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A numerical scheme for initial compliance and creep response of a system. (English) Zbl 1258.74090
Summary: This paper is an extension of a previous paper [L. Yuan and O. P. Agrawal, J. Vib. Acoust. 124, 321–324 (2002)] in which we demonstrated that a fractional differential equation (FDE) governing the dynamics of a system could be transformed into a set of differential equations with no fractional derivative terms. We developed a memory free formulation (MFF) for FDEs and eliminated the need of storing long memory. In this paper, we reexamine the MFF for FDEs and address some of the concerns raised by various authors about the formulation. The new formulation allows accurate computation of initial compliance and creep response of a system for a long duration. The performance of the scheme is demonstrated using an example. Possible future directions of research are discussed. It is hoped that the benefits of the MFF for FDEs will initiate further research in this direction.

74H15 Numerical approximation of solutions of dynamical problems in solid mechanics
74K99 Thin bodies, structures
26A33 Fractional derivatives and integrals
Full Text: DOI
[1] Agrawal, O. P.: Fractional variational calculus in terms of Riesz fractional derivatives, Journal of physics A: mathematical and theoretical 40, 1-17 (2007) · Zbl 1125.26007 · doi:10.1088/1751-8113/40/24/003
[2] Atkinson, K. E.: An introduction to numerical analysis, (1978) · Zbl 0402.65001
[3] Djordjevic, V. D.; Atanackovic, T. M.: Similarity solutions to nonlinear heat conduction and Burgers/Korteweg – de Vries fractional equations, Journal of computational and applied mathematics 222, 701-714 (2008) · Zbl 1157.35470 · doi:10.1016/j.cam.2007.12.013
[4] Gorenflo, R.; Loutchko, J.; Luchko, Y.: Computation of the Mittag – Leffler function and its derivatives, Fractional calculus and applied analysis 5, 491-518 (2002) · Zbl 1027.33016
[5] Hanyga, A., Lu, J.F., 2005. Wave field simulation for heterogeneous transversely isotropic porous media with the JKD dynamic permeability. Computational Mechanics 36, 196 – 208. · Zbl 1138.74326 · doi:10.1007/s00466-004-0652-3
[6] Lu, J. F.; Hanyga, A.: Numerical modelling method for wave propagation in a linear viscoelastic medium with singular memory, Geophysical journal international 159, 688-702 (2004)
[7] Schmidt, A.; Gaul, L.: On a critique of a numerical scheme for the calculation of fractionally damped dynamical systems, Mechanics research communications 33, 99-107 (2006) · Zbl 1192.74153 · doi:10.1016/j.mechrescom.2005.02.018
[8] Singh, S. J.; Chatterjee, A.: Galerkin projections and finite elements for fractional order derivatives, Nonlinear dynamics 45, 183-206 (2006) · Zbl 1101.65119 · doi:10.1007/s11071-005-9002-z
[9] Singh, S. J.; Chatterjee, A.: Fractional damping: stochastic origin and finite approximations, Advances in fractional calculus, theoretical developments and applications in physics and engineering, 389-402 (2007) · Zbl 1350.74010
[10] Trinks, C.; Ruge, P.: Treatment of dynamic systems with fractional derivatives without evaluating memory-integrals, Computational mechanics 29, 471-476 (2002) · Zbl 1146.76634 · doi:10.1007/s00466-002-0356-5
[11] Yuan, L., Agrawal, O.P., 1998. A numerical scheme for dynamic systems containing fractional derivatives. In: Proceedings of the DETC’98, 1998 ASME Design Engineering Technical Conferences, September 13 – 16, 1998, Atlanta, Georgia.
[12] Yuan, L.; Agrawal, O. P.: A numerical scheme for dynamic systems containing fractional derivatives, Journal of vibration and acoustics 124, 321-324 (2002)
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