Numerical approach to the Caputo derivative of the unknown function

• Authors: Fanhai Zeng , Changpin Li
• Languages of publication: EN

Abstracts

EN

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.

Keywords

EN

Caputo derivative; ordinary differential equation; fractional Adams method

• Publisher: De Gruyter Open
• Journal: Open Physics
• Year: 2013
• Volume: 11
• Issue: 10
• Pages: 1433-1439

Dates

published

1 - 10 - 2013

online

19 - 12 - 2013

Contributors

author

Fanhai Zeng

• Department of Mathematics, Shanghai University, Shanghai, 200444, PR China

author

Changpin Li

• Department of Mathematics, Shanghai University, Shanghai, 200444, PR China

References

• [1] K. Miller, B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations (Wiley, New York, 1993)
• [2] K. B. Oldham, J. Spanier, The Fractional Calculus (Academic Press, New York, 1974)
• [3] S. G. Samko, A. A. Kilbas, O. I. and Marichev, Fractional Integrals and Derivatives (Gordon and breach Science, Yverdon, Switzerland, 1993)
• [4] I. Podlubny, Fractional Differential Equations (Acdemic Press, San Dieg, 1999)
• [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)
• [6] R. Metzler, J. Klafter, Phys. Rep. 339, 1 (2000) http://dx.doi.org/10.1016/S0370-1573(00)00070-3[Crossref]
• [7] I. Podlubny, A. Chechkin, T. Skovranek, Y. Q. Chen, B. Vinagre, J. Comput. Phys. 228, 3137 (2009) http://dx.doi.org/10.1016/j.jcp.2009.01.014[Crossref]
• [8] A. Schmidt, L. Gaul, Signal Process. 86, 2592 (2006). http://dx.doi.org/10.1016/j.sigpro.2006.02.006[Crossref]
• [9] C. P. Li, F. H. Zeng, Numer. Funct. Anal. Optimiz. 34, 149 (2013) http://dx.doi.org/10.1080/01630563.2012.706673[Crossref]
• [10] K. Diethelm, N. J. Ford, A. D, Freed, Y. Luchko, Comput. Methods Appl. Mech. Engrg. 194, 743 (2005) http://dx.doi.org/10.1016/j.cma.2004.06.006[Crossref]
• [11] C. Lubich, SIAM J. Math. Anal. 17, 704 (1986) http://dx.doi.org/10.1137/0517050[Crossref]
• [12] V. E. Lynch, B. A. Carreras, D. del-Castillo-Negrete, K. M. Ferreira-Mejias, H.R. Hicks, J. Comput. Phys. 192, 406 (2003) http://dx.doi.org/10.1016/j.jcp.2003.07.008[Crossref]
• [13] C.P. Li, A. Chen, J.J. Ye, J. Comput. Phys. 230, 3352 (2011) http://dx.doi.org/10.1016/j.jcp.2011.01.030[Crossref]
• [14] Z. M. Odibat, Math. Comput. Simulat. 79, 2013 (2009) http://dx.doi.org/10.1016/j.matcom.2008.08.003[Crossref]
• [15] C. P. Li, F.H. Zeng, Int. J. Bifurcat. Chaos 22, 1230014 (2012) http://dx.doi.org/10.1142/S0218127412300145[Crossref]
• [16] E. Sousa, Int. J. Bifurcat. Chaos 22, 1250075 (2012) http://dx.doi.org/10.1142/S0218127412500757[Crossref]
• [17] C. P. Li, D. L. Qian, Y. Q. Chen, Disctete Dyn. Nat. Soc. 2011, 562494 (2011)
• [18] C. P. Li, W. H. Deng, Appl. Math. Comput. 187, 777 (2007) http://dx.doi.org/10.1016/j.amc.2006.08.163[Crossref]
• [19] C.P. Li, Z. Zhao, Euro. Phys. J.-Spec. Top. 193, 5 (2011) http://dx.doi.org/10.1140/epjst/e2011-01378-2[Crossref]
• [20] K. Diethelm, N. J. Ford, A. D. Freed, Numer. Algorithms 36, 31 (2004) http://dx.doi.org/10.1023/B:NUMA.0000027736.85078.be[Crossref]
• [21] C. Yang, F. Liu, ANZIAM J. 47, 168 (2006)

Identifiers

DOI

10.2478/s11534-013-0214-4