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A formulation of the fractional Noether-type theorem for multidimensional Lagrangians. (English) Zbl 1259.49005
Summary: This paper presents the Euler–Lagrange equations for fractional variational problems with multiple integrals. The fractional Noether-type theorem for conservative and nonconservative generalized physical systems is proved. Our approach uses the well-known notion of the Riemann–Liouville fractional derivative.
49J20Optimal control problems with PDE (existence)
49S05Variational principles of physics
26A33Fractional derivatives and integrals (real functions)
[1]Machado, A. J. Tenreiro; Kiryakova, V.; Mainardi, F.: Recent history of fractional calculus, Commun. nonlinear sci. Numer. simul. 16, 1140-1153 (2011) · Zbl 1221.26002 · doi:10.1016/j.cnsns.2010.05.027
[2]Riewe, F.: Nonconservative Lagrangian and Hamiltonian mechanics, Phys. rev. E 53, No. 2, 1890-1899 (1996)
[3]Baleanu, D.; Muslih, S.: Lagrangian formulation of classical fields within Riemann–Liouville fractional derivatives, Phy. scripta 72, 119-121 (2005) · Zbl 1122.70360 · doi:10.1238/Physica.Regular.072a00119
[4]Kobelev, V. V.: Linear non-conservative systems with fractional damping and the derivatives of critical load parameter, GAMM-mitt. 30, No. 2, 287-299 (2007) · Zbl 1156.74017 · doi:10.1002/gamm.200790019
[5]Sha, Z.; Jing-Li, F.; Yong-Song, L.: Lagrange equations of nonholonomic systems with fractional derivatives, Chinese phys. B 19, 120301 (2010)
[6]Agrawal, O. P.: Generalized Euler–Lagrange equations and transversality conditions for fvps in terms of the Caputo derivative, J. vib. Control 13, No. 9–10, 1217-1237 (2007) · Zbl 1158.49006 · doi:10.1177/1077546307077472
[7]Malinowska, A. B.; Torres, D. F. M.: Multiobjective fractional variational calculus in terms of a combined Caputo derivative, Appl. math. Comput. 218, No. 9, 5099-5111 (2012)
[8]Odzijewicz, T.; Malinowska, A. B.; Torres, D. F. M.: Fractional variational calculus with classical and combined Caputo derivatives, Nonlinear anal. 75, No. 3, 1507-1515 (2012)
[9]Almeida, R.: Fractional variational problems with the Riesz–Caputo derivative, Appl. math. Lett. 25, 142-148 (2012)
[10]Klimek, M.: Fractional sequential mechanics-models with symmetric fractional derivatives, Czech J. Phys. 52, 1247-1253 (2002)
[11]Malinowska, A. B.; Ammi, M. R. Sidi; Torres, D. F. M.: Composition functionals in fractional calculus of variations, Commun. frac. Calc. 1, No. 1, 32-40 (2010)
[12]Stanislavsky, A. A.: Hamiltonian formalism of fractional systems, Eur. phys. J. B 49, 93-101 (2006)
[13]El-Nabulsi, R. A.; Torres, D. F. M.: 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, 1931-1939 (2007) · Zbl 1177.49036 · doi:10.1002/mma.879
[14]El-Nabulsi, A. R.: The fractional calculus of variations from extended erdelyi–kober operator, Int. J. Mod. phys. B 23, No. 16, 3349-3361 (2009)
[15]El-Nabulsi, R. A.: A periodic functional approach to the calculus of variations and the problem of time-dependent damped harmonic oscillators, Appl. math. Lett. 24, 1647-1653 (2011)
[16]El-Nabulsi, A. R.: Fractional variational problems from extended exponentially fractional integral, Appl. math. Comput. 217, No. 22, 9492-9496 (2011) · Zbl 1220.26004 · doi:10.1016/j.amc.2011.04.007
[17]El-Nabulsi, A. R.: Universal fractional Euler–Lagrange equation from a generalized fractional derivate operator, Cent. eur. J. phys. 9, No. 1, 250-256 (2011)
