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Fractional difference equations with real variable. (English) Zbl 1259.39010

Summary: We independently propose a new kind of the definition of fractional difference, fractional sum, and fractional difference equation, give some basic properties of fractional difference and fractional sum, and give some examples to demonstrate several methods of how to solve certain fractional difference equations.

MSC:

39A20 Multiplicative and other generalized difference equations

References:

[1] S. G. Samko, A. A. Kilbas, and O. I. Marichev, Integrals and Derivatives of the Fractional Order and Some of Their Applications, Nauka i Tekhnika, Minsk, Russia, 1987. · Zbl 0697.26004
[2] K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, John Wiley & Sons, New York, NY, USA, 1993. · Zbl 0943.82582 · doi:10.1007/BF01048101
[3] I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, Calif, USA, 1999. · Zbl 1056.93542 · doi:10.1109/9.739144
[4] A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, vol. 204 of North-Holland Mathematics Studies, Elsevier, Amsterdam, The Netherlands, 2006. · Zbl 1206.26007 · doi:10.1016/S0304-0208(06)80001-0
[5] K. Diethelm, The Analysis of Fractional Differential Equations, Springer, Berlin, Germany, 2010. · Zbl 1222.37107 · doi:10.1016/j.cnsns.2010.01.011
[6] Z.-X. Zheng, “On the developments applications of fractional equations,” Journal of Xuzhou Normal University, vol. 26, no. 2, pp. 1-10, 2008. · Zbl 1199.34010
[7] R. P. Agarwal, Difference Equations and Inequalities: Theory, Methods, and Application, Marcel Dekker, New York, NY, USA, 2000. · Zbl 1243.76111
[8] M. Bohner and A. C. Peterson, Advances in Dynamic Equations on Time Scales, Birkhäuser, Boston, Mass, USA, 2003. · Zbl 1243.65036 · doi:10.1007/978-0-8176-8230-9
[9] F. M. Atici and P. W. Eloe, “A transform method in discrete fractional calculus,” International Journal of Difference Equations, vol. 2, no. 2, pp. 165-176, 2007.
[10] F. M. Atici and P. W. Eloe, “Discrete fractional calculus with the nabla operator,” Electronic Journal of Qualitative Theory of Differential Equations, vol. 3, pp. 1-12, 2009. · Zbl 1189.39004
[11] F. M. Atici and P. W. Eloe, “Initial value problems in discrete fractional calculus,” Proceedings of the American Mathematical Society, vol. 137, no. 3, pp. 981-989, 2009. · Zbl 1166.39005 · doi:10.1090/S0002-9939-08-09626-3
[12] F. M. Atici and S. , “Modeling with fractional difference equations,” Journal of Mathematical Analysis and Applications, vol. 369, no. 1, pp. 1-9, 2010. · Zbl 1204.39004 · doi:10.1016/j.jmaa.2010.02.009
[13] G. A. Anastassiou, “Discrete fractional calculus and inequalities,” http://arxiv.org/abs/0911.3370. · Zbl 1190.26001
[14] G. A. Anastassiou, “Nabla discrete fractional calculus and nabla inequalities,” Mathematical and Computer Modelling, vol. 51, no. 5-6, pp. 562-571, 2010. · Zbl 1190.26001 · doi:10.1016/j.mcm.2009.11.006
[15] N. R. O. Bastos, R. A. C. Ferreira, and D. F. M. Torres, “Discrete-time fractional variational problems,” Signal Processing, vol. 91, no. 3, pp. 513-524, 2011. · Zbl 1203.94022 · doi:10.1016/j.sigpro.2010.05.001
[16] T. Abdeljawad, “On riemann and caputo fractional differences,” Computers & Mathematics with Applications, vol. 62, no. 3, pp. 1602-1611, 2011. · Zbl 1228.26008 · doi:10.1016/j.camwa.2011.03.036
[17] T. Abdeljawad, “Principles of delta and nabla fractional differences,” http://arxiv.org/abs/1112.5795.
[18] F. Jarad, T. Abdeljawad, D. Baleanu, and K. Bi\ccen, “On the stability of some discrete fractional nonautonomous systems,” Abstract and Applied Analysis, vol. 2012, Article ID 476581, 9 pages, 2012. · Zbl 1235.93206 · doi:10.1155/2012/476581
[19] T. Abdeljawad and D. Baleanu, “Caputo q-fractional initial value problems and a q-analogue mittag-leffler function,” Communications in Nonlinear Science and Numerical Simulation, vol. 16, no. 12, pp. 4682-4688, 2011. · Zbl 1231.26006 · doi:10.1016/j.cnsns.2011.01.026
[20] T. Abdeljawad and D. Baleanu, “Fractional differences and integration by parts,” Journal of Computational Analysis and Applications, vol. 13, no. 3, pp. 574-582, 2011. · Zbl 1225.39008
[21] J.-F. Cheng and Y.-M. Chu, “On the fractional difference equations of order (2,q),” Abstract and Applied Analysis, vol. 2011, Article ID 497259, 16 pages, 2011. · Zbl 1227.39007 · doi:10.1155/2011/497259
[22] J.-F. Cheng, “Solutions of fractional difference equations of order (k,q),” Acta Mathematicae Applicatae Sinica, vol. 34, no. 2, pp. 313-330, 2011. · Zbl 1249.39025
[23] J.-F. Cheng, Theory of Fractional Difference Equations, Xiamen University Press, 2011.
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