Sage Journals: Discover world-class research

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 article, the formal power of this scheme is used to approximate fractional derivative operators, in the context of finite difference methods. Numerical examples are presented and analyzed, in order to show the accuracy of the Gear scheme at the power α (G^α-scheme) when compared to the classical Grünwald-Letnikov approximation. In particular, the combined G^α -Newmark scheme is shown to be second-order accurate for a fractional damped oscillator problem.

Keywords

Fractional derivatives viscoelasticity multi-step scheme linear dynamics

Get full access to this article

View all access options for this article.

References

Bagley, R. and Torvik, P. , 1983, " Fractional calculus-a different approach to the analysis of viscoelastically damped structures ," AIAA Journal 21, 741-748.

Diethelm, K. , Ford, N. , Freed, A. , and Luchko, Y. , 2005, " Algorithms for the fractional calculus: A selection of numerical methods ," Computer Methods in Applied Mechanics and Engineering 194,743-773.

Galucio, A. , Deü, J.-F. , and Ohayon, R. , 2004, " Finite element formulation of viscoelastic sandwich beams using fractional derivative operators ," Computational Mechanics 33, 282-291.

Galucio, A. , Deü, J.-F. , Mengué, S. , and Dubois, F. , 2006, " An adaptation of the Gear scheme for fractional derivatives ," Computer Methods in Applied Mechanics and Engineering 195, 6073-6085.

Lubich, C. , 1986, " Discretized fractional calculus ," SIAM Journal on Mathematical Analysis 17(3), 704-719.

Matignon, D. and Montseny, G., eds., 1998, Fractional Differential Systems: Models, Methods and Applications , ESAIM Proceedings , Vol. 5, SMAI, Paris, France.

Padovan, J. , 1987, " Computational algorithms for FE formulations involving fractional operators ," Computational Mechanics 2, 271-287.

Trinks, C. and Ruge, P. , 2002, " Treatment of dynamic systems with fractional derivatives without evaluating memory-integrals ," Computational Mechanics 29, 471-476.

Yuan, L. and Agrawal, O. , 2002, " A numerical scheme for dynamic systems containing fractional derivatives ," Journal of Vibration and Acoustics 124, 321-324.

The G α -scheme for Approximation of Fractional Derivatives: Application to the Dynamics of Dissipative Systems

Abstract

Keywords

Get full access to this article

References