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Numerical approximation of a class of discontinuous systems of fractional order

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Abstract

In this paper we investigate the possibility to formulate an implicit multistep numerical method for fractional differential equations, as a discrete dynamical system to model a class of discontinuous dynamical systems of fractional order. For this purpose, the problem is continuously transformed into a set-valued problem, to which the approximate selection theorem for a class of differential inclusions applies. Next, following the way presented in the book of Stewart and Humphries (Dynamical Systems and Numerical Analysis, Cambridge University Press, Cambridge, 1996) for the case of continuous differential equations, we prove that a variant of Adams–Bashforth–Moulton method for fractional differential equations can be considered as defining a discrete dynamical system, approximating the underlying discontinuous fractional system. For this purpose, the existence and uniqueness of solutions are investigated. One example is presented.

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References

  1. Stuart, A.M., Humphries, A.R.: Dynamical Systems and Numerical Analysis. Cambridge University Press, Cambridge (1996)

    MATH  Google Scholar 

  2. Filippov, A.F.: Differential Equations with Discontinuous Right-Hand Sides. Kluwer Academic, Dordrecht (1988)

    MATH  Google Scholar 

  3. Aubin, J.-P., Cellina, A.: Differential Inclusions: Set-Valued Maps and Viability Theory. Springer, Berlin (1984)

    MATH  Google Scholar 

  4. Aubin, J.-P., Frankowska, H.: Set-Valued Analysis. Birkhäuser, Boston (1990)

    MATH  Google Scholar 

  5. Wiercigroch, M., de Kraker, B.: Applied Nonlinear Dynamics and Chaos of Mechanical Systems with Discontinuities. World Scientific, Singapore (2000)

    Book  MATH  Google Scholar 

  6. Oldham, K.B., Spanier, J.: The Fractional Calculus: Theory and Applications of Differentiation and Integration of Arbitrary Order. Academic Press, New York (1974)

    MATH  Google Scholar 

  7. Caputo, M.: Linear models of dissipation whose Q is almost frequency independent—II. Geophys. J. R. Astron. Soc. 13, 529–539 (1967); reprinted in Fract. Calc. Appl. Anal. 10(3), 309–324 (2007)

    Google Scholar 

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

    MATH  Google Scholar 

  9. Podlubny, I.: Geometric and physical interpretation of fractional integration and fractional differentiation. J. Fract. Calc. Appl. Anal. 5(4), 367–386 (2002)

    MathSciNet  MATH  Google Scholar 

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

    MATH  Google Scholar 

  11. Diethelm, K.: The Analysis of Fractional Differential Equations: An Application-Oriented Exposition Using Differential Operators of Caputo Type. Springer, Berlin (2010)

    MATH  Google Scholar 

  12. Diethelm, K., Ford, N.J., Freed, A.D.: A predictor-corrector approach for the numerical solution of fractional differential equations. Nonlinear Dyn. 29(1–4), 3–22 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  13. Diethelm, K., Ford, N.J.: Analysis of fractional differential equations. J. Math. Anal. Appl. 265(2), 229–248 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  14. Kastner-Maresch, A., Lempio, F.: Difference methods with selection strategies for differential inclusions. Numer. Funct. Anal. Optim. 14(5–6), 555–572 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  15. Danca, M.-F.: On a class of discontinuous dynamical systems. Miskolc Math. Notes 2(2), 103–116 (2002)

    MathSciNet  Google Scholar 

  16. Gorenflo, R.: Fractional calculus: some numerical methods. In: Carpinteri, A., Mainardi, F. (eds.) Fractals and Fractional Calculus in Continuum Mechanics, pp. 277–290. Springer, Wien (1997)

    Google Scholar 

  17. Charef, A., Sun, H.H., Tsao, Y.Y., Onaral, B.: Fractal system as represented by singularity function. IEEE Trans. Autom. Control 37(9), 1465–1470 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  18. Press, W.H., Teukolsky, S.A., Vetterling, W.T., Flannery, B.P.: Numerical Recipes in C: The Art of Scientific Computing, 2nd edn. Cambridge University Press, Cambridge (1992)

    Google Scholar 

  19. Chen, G., Ueta, T.: Yet another chaotic attractor. Int. J. Bifurc. Chaos 9(7), 1465–1466 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  20. Ahmad, W.M., Sprott, J.C.: Chaos in fractional-order autonomous nonlinear systems. Chaos Solitons Fractals 16(2), 339–351 (2003)

    Article  MATH  Google Scholar 

  21. Danca, M.-F., Codreanu, S.: On a possible approximation of discontinuous dynamical systems. Chaos Solitons Fractals 13(4), 681–691 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  22. Hartley, T.T., Lorenzo, C.F., Qammer, H.K.: Chaos in a fractional order Chua’s system. IEEE Trans. Circuits Syst. 42(8), 485–490 (1995)

    Article  Google Scholar 

  23. Arena, P., Caponetto, R., Fortuna, L., Porto, D.: Chaos in a fractional order Duffing system. In: Proceedings ECCTD, Budapest, September, pp. 1259–1262 (1997)

    Google Scholar 

  24. Chang, Y.K., Nieto, J.J.: Some new existence results for fractional differential inclusions with boundary conditions. Math. Comput. Model. 49(3–4), 605–609 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  25. Henderson, J., Ouahab, A.: Impulsive differential inclusions with fractional order. Comput. Math. Appl. 59(3), 1191–1226 (2010)

    MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Dept. of Mathematics and Computer Science, Avram Iancu University, 400380, Cluj-Napoca, Romania

    Marius-F. Danca

  2. Romanian Institute of Science and Technology, 400487, Cluj-Napoca, Romania

    Marius-F. Danca

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  1. Marius-F. Danca
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Correspondence to Marius-F. Danca.

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Danca, MF. Numerical approximation of a class of discontinuous systems of fractional order. Nonlinear Dyn 66, 133–139 (2011). https://doi.org/10.1007/s11071-010-9915-z

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