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Fractional Calculus: Fundamentals and Applications

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Acoustics and Vibration of Mechanical Structures—AVMS-2017

Part of the book series: Springer Proceedings in Physics ((SPPHY,volume 198))

Abstract

This paper presents the fundamental aspects of the theory of Fractional Calculus. Several approximation methods for the calculation of fractional-order derivatives are discussed. The application of Fractional Calculus in automatic control systems and their main properties are also analyzed.

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References

  1. D. Baleanu, J.T. Machado, A. Luo, Fractional Dynamics and Control (Springer, New York, 2011)

    Google Scholar 

  2. D. Baleanu, K. Diethelm, E. Scalas, J.J. Trujillo, Fractional Calculus Models and Numerical Methods (World Scientific Publishing Company, Amsterdam, 2012)

    Book  MATH  Google Scholar 

  3. G.E. Carlson, C.A. Halijak, Approximation of fractional capacitors (1/s)(1/n) by a regular Newton process. IEEE Trans. Circuit Theory 10, 210–213 (1964)

    Article  Google Scholar 

  4. R. Hilfer, Applications of fractional calculus in physics (World Scientific, Singapore, 2000)

    Book  MATH  Google Scholar 

  5. A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and applications of fractional differential equations. North-Holland Mathematics Studies, vol. 204 (Elsevier, Amsterdam, 2006)

    Google Scholar 

  6. J.T. Machado, Analysis and design of fractional-order digital control systems. Systems Anal. Model. Simul. 27(2–3), 107–122 (1997)

    MATH  Google Scholar 

  7. J.T. Machado, Fractional-order derivative approximations in discrete-time control systems. Syst. Anal. Model. Simul. 34, 419–434 (1999)

    MATH  Google Scholar 

  8. J.T. Machado, Discrete-time fractional-order controllers. Fract. Calculus Appl. Anal. 4(1), 47–66 (2001)

    MathSciNet  MATH  Google Scholar 

  9. J.T. Machado, A.M. Galhano, Approximating fractional derivatives in the perspective of system control. Nonlinear Dyn. 56(4), 401–407 (2009)

    Article  MATH  Google Scholar 

  10. F. Mainardi, Fractional Calculus and Waves in Linear Viscoelasticity: An Introduction to Mathematical Models (Imperial College Press, London, 2010)

    Book  MATH  Google Scholar 

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

    MATH  Google Scholar 

  12. C.A. Monje, Y. Chen, B.M. Vinagre, D. Xue, V. Feliu, Fractional-Order Systems and Con- trols (Springer, London, 2010)

    Book  MATH  Google Scholar 

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

    MATH  Google Scholar 

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

    Google Scholar 

  15. I. Petráš: Fractional-Order Nonlinear Systems: Modeling, Analysis and Simulation (Springer, Heidelberg, 2011)

    Google Scholar 

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

    MATH  Google Scholar 

  17. S.G. Samko, A.A. Kilbas, O.I. Marichev, Fractional Integrals and Derivatives (Gordon and Breach Science Publishers, Yverdon, 1993)

    MATH  Google Scholar 

  18. V.E. Tarasov, Fractional Dynamics: Applications of Fractional Calculus to Dynamics of Particles (Fields and Media, Springer, 2010)

    Book  MATH  Google Scholar 

  19. D. Valerio, J.S. da Costa, An Introduction to Fractional Control (IET, Stevenage, 2012)

    Book  MATH  Google Scholar 

  20. S. Westerlund, Dead Matter Has Memory (Causal Consulting, Kalmar, 2002)

    Google Scholar 

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Correspondence to J. A. Tenreiro Machado .

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Machado, J.A.T. (2018). Fractional Calculus: Fundamentals and Applications. In: Herisanu, N., Marinca, V. (eds) Acoustics and Vibration of Mechanical Structures—AVMS-2017. Springer Proceedings in Physics, vol 198. Springer, Cham. https://doi.org/10.1007/978-3-319-69823-6_1

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