Skip Navigation Links
Journal of Applied Nonlinear Dynamics
Miguel A. F. Sanjuan (editor), Albert C.J. Luo (editor)
Miguel A. F. Sanjuan (editor)

Department of Physics, Universidad Rey Juan Carlos, 28933 Mostoles, Madrid, Spain


Albert C.J. Luo (editor)

Department of Mechanical and Industrial Engineering, Southern Illinois University Ed-wardsville, IL 62026-1805, USA

Fax: +1 618 650 2555 Email:

Entropy Analysis of Fractional Derivatives and Their Approximation

Journal of Applied Nonlinear Dynamics 1(1) (2012) 109--112 | DOI:10.5890/JAND.2012.03.001

J.A. Tenreiro Machado

Institute of Engineering of Polytechnic of Porto, Dept. of Electrical Engineering, Rua Dr. Antonio Bernardino de Almeida, 431, 4200-072 Porto, Portugal

Download Full Text PDF



Fractional derivatives (FDs) need to be calculated using series or rational fractions. The effect of such approximations are usually analysed by means of the frequency or the time responses. This paper takes advantage of a probabilistic interpretation of FDs and adopts the Shannon entropy for assessing the truncation effect produced by the approximations.


  1. [1]  Oldham, K.B. and Spanier, J. (1974), The Fractional Calculus: Theory and Application of Differentiation and Integration to Arbitrary Order, Academic Press, New York-London.
  2. [2]  Samko, S.G,, Kilbas, A.A. and Marichev, O.I. (1993), Fractional Integrals and Derivatives: The,ory and Applications, Gordon & Breach Science Publishers.
  3. [3]  Miller, K.S. and Ross, B. (1993), An Introduction to the Fractional Calculus and Fractional Differential Equations, JohnWiley and Sons.
  4. [4]  Kilbas, A.A., Srivastava, H.M. and Trujillo, J.J. (2006), Theory and Applications of the Fractional Differential Equations, Elsevier (North-Holland),Math. Studies, Vol. 204.
  5. [5]  Oustaloup, A. (1991), La Commande CRONE: Commande Robuste d'Ordre Non Entier, Hermes, Paris.
  6. [6]  Machado, J. T. (1997),Analysis and Design of Fractional-order Digital Control Systems, Systems Analysis, Modelling, Simulation, 27(2-3), 107-122.
  7. [7]  Podlubny, I. (1999), Fractional Differential Equations, Academic Press, San Diego-Boston-New York-London- Tokyo-Toronto.
  8. [8]  Westerlund, S. (2002), Dead matter has memory!, Causal Consulting, Kalmar.
  9. [9]  Podlubny, I.(2002), Geometric and Physical Interpretation of Fractional Integration and Fractional Differentiation, Journal of Fractional Calculus & Applied Analysis, 5(4) 357-366.
  10. [10]  Zaslavsky, G. M. (2005), Hamiltonian Chaos and Fractional Dynamics, Oxford University Press, Oxford.
  11. [11]  Magin, R. (2006), Fractional Calculus in Bioengineering. Begell House Inc, Redding.
  12. [12]  Mainardi, F.(2010), Fractional Calculus and Waves in Linear Viscoelasticity: An Introduction to Mathematical Models. Imperial College Press, London.
  13. [13]  Machado, J.T. (2003), A Probabilistic Interpretation of the Fractional-Order Differentiation, Journal of Fractional Calculus & Applied Analysis, 6(1), 73-80.
  14. [14]  Machado, J.T. (2009), Fractional Derivatives: Probability Interpretation and Frequency Response of Rational Approximations, Communications in Nonlinear Science and Numerical Simulations, 14(9-10), 3492-3497.
  15. [15]  Shannon, C.E. (1948), A Mathematical Theory of Communication, Bell System Technical Journal, 27, 379-423 & 623-656.
  16. [16]  Jaynes, E.T. (1957), Information Theory and Statistical Mechanics, Phys. Rev., 106, 620-630.
  17. [17]  Khinchin, A.I. (1957), Mathematical Foundations of Information Theory, Dover, New York.
  18. [18]  Beck, C. (2009), Generalised Information and EntropyMeasures in Physics, Contemporary Physics, 50(4), 495- 510.
  19. [19]  Gray, R.M. (2009), Entropy and Information Theory, Springer-Verlag.
  20. [20]  Machado, J.T. (2010), Entropy Analysis of Integer and Fractional Dynamical Systems, Nonlinear Dynamics, 62, (1-2) 371-378.