About this Journal Submit a Manuscript Table of Contents
Mathematical Problems in Engineering
Volume 2011 (2011), Article ID 657839, 9 pages
http://dx.doi.org/10.1155/2011/657839
Research Article

Exact Solution of Impulse Response to a Class of Fractional Oscillators and Its Stability

1School of Information Science and Technology, East China Normal University, no. 500, Dong-Chuan Road, Shanghai 200241, China
228 Farrer Road, #05-01, Sutton Place, Singapore 268831
3College of Computer Science, Zhejiang University of Technology, Hangzhou 310023, China

Received 18 August 2010; Accepted 15 September 2010

Academic Editor: Cristian Toma

Copyright © 2011 Ming Li et al. 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.

Linked References

  1. M. D. Ortigueira and A. G. Batista, “On the relation between the fractional Brownian motion and the fractional derivatives,” Physics Letters A, vol. 372, no. 7, pp. 958–968, 2008. View at Publisher · View at Google Scholar · View at Scopus
  2. M. D. Ortigueira, “An introduction to the fractional continuous-time linear systems: the 21st century systems,” IEEE Circuits and Systems Magazine, vol. 8, no. 3, pp. 19–26, 2008. View at Publisher · View at Google Scholar · View at Scopus
  3. Y. Luo and Y. Chen, “Fractional order [proportional derivative] controller for a class of fractional order systems,” Automatica, vol. 45, no. 10, pp. 2446–2450, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  4. Y. Q. Chen and K. L. Moore, “Discretization schemes for fractional-order differentiators and integrators,” IEEE Transactions on Circuits and Systems I, vol. 49, no. 3, pp. 363–367, 2002. View at Publisher · View at Google Scholar · View at Scopus
  5. J. A. Tenreiro Machado, M. F. Silva, R. S. Barbosa, et al., “Some applications of fractional calculus in engineering,” Mathematical Problems in Engineering, vol. 2010, Article ID 639801, 34 pages, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  6. O. P. Agrawal, “Solution for a fractional diffusion-wave equation defined in a bounded domain,” Nonlinear Dynamics, vol. 29, no. 1–4, pp. 145–155, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  7. I. Podlubny, I. Petráš, B. M. Vinagre, P. O'Leary, and L. Dorčák, “Analogue realizations of fractional-order controllers,” Nonlinear Dynamics, vol. 29, no. 1–4, pp. 281–296, 2002. View at Publisher · View at Google Scholar · View at Scopus
  8. C. H. Eab and S. C. Lim, “Path integral representation of fractional harmonic oscillator,” Physica A, vol. 371, no. 2, pp. 303–316, 2006. View at Publisher · View at Google Scholar · View at Scopus
  9. C. H. Eab and S. C. Lim, “Fractional generalized Langevin equation approach to single-file diffusion,” Physica A, vol. 389, no. 13, pp. 2510–2521, 2010. View at Publisher · View at Google Scholar
  10. C.-C. Tseng and S.-L. Lee, “Digital IIR integrator design using recursive Romberg integration rule and fractional sample delay,” Signal Processing, vol. 88, no. 9, pp. 2222–2233, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  11. V. Lakshmikantham, S. Leela, and J. Vasundhara Devi, Theory of Fractional Dynamic Systems, Cambridge Scientific Publishers, Cambridge, UK, 2009.
  12. C. M. Harris, Ed., Shock and Vibration Handbook, McGraw-Hill, New York, NY, USA, 4th edition, 1995.
  13. Y. Chen and K. L. Moore, “Analytical stability bound for a class of delayed fractional-order dynamic systems,” Nonlinear Dynamics, vol. 29, no. 1–4, pp. 191–200, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  14. H.-S. Ahn and Y. Chen, “Necessary and sufficient stability condition of fractional-order interval linear systems,” Automatica, vol. 44, no. 11, pp. 2985–2988, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  15. D. Qian, C. Li, R. P. Agarwal, and P. J. Y. Wong, “Stability analysis of fractional differential system with Riemann-Liouville derivative,” Mathematical and Computer Modelling, vol. 52, no. 5-6, pp. 862–874, 2010. View at Publisher · View at Google Scholar
  16. S. S. Antman, J. E. Marsden, and L. Sirovich, Eds., Applied Delay Differential Equations, Springer, New York, NY, USA, 2009.
  17. B. N. N. Achar, J. W. Hanneken, T. Enck, and T. Clarke, “Dynamics of the fractional oscillator,” Physica A, vol. 297, no. 3-4, pp. 361–367, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  18. B. N. N. Achar, J. W. Hanneken, and T. Clarke, “Damping characteristics of a fractional oscillator,” Physica A, vol. 339, no. 3-4, pp. 311–319, 2004. View at Publisher · View at Google Scholar · View at Scopus
  19. B. N. Narahari Achar, J. W. Hanneken, T. Enck, and T. Clarke, “Response characteristics of a fractional oscillator,” Physica A, vol. 309, no. 3-4, pp. 275–288, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  20. A. Al-rabtah, V. S. Ertürk, and S. Momani, “Solutions of a fractional oscillator by using differential transform method,” Computers and Mathematics with Applications, vol. 59, no. 3, pp. 1356–1362, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  21. S. C. Lim, M. Li, and L. P. Teo, “Locally self-similar fractional oscillator processes,” Fluctuation and Noise Letters, vol. 7, no. 2, pp. L169–L179, 2007. View at Publisher · View at Google Scholar · View at Scopus
