Table of Contents Author Guidelines Submit a Manuscript
Journal of Applied Mathematics
Volume 2012 (2012), Article ID 417942, 14 pages
http://dx.doi.org/10.1155/2012/417942
Research Article

Numerical Solutions of a Variable-Order Fractional Financial System

1School of Business, Central South University, Hunan, Changsha 410083, China
2Department of Applied Mathematics, Central South University, Hunan, Changsha 410083, China

Received 10 May 2012; Revised 21 July 2012; Accepted 6 August 2012

Academic Editor: Changbum Chun

Copyright © 2012 Shichang Ma 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. S. Umarov and S. Steinberg, “Variable order differential equations and diffusion processes with changing modes,” submitted, http://www.arxiv.org/pdf/0903.2524.pdf.
  2. S. G. Samko, “Fractional integration and differentiation of variable order,” Analysis Mathematica, vol. 21, no. 3, pp. 213–236, 1995. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  3. 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  4. C. F. M. Coimbra, “Mechanics with variable-order differential operators,” Annalen der Physik, vol. 12, no. 11-12, pp. 692–703, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  5. H. G. Sun, W. Chen, and Y. Q. Chen, “Variable-order fractional differential operators in anomalous diffusion modeling,” Physica A, vol. 388, no. 21, pp. 4586–4592, 2009. View at Publisher · View at Google Scholar · View at Scopus
  6. D. Valério and J. S. Costa, “Variable-order fractional derivatives and their numerical approximations,” Signal Processing, vol. 91, no. 3, pp. 470–483, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  7. H. Sheng, H. G. Sun, Y. Q. Chen, and T. S. Qiu, “Synthesis of multifractional Gaussian noises based on variable-order fractional operators,” Signal Processing, vol. 91, no. 7, pp. 1645–1650, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  8. C. C. Tseng, “Design of variable and adaptive fractional order FIR differentiators,” Signal Processing, vol. 86, no. 10, pp. 2554–2566, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  9. H. Sheng, H. G. Sun, C. Coopmans, Y. Q. Chen, and G. W. Bohannan, “A physical experimental study of variable-order fractional integrator and differentiator,” The European Physical Journal, vol. 193, no. 1, pp. 93–104, 2011. View at Publisher · View at Google Scholar · View at Scopus
  10. K. B. Oldham and J. Spanier, The Fractional Calculus: Theory and Applications of Differentiation and Integration to Arbitrary Order, Academic Press, 1974.
  11. I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, Calif, USA, 1999.
  12. P. L. Butzer and U. Westphal, An Introduction to Fractional Calculus, World Scientific, Singapore, 2000.
  13. K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, Wiley-Interscience, New York, NY, USA, 1993.
  14. K. Diethelm, The Analysis of Fractional Differential Equations, Springer, Berlin, Germany, 2010.
  15. K. Diethelm, N. J. Ford, and A. D. Freed, “A predictor-corrector approach for the numerical solution of fractional differential equations,” Nonlinear Dynamics, vol. 29, no. 1–4, pp. 3–22, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  16. K. Yang and J. M. Wang, “A temperature prediction-correction method for estimating surface soil heat flux from soil temperature and moisture data,” Science in China D, vol. 51, no. 5, pp. 721–729, 2008. View at Publisher · View at Google Scholar · View at Scopus
  17. A. Bnouhachem and M. A. Noor, “Numerical comparison between prediction-correction methods for general variational inequalities,” Applied Mathematics and Computation, vol. 186, no. 1, pp. 496–505, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  18. X. L. Fu, “A two-stage prediction-correction method for solving monotone variational inequalities,” Journal of Computational and Applied Mathematics, vol. 214, no. 2, pp. 345–355, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  19. P. Midya, B. Roeckner, P. Rakers, and P. Wagh, “Prediction correction algorithm for natural pulse width modulation,” in Proceedings of the 109th AES Convention, September 2000. View at Publisher · View at Google Scholar
  20. X. M. Yuan, “The prediction-correction approach to nonlinear complementarity problems,” European Journal of Operational Research, vol. 176, no. 3, pp. 1357–1370, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  21. L. J. Wicker, “A two-step Adams-Bashforth-Moulton split-explicit integrator for compressible atmospheric models,” Monthly Weather Review, vol. 137, no. 10, pp. 3588–3595, 2009. View at Publisher · View at Google Scholar · View at Scopus
  22. K. H. Sun, X. Wang, and J. C. Sprott, “Bifurcations and chaos in fractional-order simplified Lorenz system,” International Journal of Bifurcation and Chaos, vol. 20, no. 4, pp. 1209–1219, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  23. K. Diethelm, N. J. Ford, and A. D. Freed, “Detailed error analysis for a fractional Adams method,” Numerical Algorithms, vol. 36, no. 1, pp. 31–52, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  24. E. Misirli and Y. Gurefe, “Multiplicative Adams Bashforth-Moulton methods,” Numerical Algorithms, vol. 57, no. 4, pp. 425–439, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  25. S. K. Agrawal, M. Srivastava, and S. Das, “Synchronization of fractional order chaotic systems using active control method,” Chaos, Solitons & Fractals, vol. 45, no. 6, pp. 737–752, 2012. View at Publisher · View at Google Scholar
  26. G. Psihoyios and T. E. Simos, “Trigonometrically fitted predictor-corrector methods for IVPs with oscillating solutions,” Journal of Computational and Applied Mathematics, vol. 158, no. 1, pp. 135–144, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  27. P. Zhuang, F. Liu, V. Anh, and I. Turner, “Numerical methods for the variable-order fractional advection-diffusion equation with a nonlinear source term,” SIAM Journal on Numerical Analysis, vol. 47, no. 3, pp. 1760–1781, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  28. Y. F. Xu and Z. M. He, “The short memory principle for solving Abel differential equation of fractional order,” Computer & Mathematics with Applications, vol. 62, no. 12, pp. 4796–4805, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  29. A. Razminia, A. F. Dizaji, and V. J. Majd, “Solution existence for non-autonomous variable-order fractional differential equations,” Mathematical and Computer Modelling, vol. 55, no. 3-4, pp. 1106–1117, 2012. View at Publisher · View at Google Scholar
  30. W. C. Chen, “Nonlinear dynamics and chaos in a fractional-order financial system,” Chaos, Solitons & Fractals, vol. 36, no. 5, pp. 1305–1314, 2008. View at Publisher · View at Google Scholar · View at Scopus
  31. X. S. Zhao, Z. B. Li, and S. Li, “Synchronization of a chaotic finance system,” Applied Mathematics and Computation, vol. 217, no. 13, pp. 6031–6039, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus