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On Riemann and Caputo fractional differences. (English) Zbl 1228.26008
Summary: We define left and right Caputo fractional sums and differences, study some of their properties and then relate them to Riemann-Liouville ones studied before. Also, the discrete version of the \(Q\)-operator is used to relate the left and right Caputo fractional differences. A Caputo fractional difference equation is solved. The solution proposes discrete versions of Mittag-Leffler functions.

MSC:
26A33 Fractional derivatives and integrals
34A08 Fractional ordinary differential equations
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[1] Samko, S.G.; Kilbas, A.A.; Marichev, O.I., Fractional integrals and derivatives — theory and applications, (1993), Gordon and Breach Linghorne, PA · Zbl 0818.26003
[2] Podlubny, I., Fractional differential equations, (1999), Academic Press San Diego, CA · Zbl 0918.34010
[3] Kilbas, A.; Srivastava, M.H.; Trujillo, J.J., ()
[4] Zaslavsky, G.M., Hamiltonian chaos and fractional dynamics, (2005), Oxford University Press Oxford · Zbl 1080.37082
[5] Magin, R.L., Fractional calculus in bioengineering, (2006), Begell House Publisher, Inc. Connecticut
[6] Mainardi, F., Fractional relaxation-oscillation and fractional diffusion-wave phenomena, Chaos, solitons and fractals, 7, 9, (1996), 1461-1447 · Zbl 1080.26505
[7] Kilbas, A.A.; Rivero, M.; Trujillo, J.J., Existence and uniqueness theorems for differential equations of fractional order in weighted spaces of continuous functions, Fractional calculus & applied analysis, 6, 4, 363-400, (2003) · Zbl 1085.34002
[8] Silva, M.F.; Machado, J.A.T.; Lopes, A.M., Modelling and simulation of artificial locomotion systems, Robotica, 23, 595-606, (2005)
[9] Agrawal, O.P.; Baleanu, D., A Hamiltonian formulation and a direct numerical scheme for fractional optimal control problems, Journal of vibration and control, 13, 9-10, 1269-1281, (2007) · Zbl 1182.70047
[10] Scalas, E., Mixtures of compound Poisson processes as models of tick-by-tick financial data, Chaos, solitons and fractals, 34, 1, 33-40, (2007) · Zbl 1142.60392
[11] Diethelm, K.; Ford, N.J.; Freed, A.D.; Luchko, Y., Algorithms for the fractional calculus: a selection of numerical methods, Computer methods in applied mechanics and engineering, 194, 6-8, 743-773, (2005) · Zbl 1119.65352
[12] Atanackovic, T.M.; Stankovic, B., On a class of differential equations with left and right fractional derivatives, Zeitschrift fur angewandte Mathematik und mechanik, 87, 7, 537-546, (2007) · Zbl 1131.34003
[13] Abdeljawad (Maraaba), T.; Baleanu, D.; Jarad, F., Existence and uniqueness theorem for a class of delay differential equations with left and right Caputo fractional derivatives, Journal of mathematical physics, 49, 083507, (2008) · Zbl 1152.81550
[14] Maraaba (Abdeljawad), T.; Jarad, F.; Baleanu, D., On the existence and the uniqueness theorem for fractional differential equations with bounded delay within Caputo derivatives, Science in China series A: mathematics, 51, 10, 1775-1786, (2008) · Zbl 1179.26024
[15] Baleanu, D.; Trujillo, J.J., On exact solutions of a class of fractional euler – lagrange equations, Nonlinear dynamics, 52, 4, 331-335, (2008) · Zbl 1170.70328
[16] K.S. Miller, B. Ross, Fractional difference calculus, in: Proceedings of the International Symposium on Univalent Functions, Fractional Calculus and Their Applications, Nihon University, Koriyama, Japan, 1989, pp. 139-152. · Zbl 0693.39002
[17] Atıcı, F.M.; Eloe, P.W., A transform method in discrete fractional calculus, International journal of difference equations, 2, 2, 165-176, (2007)
[18] Atıcı, F.M.; Eloe, P.W., Initial value problems in discrete fractional calculus, Proceedings of the American mathematical society, 137, 981-989, (2009) · Zbl 1166.39005
[19] Abdeljawad, T.; Baleanu, D, Fractional differences and integration by parts, Journal of computational analysis and applications, 13, 3, 574-582, (2011) · Zbl 1225.39008
[20] Kilbas, Samko G.; Marichev, A.A., Fractional integrals and derivatives: theory and applications, (1993), Gordon and Breach Yverdon · Zbl 0818.26003
[21] Bastos, Nuno R.O.; Ferreira, Rui A.C.; Torres, Delfim F.M., Discrete-time fractional variational problems, Signal processing, 91, 3, 513-524, (2011) · Zbl 1203.94022
[22] Atıcı, Ferhan M.; Eloe, Paul W., Discrete fractional calculus with the nabla operator, Electronic journal of qualitative theory of differential equations, 3, 1-12, (2009), Spec. Ed. I · Zbl 1166.39005
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