  22. S. C. Lim, M. Li, and L. P. Teo, “Langevin equation with two fractional orders,” Physics Letters A, vol. 372, no. 42, pp. 6309–6320, 2008. View at Publisher · View at Google Scholar · View at Scopus
  23. S. C. Lim and S. V. Muniandy, “Self-similar Gaussian processes for modeling anomalous diffusion,” Physical Review E, vol. 66, no. 2, Article ID 021114, pp. 021114/1–021114/14, 2002. View at Publisher · View at Google Scholar · View at Scopus
  24. S. C. Lim and L. P. Teo, “The fractional oscillator process with two indices,” Journal of Physics A: Mathematical and Theoretical, vol. 42, no. 6, Article ID 065208, 34 pages, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  25. M. Li, “Fractal time series—a tutorial review,” Mathematical Problems in Engineering, vol. 2010, Article ID 157264, 26 pages, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  26. I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, Calif, USA, 1999.
  27. I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, Elsevier, Singapore, 7th edition, 2007, edited by A. Jeffrey and D. Zwillinger.
  28. R. C. Dorf and R. H. Bishop, Modern Control Systems, Prentice Hall, Upper Saddle River, NJ, USA, 9th edition, 2002.
  29. H. K. Khalil, Nonlinear Systems, Prentice Hall, Upper Saddle River, NJ, USA, 2nd edition, 2002.
  30. M. Li and W. Zhao, “Representation of a stochastic traffic bound,” IEEE Transactions on Parallel and Distributed Systems, vol. 21, no. 9, pp. 1368–1372, 2010. View at Publisher · View at Google Scholar
  31. M. Li and S. C. Lim, “Power spectrum of generalized Cauchy process,” Telecommunication Systems, vol. 43, no. 3-4, pp. 219–222, 2010. View at Publisher · View at Google Scholar
  32. M. Li, “Generation of teletraffic of generalized Cauchy type,” Physica Scripta, vol. 81, no. 2, Article ID 025007, 10 pages, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  33. M. Li and S. C. Lim, “Modeling network traffic using generalized Cauchy process,” Physica A, vol. 387, no. 11, pp. 2584–2594, 2008. View at Publisher · View at Google Scholar · View at Scopus
  34. S. C. Lim and M. Li, “A generalized Cauchy process and its application to relaxation phenomena,” Journal of Physics A: Mathematical and General, vol. 39, no. 12, pp. 2935–2951, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  35. M. Li, “Modeling autocorrelation functions of long-range dependent teletraffic series based on optimal approximation in Hilbert space-A further study,” Applied Mathematical Modelling, vol. 31, no. 3, pp. 625–631, 2007. View at Publisher · View at Google Scholar · View at Scopus
  36. S. Y. Chen, Y. F. Li, and J. Zhang, “Vision processing for realtime 3-D data acquisition based on coded structured light,” IEEE Transactions on Image Processing, vol. 17, no. 2, pp. 167–176, 2008. View at Publisher · View at Google Scholar · View at Scopus
  37. S. Y. Chen, Y. F. Li, Q. Guan, and G. Xiao, “Real-time three-dimensional surface measurement by color encoded light projection,” Applied Physics Letters, vol. 89, no. 11, Article ID 111108, 2006. View at Publisher · View at Google Scholar · View at Scopus
  38. E. G. Bakhoum and C. Toma, “Mathematical transform of traveling-wave equations and phase aspects of quantum interaction,” Mathematical Problems in Engineering, vol. 2010, Article ID 695208, 15 pages, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  39. C. Cattani, “Harmonic wavelet approximation of random, fractal and high frequency signals,” Telecommunication Systems, vol. 43, no. 3-4, pp. 207–217, 2010. View at Publisher · View at Google Scholar
  40. O. M. Abuzeid, A. N. Al-Rabadi, and H. S. Alkhaldi, “Fractal geometry-based hypergeometric time series solution to the hereditary thermal creep model for the contact of rough surfaces using the Kelvin-Voigt medium,” Mathematical Problems in Engineering, vol. 2010, Article ID 652306, 22 pages, 2010. View at Publisher · View at Google Scholar
  41. C. Cattani, “Fractals and hidden symmetries in DNA,” Mathematical Problems in Engineering, vol. 2010, Article ID 507056, 31 pages, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  42. J. Chen, C. Hu, and Z. Ji, “An improved ARED algorithm for congestion control of network transmission,” Mathematical Problems in Engineering, vol. 2010, Article ID 329035, 14 pages, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  43. E. G. Bakhoum and C. Toma, “Dynamical aspects of macroscopic and quantum transitions due to coherence function and time series events,” Mathematical Problems in Engineering, vol. 2010, Article ID 428903, 13 pages, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH