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Fractional Calculus and Applied Analysis

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Fractional calculus of variations for a combined Caputo derivative

1Białystok University of Technology

2University of Aveiro

© 2011 Versita Warsaw. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License. (CC BY-NC-ND 3.0)

Citation Information: Fractional Calculus and Applied Analysis. Volume 14, Issue 4, Pages 523–537, ISSN (Online) 1314-2224, ISSN (Print) 1311-0454, DOI: 10.2478/s13540-011-0032-6, September 2011

Publication History

Published Online:
2011-09-29

Abstract

We generalize the fractional Caputo derivative to the fractional derivative C D γα,β, which is a convex combination of the left Caputo fractional derivative of order α and the right Caputo fractional derivative of order β. The fractional variational problems under our consideration are formulated in terms of C D γα,β. The Euler-Lagrange equations for the basic and isoperimetric problems, as well as transversality conditions, are proved.

MSC: Primary 26A33; Secondary 49K05

Keywords: fractional derivatives; Caputo derivatives; fractional variational principles; Euler-Lagrange equations; isoperimetric constraints; transversality conditions

  • [1] O.P. Agrawal, Formulation of Euler-Lagrange equations for fractional variational problems. J. Math. Anal. Appl. 272, No 1 (2002), 368–379. http://dx.doi.org/10.1016/S0022-247X(02)00180-4 [CrossRef]

  • [2] O.P. Agrawal, Generalized Euler-Lagrange equations and transversality conditions for FVPs in terms of the Caputo derivative. J. Vib. Control 13, No 9–10 (2007), 1217–1237. http://dx.doi.org/10.1177/1077546307077472 [CrossRef] [Web of Science]

  • [3] O.P. Agrawal, Fractional variational calculus in terms of Riesz fractional derivatives. J. Phys. A 40, No 24 (2007), 6287–6303. http://dx.doi.org/10.1088/1751-8113/40/24/003 [CrossRef]

  • [4] R. Almeida, A.B. Malinowska, D.F.M. Torres, A fractional calculus of variations for multiple integrals with application to vibrating string. J. Math. Phys. 51, No 3 (2010), 033503, 12 pp. http://dx.doi.org/10.1063/1.3319559 [Web of Science] [CrossRef]

  • [5] R. Almeida, D.F.M. Torres, Calculus of variations with fractional derivatives and fractional integrals. Appl. Math. Lett. 22, No 12 (2009), 1816–1820. http://dx.doi.org/10.1016/j.aml.2009.07.002 [CrossRef] [Web of Science]

  • [6] R. Almeida, D.F.M. Torres, Leitmann’s direct method for fractional optimization problems. Appl. Math. Comput. 217, No 3 (2010), 956–962. http://dx.doi.org/10.1016/j.amc.2010.03.085 [Web of Science] [CrossRef]

  • [7] R. Almeida, D.F.M. Torres, Fractional variational calculus for nondifferentiable functions. Comput. Math. Appl. 61, No 10 (2011), 3097–3104. http://dx.doi.org/10.1016/j.camwa.2011.03.098 [Web of Science] [CrossRef]

  • [8] T.M. Atanacković, S. Konjik, S. Pilipović, Variational problems with fractional derivatives: Euler-Lagrange equations. J. Phys. A 41, No 9 (2008), 095201, 12 pp.

  • [9] D. Baleanu, Fractional variational principles in action. Phys. Scripta T136 (2009), Article Number: 014006. [Web of Science]

  • [10] D. Baleanu, O.P. Agrawal, Fractional Hamilton formalism within Caputo’s derivative. Czechoslovak J. Phys. 56, No 10–11 (2006), 1087–1092. http://dx.doi.org/10.1007/s10582-006-0406-x [CrossRef]

  • [11] D. Baleanu, A.K. Golmankhaneh, R. Nigmatullin, A.K. Golmankhaneh, Fractional Newtonian mechanics. Cent. Eur. J. Phys. 8, No 1 (2010), 120–125. http://dx.doi.org/10.2478/s11534-009-0085-x [CrossRef]

  • [12] D. Baleanu, S.I. Muslih, Lagrangian formulation of classical fields within Riemann-Liouville fractional derivatives. Phys. Scripta 72, No 2–3 (2005), 119–121. http://dx.doi.org/10.1238/Physica.Regular.072a00119 [CrossRef]

  • [13] N.R.O. Bastos, R.A.C. Ferreira, D.F.M. Torres, Discrete-time fractional variational problems. Signal Process. 91, No 3 (2011), 513–524. http://dx.doi.org/10.1016/j.sigpro.2010.05.001 [Web of Science] [CrossRef]

  • [14] R. Brunetti, D. Guido, R. Longo, Modular structure and duality in conformal quantum field theory. Comm. Math. Phys. 156, No 1 (1993), 201–219. http://dx.doi.org/10.1007/BF02096738 [CrossRef]

  • [15] J. Cresson, Fractional embedding of differential operators and Lagrangian systems. J. Math. Phys. 48, No 3 (2007), 033504, 34 pp. http://dx.doi.org/10.1063/1.2483292 [CrossRef]

  • [16] R.A. El-Nabulsi, D.F.M. Torres, Necessary optimality conditions for fractional action-like integrals of variational calculus with Riemann-Liouville derivatives of order (α, β). Math. Methods Appl. Sci. 30, No 15 (2007), 1931–1939. http://dx.doi.org/10.1002/mma.879 [CrossRef]

  • [17] R.A. El-Nabulsi, D.F.M. Torres, Fractional actionlike variational problems. J. Math. Phys. 49, No 5 (2008), 053521, 7 pp. [CrossRef]

  • [18] Fract. Calc. Appl. Anal., p ISSN 1311-0454, e ISSN 1314-2224, Vol. 1 (1998) — Vol. 13 (2010) at http://www.math.bas.bg/~fcaa; Vol. 14 (2011) at http://www.springerlink.com/content/1311-0454.

  • [19] G.S.F. Frederico, D.F.M. Torres, A formulation of Noether’s theorem for fractional problems of the calculus of variations. J. Math. Anal. Appl. 334, No 2 (2007), 834–846. http://dx.doi.org/10.1016/j.jmaa.2007.01.013 [CrossRef]

  • [20] G.S.F. Frederico, D.F.M. Torres, Fractional conservation laws in optimal control theory. Nonlinear Dynam. 53, No 3 (2008), 215–222. http://dx.doi.org/10.1007/s11071-007-9309-z [Web of Science] [CrossRef]

  • [21] G.S.F. Frederico, D.F.M. Torres, Fractional Noether’s theorem in the Riesz-Caputo sense. Appl. Math. Comput. 217, No 3 (2010), 1023–1033. http://dx.doi.org/10.1016/j.amc.2010.01.100 [Web of Science] [CrossRef]

  • [22] R. Hilfer, Applications of Fractional Calculus in Physics. World Sci. Publishing, River Edge, NJ (2000). http://dx.doi.org/10.1142/9789812817747 [CrossRef]

  • [23] G. Jumarie, Fractional Hamilton-Jacobi equation for the optimal control of nonrandom fractional dynamics with fractional cost function. J. Appl. Math. Comput. 23, No 1–2 (2007), 215–228. http://dx.doi.org/10.1007/BF02831970 [CrossRef]

  • [24] G. Jumarie, An approach via fractional analysis to non-linearity induced by coarse-graining in space. Nonlinear Anal. Real World Appl. 11, No 1 (2010), 535–546. http://dx.doi.org/10.1016/j.nonrwa.2009.01.003 [CrossRef] [Web of Science]

  • [25] A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam (2006).

