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International Journal of Differential Equations
Volume 2010 (2010), Article ID 846107, 16 pages
On the Selection and Meaning of Variable Order Operators for Dynamic Modeling
School of Engineering, University of California, P.O. Box 2039, Merced, CA 95344, USA
Received 4 August 2009; Accepted 8 October 2009
Academic Editor: Nikolai Leonenko
Copyright © 2010 Lynnette E. S. Ramirez and Carlos F. M. Coimbra. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
- C. F. M. Coimbra, “Mechanics with variable-order differential operators,” Annalen der Physik, vol. 12, no. 11-12, pp. 692–703, 2003.
- D. Ingman and J. Suzdalnitsky, “Control of damping oscillations by fractional differential operator with time-dependent order,” Computer Methods in Applied Mechanics and Engineering, vol. 193, no. 52, pp. 5585–5595, 2004.
- C. M. Soon, C. F. M. Coimbra, and M. H. Kobayashi, “The variable viscoelasticity oscillator,” Annalen der Physik, vol. 14, no. 6, pp. 378–389, 2005.
- I. Podlubny, Fractional Differential Equations, vol. 198 of Mathematics in Science and Engineering, Academic Press, San Diego, Calif, USA, 1999.
- K. B. Oldham and J. Spanier, The Fractional Calculus: Theory and Applications of Differentiation and Integration to Arbitrary Order, vol. 11, Academic Press, London, UK, 1974.
- S. G. Samko, A. A. Kilbas, and O. I. Marichev, Integraly i proizvodnye drobnogo poryadka i nekotorye ikh prilozheniya, “Nauka i Tekhnika”, Minsk, Belarus, 1987.
- K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, John Wiley & Sons, New York, NY, USA, 1993.
- A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, vol. 204 of North-Holland Mathematics Studies, Elsevier, Amsterdam, The Netherlands, 2006.
- C. F. M. Coimbra and R. H. Rangel, “General solution of the particle momentum equation in unsteady Stokes flows,” Journal of Fluid Mechanics, vol. 370, pp. 53–72, 1998.
- C. F. M. Coimbra and R. H. Rangel, “Spherical particle motion in harmonic Stokes flows,” AIAA Journal, vol. 39, no. 9, pp. 1673–1682, 2001.
- C. F. M. Coimbra and M. H. Kobayashi, “On the viscous motion of a small particle in a rotating cylinder,” Journal of Fluid Mechanics, vol. 469, pp. 257–286, 2002.
- L. E. S. Ramirez, E. A. Lim, C. F. M. Coimbra, and M. H. Kobayashi, “On the dynamics of a spherical scaffold in rotating bioreactors,” Biotechnology and Bioengineering, vol. 84, no. 3, pp. 382–389, 2003.
- R. L. Bagley and P. J. Torvik, “A theoretical basis for the application of fractional calculus to viscoelasticity,” Journal of Rheology, vol. 27, no. 3, pp. 201–210, 1983.
- R. L. Bagley and P. J. Torvik, “On the fractional calculus model of viscoelastic behavior,” Journal of Rheology, vol. 30, no. 1, pp. 133–155, 1986.
- N. Heymans and J.-C. Bauwens, “Fractal rheological models and fractional differential equations for viscoelastic behavior,” Rheologica Acta, vol. 33, no. 3, pp. 210–219, 1994.
- T. Pritz, “Five-parameter fractional derivative model for polymeric damping materials,” Journal of Sound and Vibration, vol. 265, no. 5, pp. 935–952, 2003.
- R. Metzler and J. Klafter, “The random walk's guide to anomalous diffusion: a fractional dynamics approach,” Physics Reports, vol. 339, no. 1, pp. 1–77, 2000.
- I. Podlubny, “Fractional-order systems and -controllers,” IEEE Transactions on Automatic Control, vol. 44, no. 1, pp. 208–214, 1999.
- D. Ingman, J. Suzdalnitsky, and M. Zeifman, “Constitutive dynamic-order model for nonlinear contact phenomena,” Journal of Applied Mechanics, vol. 67, no. 2, pp. 383–390, 2000.
- D. Ingman and J. Suzdalnitsky, “Application of differential operator with servo-order function in model of viscoelastic deformation process,” Journal of Engineering Mechanics, vol. 131, no. 7, pp. 763–767, 2005.
- L. E. S. Ramirez and C. F. M. Coimbra, “A variable order constitutive relation for viscoelasticity,” Annalen der Physik, vol. 16, no. 7-8, pp. 543–552, 2007.
- D. Ingman and J. Suzdalnitsky, “Response of viscoelastic plate to impact,” Journal of Vibration and Acoustics, vol. 130, no. 1, Article ID 011010, 8 pages, 2008.
- S. G. Samko and B. Ross, “Integration and differentiation to a variable fractional order,” Integral Transforms and Special Functions, vol. 1, no. 4, pp. 277–300, 1993.
- C. F. Lorenzo and T. T. Hartley, “Variable order and distributed order fractional operators,” Nonlinear Dynamics, vol. 29, no. 1–4, pp. 57–98, 2002.
- C. F. Lorenzo and T. T. Hartley, “Initialization, conceptualization, and application in the generalized fractional calculus,” NASA Technical Publication 98-208415, Lewis Research Center, NASA, Cleveland, Ohio, USA, 1998.
- K. Diethelm, N. J. Ford, A. D. Freed, and Y. Luchko, “Algorithms for the fractional calculus: a selection of numerical methods,” Computer Methods in Applied Mechanics and Engineering, vol. 194, no. 6–8, pp. 743–773, 2005.
- Y. A. Rossikhin and M. V. Shitikova, “Applications of fractional calculus to dynamic problems of linear and nonlinear hereditary mechanics of solids,” Applied Mechanics Reviews, vol. 50, no. 1, pp. 15–67, 1997.
- Y. A. Rossikhin and M. V. Shitikova, “Application of fractional derivatives to the analysis of damped vibrations of viscoelastic single mass systems,” Acta Mechanica, vol. 120, no. 1–4, pp. 109–125, 1997.