Skip to main content
SpringerLink
Log in

Numerical approach to the Caputo derivative of the unknown function

  • Research Article
  • Published:
Central European Journal of Physics

Abstract

If a function can be explicitly expressed, then one can easily compute its Caputo derivative by the known methods. If a function cannot be explicitly expressed but it satisfies a differential equation, how to seek Caputo derivative of such a function has not yet been investigated. In this paper, we propose a numerical algorithm for computing the Caputo derivative of a function defined by a classical (integer-order) differential equation. By the properties of Caputo derivative derived in this paper, we can change the original typical differential system into an equivalent Caputo-type differential system. Numerical examples are given to support the derived numerical method.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. K. Miller, B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations (Wiley, New York, 1993)

    MATH  Google Scholar 

  2. K. B. Oldham, J. Spanier, The Fractional Calculus (Academic Press, New York, 1974)

    MATH  Google Scholar 

  3. S. G. Samko, A. A. Kilbas, O. I. and Marichev, Fractional Integrals and Derivatives (Gordon and breach Science, Yverdon, Switzerland, 1993)

    MATH  Google Scholar 

  4. I. Podlubny, Fractional Differential Equations (Acdemic Press, San Dieg, 1999)

    MATH  Google Scholar 

  5. C. P. Li, Y. J. Wu, R. S. Ye eds., Recent Advances in Applied Nonlinear Dynamics with Numerical Analysis: Fractional Dynamics, Network Dynamics, Classical Dynamics and Fractal Dynamics with Numerical Simulations (World Scientific, 2013)

    Google Scholar 

  6. R. Metzler, J. Klafter, Phys. Rep. 339, 1 (2000)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  7. I. Podlubny, A. Chechkin, T. Skovranek, Y. Q. Chen, B. Vinagre, J. Comput. Phys. 228, 3137 (2009)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  8. A. Schmidt, L. Gaul, Signal Process. 86, 2592 (2006).

    Article  MATH  Google Scholar 

  9. C. P. Li, F. H. Zeng, Numer. Funct. Anal. Optimiz. 34, 149 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  10. K. Diethelm, N. J. Ford, A. D, Freed, Y. Luchko, Comput. Methods Appl. Mech. Engrg. 194, 743 (2005)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  11. C. Lubich, SIAM J. Math. Anal. 17, 704 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  12. V. E. Lynch, B. A. Carreras, D. del-Castillo-Negrete, K. M. Ferreira-Mejias, H.R. Hicks, J. Comput. Phys. 192, 406 (2003)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  13. C.P. Li, A. Chen, J.J. Ye, J. Comput. Phys. 230, 3352 (2011)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  14. Z. M. Odibat, Math. Comput. Simulat. 79, 2013 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  15. C. P. Li, F.H. Zeng, Int. J. Bifurcat. Chaos 22, 1230014 (2012)

    Article  MathSciNet  Google Scholar 

  16. E. Sousa, Int. J. Bifurcat. Chaos 22, 1250075 (2012)

    Article  MathSciNet  Google Scholar 

  17. C. P. Li, D. L. Qian, Y. Q. Chen, Disctete Dyn. Nat. Soc. 2011, 562494 (2011)

    MathSciNet  Google Scholar 

  18. C. P. Li, W. H. Deng, Appl. Math. Comput. 187, 777 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  19. C.P. Li, Z. Zhao, Euro. Phys. J.-Spec. Top. 193, 5 (2011)

    Article  ADS  Google Scholar 

  20. K. Diethelm, N. J. Ford, A. D. Freed, Numer. Algorithms 36, 31 (2004)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  21. C. Yang, F. Liu, ANZIAM J. 47, 168 (2006)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Shanghai University, Shanghai, 200444, PR China

    Fanhai Zeng & Changpin Li

Authors
  1. Fanhai Zeng
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Changpin Li
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Changpin Li.

About this article

Cite this article

Zeng, F., Li, C. Numerical approach to the Caputo derivative of the unknown function. centr.eur.j.phys. 11, 1433–1439 (2013). https://doi.org/10.2478/s11534-013-0214-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.2478/s11534-013-0214-4

Keywords

Search

Navigation