An adaptation of the Gear scheme for fractional derivatives

• Published in: Computer methods in applied mechanics and engineering Vol. 195; no. 44; pp. 6073 - 6085
• Main Authors: Galucio, A.C., Deü, J.-F., Mengué, S., Dubois, F.
• Format: Journal Article
• Language: English
• Published: Amsterdam Elsevier B.V 01-01-2006; Elsevier
• Subjects: Computational techniques; Engineering Sciences; Exact sciences and technology; Formal power series; Fractional derivatives; Fundamental areas of phenomenology (including applications); Mathematical methods in physics; Multi-step scheme; Physics; Solid mechanics; Structural and continuum mechanics; Vibration, mechanical wave, dynamic stability (aeroelasticity, vibration control...); Multi-step scheme; Fractional derivatives; Formal power series; Vibration damping; Series expansion; Trapezoidal method; Function approximation; Power series; Gear; Fractional derivative; Finite difference method
• ISSN: 0045-7825; 1879-2138
• DOI: 10.1016/j.cma.2005.10.013

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
α (G
α
-scheme) when compared to the classical Grünwald–Letnikov approximation. In particular, for a fractional damped oscillator problem, the combined G
α
-Newmark scheme is shown to be second-order accurate.