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Comparison of h-Difference Fractional Operators

  • Dorota Mozyrska
  • Ewa Girejko
  • Małgorzata Wyrwas
Part of the Lecture Notes in Electrical Engineering book series (LNEE, volume 257)

Abstract

We compare three different types of h-difference fractional operators: Grünwald-Letnikov, Caputo, Riemann-Liouville types of operators. There is introduced the formula for fundamental matrix of solutions for linear systems of h-difference fractional equations with Grünwald-Letnikov type operator while the one with Caputo type or Riemann-Liouville type is well known. We present new formulas for linear control systems with the mentioned operators.

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References

  1. 1.
    Abdeljawad, T.: On Riemann and Caputo fractional differences. Comp. and Math. with Appl. (2011), doi:10.1016/j.camwa.2011.03.036Google Scholar
  2. 2.
    Anastassiou, G.A.: Intelligent Mathematics: Computational Analysis. Springer (2011)Google Scholar
  3. 3.
    Atici, F.M., Eloe, P.W.: A Transform Method in Discrete Fractional Calculus. International Journal of Difference Equations 2, 165–176 (2007)MathSciNetGoogle Scholar
  4. 4.
    Atici, F.M., Eloe, P.W.: Initial value problems in discrete fractional calculus. In: Proceedings of the American Mathematical Society, 9 pages (2009), S 0002-9939(08)09626-3Google Scholar
  5. 5.
    Bastos, N.R.O., Ferreira, R.A.C., Torres, D.F.M.: Necessary optimality conditions for fractional difference problems of the calculus of variations. Discrete Contin. Dyn. Syst. 29(2), 417–437 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Ferreira, R.A.C., Torres, D.F.M.: Fractional h-difference equations arising from the calculus of variations. Appl. Anal. Discrete Math. 5(1), 110–121 (2011)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Chen, F., Luo, X., Zhou, Y.: Existence results for nonlinear fractional difference equation. Advances in Difference Eq., article ID 713201, 12 p. (2011), doi: 10.1155/2011/713201Google Scholar
  8. 8.
    Busłowicz, M., Nartowicz, N.: Design of fractional order controller for a class of plants with delay. Measurment Automation and Robotics 2, 398–405 (2009) (in Polish)Google Scholar
  9. 9.
    Ferreira, R.A.C., Torres, D.F.M.: Fractional h-difference equations arising from the calculus of variations. Appl. Anal. Discrete Math. 5(1), 110–121 (2011)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Mozyrska, D., Girejko, E.: Overview of the fractional h-difference operators. In: Almeida, A., Castro, L., Speck, F.O.: Advances in Harmonic Analysis and Operator Theory – The Stefan Samko Anniversary Volume, Operator Theory: Advances and Applications, vol. 229, XII, 388 p. Birkhäuser (2013) ISBN: 978-3-0348-0515-5Google Scholar
  11. 11.
    Holm, M.T.: The theory of discrete fractional calculus: Development and application. University of Nebraska, Lincoln (2011)Google Scholar
  12. 12.
    Kaczorek, T.: Fractional positive linear systems. Kybernetes 38(7/8), 1059–1078 (2009)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Kaczorek, T.: Reachability of cone fractional continuous-time linear systems. Int. J. Appl. Math. Comput. Sci. 19(1), 89–93 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Kaczorek, T.: Reachability and controllability to zero of positive fractional discrete-time systems. Machine Intelligence and Robotic Control 6(4), 139–143 (2007)Google Scholar
  15. 15.
    Mozyrska, D., Pawluszewicz, E.: Local controllability of nonlinear discrete-time fractional order systems. Bull. Pol. Acad. Sci. Tech. Sci. (2012) (series submitted)Google Scholar
  16. 16.
    Podlubny, I.: Fractional differential systems. Academic Press, San Diego (1999)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2013

Authors and Affiliations

  • Dorota Mozyrska
    • 1
  • Ewa Girejko
    • 1
  • Małgorzata Wyrwas
    • 1
  1. 1.Bialystok University of TechnologyBiałystokPoland

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