Abstract
This paper addresses the calculation of derivatives of fractional order for non-smooth data. The noise is avoided by adopting an optimization formulation using genetic algorithms (GA). Given the flexibility of the evolutionary schemes, a hierarchical GA composed by a series of two GAs, each one with a distinct fitness function, is established.
Similar content being viewed by others
References
Oldham, K.B., Spanier, J.: The Fractional Calculus: Theory and Application of Differentiation and Integration to Arbitrary Order. Academic Press, San Diego (1974)
Ross, B.: Fractional calculus. Math. Mag. 50, 15–122 (1977)
Samko, S.G., Kilbas, A.A., Marichev, O.I.: Fractional Integrals and Derivatives: Theory and Applications. Gordon and Breach, New York (1993)
Miller, K.S., Ross, B.: An Introduction to the Fractional Calculus and Fractional Differential Equations. Wiley, New York (1993)
Bagley, R.L., Torvik, P.J.: Fractional calculus—a different approach to the analysis of viscoelastically damped structures. AIAA J. 21, 741–748 (1983)
Oustaloup, A.: La commande CRONE: Commande Robuste d’Ordre Non Entier. Hermes (1991)
Mainardi, F.: Fractional relaxation-oscillation and fractional diffusion-wave phenomena. Chaos Solitons Fractals 7, 1461–1477 (1996)
Machado, J.T.: Analysis and design of fractional-order digital control systems. J. Syst. Anal. Model. Simul. 27, 107–122 (1997)
Podlubny, I.: Fractional Differential Equations. Academic Press, San Diego (1999)
Machado, J.T.: Discrete-time fractional-order controllers. J. Fract. Calc. Appl. Anal. 4, 47–66 (2001)
Nigmatullin, R.R.: A fractional integral and its physical interpretation. Theor. Math. Phys. 90, 242–251 (1992)
Rutman, R.S.: On the paper by R.R. Nigmatullin “A fractional integral and its physical interpretation”. Theor. Math. Phys. 100, 1154–1156 (1994)
Tatom, F.B.: The relationship between fractional calculus and fractals. Fractals 3, 217–229 (1995)
Yu, Z., Ren, F., Zhou, J.: Fractional integral associated to generalized cookie-cutter set and its physical interpretation. J. Phys. A: Math. Gen. 30, 5569–5577 (1997)
Adda, F.B.: Geometric interpretation of the fractional derivative. J. Fract. Calc. 11, 21–52 (1997)
Moshrefi-Torbati, M., Hammond, J.K.: Physical and geometrical interpretation of fractional operators. J. Franklin Inst. B 335, 1077–1086 (1998)
Podlubny, I.: Geometrical and physical interpretation of fractional integration and fractional differentiation. J. Fract. Calc. Appl. Anal. 5, 357–366 (2002)
Machado, J.T.: A probabilistic interpretation of the fractional-order differentiation. J. Fract. Calc. Appl. Anal. 6, 73–80 (2003)
Stanislavsky, A.A.: Probabilistic interpretation of the integral of fractional order. Theor. Math. Phys. 138, 418–431 (2004)
Li, J.: General explicit difference formulas for numerical differentiation. J. Comput. Appl. Math. 183, 29–52 (2005)
Knowles, I., Wallace, R.: A variational method for numerical differentiation. Numer. Math. 70, 91–110 (1995)
Chartrand, R.: Numerical differentiation of noisy, non-smooth data. Los Alamos National Laboratory, December 13 (2005)
Ahnert, K., Abel, M.: Numerical differentiation: local versus global methods. Comput. Phys. (2006)
Le, T., Chartrand, R., Asaki, T.J.: A variational approach to reconstructing images corrupted by Poisson noise. J. Math. Imaging Vis. 27, 257–263 (2007)
Holland, J.H.: Adaptation in Natural and Artificial Systems. University of Michigan Press, Ann Arbor (1975)
Goldenberg, D.E.: Genetic Algorithms in Search Optimization, and Machine Learning. Addison–Wesley, Reading (1989)
Machado, J.T., Galhano, A.: Numerical calculation of fractional derivatives of non-smooth data. In: ENOC 2008—6th EUROMECH Conference, Saint Petersburg, Russia (2008)
Hanke, M., Scherzer, O.: Inverse problems light: numerical differentiation. Am. Math. Mon. 108, 512–521 (2001)
Wang, J.: Wavelet approach to numerical differentiation of noisy functions. Commun. Pure Appl. Anal. 6, 873–897 (2007)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Machado, J.A.T. Calculation of fractional derivatives of noisy data with genetic algorithms. Nonlinear Dyn 57, 253–260 (2009). https://doi.org/10.1007/s11071-008-9436-1
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11071-008-9436-1