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Calculation of fractional derivatives of noisy data with genetic algorithms

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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.

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References

  1. Oldham, K.B., Spanier, J.: The Fractional Calculus: Theory and Application of Differentiation and Integration to Arbitrary Order. Academic Press, San Diego (1974)

    Google Scholar 

  2. Ross, B.: Fractional calculus. Math. Mag. 50, 15–122 (1977)

    Article  Google Scholar 

  3. Samko, S.G., Kilbas, A.A., Marichev, O.I.: Fractional Integrals and Derivatives: Theory and Applications. Gordon and Breach, New York (1993)

    MATH  Google Scholar 

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

    MATH  Google Scholar 

  5. Bagley, R.L., Torvik, P.J.: Fractional calculus—a different approach to the analysis of viscoelastically damped structures. AIAA J. 21, 741–748 (1983)

    Article  MATH  Google Scholar 

  6. Oustaloup, A.: La commande CRONE: Commande Robuste d’Ordre Non Entier. Hermes (1991)

  7. Mainardi, F.: Fractional relaxation-oscillation and fractional diffusion-wave phenomena. Chaos Solitons Fractals 7, 1461–1477 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  8. Machado, J.T.: Analysis and design of fractional-order digital control systems. J. Syst. Anal. Model. Simul. 27, 107–122 (1997)

    MATH  Google Scholar 

  9. Podlubny, I.: Fractional Differential Equations. Academic Press, San Diego (1999)

    MATH  Google Scholar 

  10. Machado, J.T.: Discrete-time fractional-order controllers. J. Fract. Calc. Appl. Anal. 4, 47–66 (2001)

    MATH  Google Scholar 

  11. Nigmatullin, R.R.: A fractional integral and its physical interpretation. Theor. Math. Phys. 90, 242–251 (1992)

    Article  MathSciNet  Google Scholar 

  12. Rutman, R.S.: On the paper by R.R. Nigmatullin “A fractional integral and its physical interpretation”. Theor. Math. Phys. 100, 1154–1156 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  13. Tatom, F.B.: The relationship between fractional calculus and fractals. Fractals 3, 217–229 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  14. 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)

    Article  MATH  MathSciNet  Google Scholar 

  15. Adda, F.B.: Geometric interpretation of the fractional derivative. J. Fract. Calc. 11, 21–52 (1997)

    MATH  MathSciNet  Google Scholar 

  16. Moshrefi-Torbati, M., Hammond, J.K.: Physical and geometrical interpretation of fractional operators. J. Franklin Inst. B 335, 1077–1086 (1998)

    Article  MathSciNet  Google Scholar 

  17. Podlubny, I.: Geometrical and physical interpretation of fractional integration and fractional differentiation. J. Fract. Calc. Appl. Anal. 5, 357–366 (2002)

    MathSciNet  Google Scholar 

  18. Machado, J.T.: A probabilistic interpretation of the fractional-order differentiation. J. Fract. Calc. Appl. Anal. 6, 73–80 (2003)

    MATH  MathSciNet  Google Scholar 

  19. Stanislavsky, A.A.: Probabilistic interpretation of the integral of fractional order. Theor. Math. Phys. 138, 418–431 (2004)

    Article  MATH  Google Scholar 

  20. Li, J.: General explicit difference formulas for numerical differentiation. J. Comput. Appl. Math. 183, 29–52 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  21. Knowles, I., Wallace, R.: A variational method for numerical differentiation. Numer. Math. 70, 91–110 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  22. Chartrand, R.: Numerical differentiation of noisy, non-smooth data. Los Alamos National Laboratory, December 13 (2005)

  23. Ahnert, K., Abel, M.: Numerical differentiation: local versus global methods. Comput. Phys. (2006)

  24. 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)

    Article  MathSciNet  Google Scholar 

  25. Holland, J.H.: Adaptation in Natural and Artificial Systems. University of Michigan Press, Ann Arbor (1975)

    Google Scholar 

  26. Goldenberg, D.E.: Genetic Algorithms in Search Optimization, and Machine Learning. Addison–Wesley, Reading (1989)

    Google Scholar 

  27. Machado, J.T., Galhano, A.: Numerical calculation of fractional derivatives of non-smooth data. In: ENOC 2008—6th EUROMECH Conference, Saint Petersburg, Russia (2008)

  28. Hanke, M., Scherzer, O.: Inverse problems light: numerical differentiation. Am. Math. Mon. 108, 512–521 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  29. Wang, J.: Wavelet approach to numerical differentiation of noisy functions. Commun. Pure Appl. Anal. 6, 873–897 (2007)

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Electrical Engineering, Institute of Engineering of Porto, Rua Dr. Antonio Bernardino de Almeida, 4200-072, Porto, Portugal

    J. A. Tenreiro Machado

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  1. J. A. Tenreiro Machado
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Correspondence to J. A. Tenreiro Machado.

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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

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  • DOI: https://doi.org/10.1007/s11071-008-9436-1

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