  • [26] M. Klimek, Stationarity-conservation laws for fractional differential equations with variable coefficients. J. Phys. A 35, No 31 (2002), 6675–6693. http://dx.doi.org/10.1088/0305-4470/35/31/311 [CrossRef]

  • [27] A.B. Malinowska, D.F.M. Torres, On the diamond-alpha Riemann integral and mean value theorems on time scales. Dynam. Systems Appl. 18, No 3–4 (2009), 469–481.

  • [28] A.B. Malinowska, D.F.M. Torres, Generalized natural boundary conditions for fractional variational problems in terms of the Caputo derivative. Comput. Math. Appl. 59, No 9 (2010), 3110–3116. http://dx.doi.org/10.1016/j.camwa.2010.02.032 [CrossRef]

  • [29] A.B. Malinowska, D.F.M. Torres, Natural boundary conditions in the calculus of variations. Math. Methods Appl. Sci. 33, No 14 (2010), 1712–1722. http://dx.doi.org/10.1002/mma.1289 [CrossRef]

  • [30] K.S. Miller, B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations. Wiley, New York (1993).

  • [31] D. Mozyrska, D.F.M. Torres, Minimal modified energy control for fractional linear control systems with the Caputo derivative. Carpathian J. Math. 26, No 2 (2010), 210–221.

  • [32] T. Odzijewicz, D.F.M. Torres, Fractional calculus of variations for double integrals. Balkan J. Geom. Appl. 16, No 2 (2011), 102–113.

  • [33] K.B. Oldham, J. Spanier, The Fractional Calculus. Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York (1974).

  • [34] A.Yu. Plakhov, D.F.M. Torres, Newton’s aerodynamic problem in media of chaotically moving particles. Mat. Sb. 196, No 6 (2005), 111–160 (In Russian); transl. in EN: Sb. Math. 196, No 5–6 (2005), 885–933.

  • [35] I. Podlubny, Fractional Differential Equations. Academic Press, San Diego, CA (1999).

  • [36] E.M. Rabei, B.S. Ababneh, Hamilton-Jacobi fractional mechanics. J. Math. Anal. Appl. 344, No 2 (2008), 799–805. http://dx.doi.org/10.1016/j.jmaa.2008.03.011 [CrossRef]

  • [37] E.M. Rabei, K.I. Nawafleh, R.S. Hijjawi, S.I. Muslih, D. Baleanu, The Hamilton formalism with fractional derivatives. J. Math. Anal. Appl. 327, No 2 (2007), 891–897. http://dx.doi.org/10.1016/j.jmaa.2006.04.076 [CrossRef]

  • [38] F. Riewe, Nonconservative Lagrangian and Hamiltonian mechanics. Phys. Rev. E (3) 53, No 2 (1996), 1890–1899. http://dx.doi.org/10.1103/PhysRevE.53.1890 [CrossRef]

  • [39] B. Ross, S.G. Samko, E.R. Love, Functions that have no first order derivative might have fractional derivatives of all orders less than one. Real Anal. Exchange 20, No 1 (1994/95), 140–157.

  • [40] S.G. Samko, A.A. Kilbas, O.I. Marichev, Fractional Integrals and Derivatives. Transl. from the 1987 Russian original, Gordon and Breach, Yverdon (1993).

  • [41] M.R. Sidi Ammi, R.A.C. Ferreira, D.F.M. Torres, Diamond-α Jensen’s inequality on time scales. J. Inequal. Appl. 2008, Art. ID 576876 (2008), 13 pp.

  • [42] V.E. Tarasov, Fractional vector calculus and fractional Maxwell’s equations. Ann. Physics 323, No 11 (2008), 2756–2778. http://dx.doi.org/10.1016/j.aop.2008.04.005 [CrossRef] [Web of Science]

  • [43] J.L. Troutman, Variational Calculus and Optimal Control. Second Ed., Springer, New York (1996). http://dx.doi.org/10.1007/978-1-4612-0737-5 [CrossRef]

  • [44] B. van Brunt, The Calculus of Variations. Springer, New York (2004).

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