On Dynamical Systems with Nabla Half Derivative on Time Scales

Abstract

This paper is devoted to study of dynamical systems involving nabla half derivative on an arbitrary time scale. We prove existence and uniqueness of the solution of such system supplied with a suitable initial condition. Both Riemann–Liouville and Caputo approaches to noninteger-order derivatives are covered. Under special conditions we present an explicit form of the solution involving a time scales analogue of Mittag–Leffler function. Also an algorithm for solving of such problems on isolated time scales is established. Moreover, we show that half power functions are positive and decreasing with respect to \(t-s\) on an arbitrary time scale.

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References

  1. 1.

    Bastos, N.R.O., Mozyrska, D., Torres, D.F.M.: Fractional derivatives and integrals on time scales via the inverse generalized Laplace transform. Int. J. Math. Comput. 11, 1–9 (2011)

    MathSciNet  Google Scholar 

  2. 2.

    Benkhettou, N., Brito da Cruz, A.M.C., Torres, D.F.M.: A fractional calculus on arbitrary time scales: fractional differentiation and fractional integration. Signal Process. 107, 230–237 (2015)

    Article  Google Scholar 

  3. 3.

    Bohner, M., Lutz, D.A.: Asymptotic expansions and analytic dynamic equations. Z. Angew. Math. Mech. 86, 37–45 (2006)

    MathSciNet  Article  Google Scholar 

  4. 4.

    Bohner, M., Peterson, A.C.: Advances in Dynamic Equations on Time Scales, 1st edn. Birkhäuser, Basel (2003)

    Google Scholar 

  5. 5.

    Čermák, J., Kisela, T., Nechvátal, L.: Discrete Mittag–Leffler functions in linear fractional difference equations. Abstr. Appl. Anal. 2011, 1–21 (2011)

    MathSciNet  Article  Google Scholar 

  6. 6.

    Čermák, J., Nechvátal, L.: On \((q, h)\)-analogue of fractional calculus. J. Nonlinear Math. Phys. 17, 51–68 (2010)

    MathSciNet  Article  Google Scholar 

  7. 7.

    Kisela, T.: Power functions and essentials of fractional calculus on isolated time scales. Adv. Differ. Equ. 2013, 1–18 (2013)

    MathSciNet  Article  Google Scholar 

  8. 8.

    Mainardi, F.: Fractional calculus: some basic problems in continuum and statistical mechanics. In: Carpinteri, A., Mainardi, F. (eds.) Fractals and Fractional Calculus in Continuum Mechanics, pp. 291–348. Springer, Wien (1997)

    Google Scholar 

  9. 9.

    Mozyrska, D., Pawluszewicz, E.: Delta and nabla monomials and generalized polynomial series on time scales. In: Leizarowitz A., Mordukhovich, B.S., Shafrir I., Zaslavski A.J. (eds.) Nonlinear Analysis and Optimization II: Optimization, Contemporary Mathematics, pp. 199–213 (2010)

  10. 10.

    Mozyrska, D., Pawluszewicz, E., Wyrwas, M.: Local observability and controllability of nonlinear discrete-time fractional order systems based on their linearisation. Int. J. Syst. Sci. 48, 788–794 (2017)

    MathSciNet  Article  Google Scholar 

  11. 11.

    Podlubný, I.: Fractional Differential Equations, 1st edn. Academic Press, New York (1999)

    Google Scholar 

  12. 12.

    Segi Rahmat, R.M.: A new definition of conformable fractional derivative on arbitrary time scales. Adv. Differ. Equ. 2019, 1–16 (2019)

    MathSciNet  Article  Google Scholar 

  13. 13.

    Torvik, P.J., Bagley, R.L.: On the appearance of the fractional derivative in the behavior of real materials. J. Appl. Mech. T. ASME 51, 294–298 (1984)

    Article  Google Scholar 

  14. 14.

    Williams, P.A.: Fractional calculus on time scales with Taylor’s theorem. Fract. Calc. Appl. Anal. 15, 616–638 (2012)

    MathSciNet  Article  Google Scholar 

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Acknowledgements

The research has been supported by the Grant GA17-03224S of the Czech Science Foundation. The preparation of the final version was supported by the Grant GA20-11846S of the Czech Science Foundation.

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Correspondence to Tomáš Kisela.

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Kisela, T. On Dynamical Systems with Nabla Half Derivative on Time Scales. Mediterr. J. Math. 17, 187 (2020). https://doi.org/10.1007/s00009-020-01629-w

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Keywords

  • Fractional calculus
  • time scales
  • Nabla half derivative
  • dynamical systems
  • Mittag–Leffler function
  • existence and uniqueness

Mathematics Subject Classification

  • Primary 26A33
  • 26E70
  • Secondary 39A12
  • 39A13