-Sud - Paris 11 - Faculté des Sciences

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.

Mots clés: Fractional derivatives; Multi-step scheme; Formal power series

Domaines: Sciences de l'ingénieur [physics]

Ana Cristina Galucio, Jean-François Deü, Stéphanie Mengué, François Dubois. An adaptation of the Gear scheme for fractional derivatives. Computer Methods in Applied Mechanics and Engineering, 2006, 195 (44-47), pp.6073-6085. ⟨10.1016/j.cma.2005.10.013⟩. ⟨hal-01516404⟩