Abstract
The linear Caputo–type sequential difference fractional-order systems are discussed. The classical \(\mathcal {Z}\)-transform method is used to show the general solutions of sequential systems in the form \(\left( \varDelta _*^\alpha (\varDelta _*^\alpha x)\right) (n)+b\left( \varDelta _*^\alpha x\right) (n)+cx(n)=0\), where \(b,c\in \mathbb {R}\). In proofs we base on the formula for the image of the discrete Mittag-Leffler function in the \(\mathcal {Z}\)-transform.
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References
Bonilla, B., Rivero, M., Rodriguez-Germá, L., Trujillo, J.: Fractional differential equations as alternative models to nonlinear differential equations. Appl. Math. Comput. 187, 79–88 (2007)
Chen, F., Luo, X., Zhou, Y.: Existence results for nonlinear fractional difference equation. Adv. Differ. Equ. 2011, 12 (2011). doi:10.1155/2011/713201
Elaydi, S.N.: An Introduction to Difference Equations. Springer, New York (1967)
Ferreira, R.A.C., Torres, D.F.M.: Fractional \(h\)-difference equations arising from the calculus of variations. Appl. Anal. Discret. Math. 5(1), 110–121 (2011)
Furati, K.M.: A Cauchy-type problem with a sequential fractional derivative in the space of continuous functions. Bound. Value Probl. (58) (2012)
Herrmann, R.: Fractional Calculus: An Introduction for Physicists. World Scientific Publishing Company, Singapore (2011)
Kaczorek, T.: Selected Problems of Fractional Systems Theory, vol. 411. Springer, Berlin (2011)
Klimek, M.: Sequential fractional differential equations with hadamard derivative. Commun. Nonlinear Sci. Numer. Simul. 16(12), 4689–4697 (2011)
Klimek, M., Błasik, M.: Existence and uniqness of solution for a class of nonlinear sequential differential equations of fractional order. Cent. Eur. J. Math. 10(6), 1–14 (2012). doi:10.2478/s11533-012-0112-9
Machado, J.A.T.: Analysis and design of fractional-order digital control systems. Syst. Anal. Model. Simul. 27(2–3), 107–122 (1997)
Miller, K.S., Ross, B.: Fractional difference calculus. In: Proceedings of the International Symposium on Univalent Functions, Fractional Calculus and their Applications, pp. 139–152, Nihon University, Kōriyama, Japan (1988)
Mozyrska, D., Girejko, E., Wyrwas, M.: Fractional nonlinear systems with sequential operators. Cent. Eur. J. Phys. 11(10), 9 (2013)
Mozyrska, D., Pawluszewicz, E.: Fractional discrete-time linear control systems with initialisation. Int. J. Control 85(2), 213–219 (2012)
Mozyrska, D., Wyrwas, M.: In: Latawiec, K.J., Łukaniszyn, M., Stanisławski, R. (eds.) Advances in the Thoery and Applications of Non-integer Order Systems. Lecture Notes in Electrical Engineering, vol. 320, pp. 47–58. Springer, Berlin (2014)
Mozyrska, D., Wyrwas, M.: Solutions of fractional linear difference systems with Caputo-type operator via transform method. In: Proceedings of International Conference on Fractional Differentiation and its Applications: FDA’2014, 6 (2014)
Mozyrska, D., Wyrwas, M.: The Z-transform method and delta type fractional difference operators. Discret. Dyn. Nat. Soc. 2015, 12 (2015)
Podlubny, I.: Fractional Differential Equations. Academic Press, San Diego (1999)
Sierociuk, D., Dzielinski, A., Sarwas, G., Petras, I., Podlubny, I., Skovranek, T.: Modelling heat transfer media using fractional calculus. Philos. Trans. R. Soc. 371, 1990 (2013)
Acknowledgments
The work was supported by Bialystok University of Technology grant G/WM/3/2012. The project was supported by the founds of National Science Centre granted on the bases of the decision number \(DEC-2011/03/B/ST7/03476.\)
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Girejko, E., Pawłuszewicz, E., Wyrwas, M. (2016). The Z-Transform Method for Sequential Fractional Difference Operators. In: Domek, S., Dworak, P. (eds) Theoretical Developments and Applications of Non-Integer Order Systems. Lecture Notes in Electrical Engineering, vol 357. Springer, Cham. https://doi.org/10.1007/978-3-319-23039-9_5
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DOI: https://doi.org/10.1007/978-3-319-23039-9_5
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