[18]El-Nabulsi, R. A.; Torres, D. F. M.: Fractional actionlike variational problems, J. math. Phys. 49, No. 5, 053521 (2008) · Zbl 1152.81422 · doi:10.1063/1.2929662
[19]El-Nabulsi, A. R.: Fractional field theories from multi-dimensional fractional variational problems, J. mod. Geom. meth. Mod. phys. 5, No. 6, 863-892 (2008) · Zbl 1172.26305 · doi:10.1142/S0219887808003119
[20]Cresson, J.: Fractional embedding of differential operators and Lagrangian systems, J. math. Phys. 48, No. 3, 033504 (2007) · Zbl 1137.37322 · doi:10.1063/1.2483292
[21]Cresson, J.: Inverse problem of fractional calculus of variations for partial differential equations, Commun. nonlinear sci. Numer. simul. 15, No. 4, 987-996 (2010) · Zbl 1221.35447 · doi:10.1016/j.cnsns.2009.05.036
[22]Almeida, R.; Malinowska, A. B.; Torres, D. F. M.: A fractional calculus of variations for multiple integrals with application to vibrating string, J. math. Phys. 51, No. 3, 033503 (2010)
[23]Frederico, G. S. F.; Torres, D. F. M.: A formulation of Noether’s theorem for fractional problems of the calculus of variations, J. math. Anal. appl. 334, No. 2, 834-846 (2007) · Zbl 1119.49035 · doi:10.1016/j.jmaa.2007.01.013
[24]Atanacković, T. M.; Konjik, S.; Pilipović, S.; Simic, S.: Variational problems with fractional derivatives: invariance conditions and Noether’s theorem, Nonlinear anal., theory methods appl. 71, No. 5–6, 1504-1517 (2009) · Zbl 1163.49022 · doi:10.1016/j.na.2008.12.043
[25]Frederico, G. S. F.; Torres, D. F. M.: Fractional optimal control in the sense of Caputo and the fractional Noether’s theorem, Int. math. Forum 3, No. 9-12, 479-493 (2008) · Zbl 1154.49016
[26]Frederico, G. S. F.; Torres, D. F. M.: Fractional Noether’s theorem in the Riesz–Caputo sense, Appl. math. Comput. 217, No. 3, 1023-1033 (2010) · Zbl 1200.49019 · doi:10.1016/j.amc.2010.01.100
[27]Kilbas, A. A.; Srivastava, H. M.; Trujillo, J. J.: Theory and applications of fractional differential equations, (2006)
[28]Podlubny, I.: Fractional differential equations, (1999)
[29]Klimek, M.: On solutions of linear fractional differential equations of a variational type, (2009)
[30]Love, E. R.; Young, L. C.: On fractional integration by parts, Proc. lond. Math. soc. 44, 1-35 (1938) · Zbl 0019.01006 · doi:10.1112/plms/s2-44.1.1
[31]Das, S.: Functional fractional calculus for system identification and controls, (2008)
[32]Giaquinta, M.; Hildebrandt, S.: Calculus of variations. I, (1996)
[33]Blackledge, J. M.: A generalized nonlinear model for the evolution of low frequency freak waves, Int. J. Appl. math. 41, No. 1, 06 (2011) · Zbl 1229.86002
[34]Modes, C. D.; Bhattacharya, K.; Warner, M.: Gaussian curvature from flat elastica sheets, Proc. R. Soc. A 467, No. 2128, 1121-1140 (2011) · Zbl 1219.74026 · doi:10.1098/rspa.2010.0352
[35]Momani, S.: General solutions for the space-and time-fractional diffusion-wave equation, J. phys. Sci. 10, 30-43 (2006)
[36]Murillo, J. Q.; Yuste, S. B.: An explicit difference method for solving fractional diffusion and diffusion-wave equations in the Caputo form, J. comput. Nonlinear dynam. 6, No. 2, 021014 (2011)
[37]Schneider, W. R.; Wyss, W.: Fractional difussion and wave equations, J. math. Phys. 30, 134-144 (1989) · Zbl 0692.45004 · doi:10.1063/1.528578
[38]Parsian, H.: Time fractional wave equation: Caputo sense, Adv. stud. Theor. phys. 6, No. 2, 95-100 (2012)
[39]Goldstein, H.: Classical mechanics, (1951) · Zbl 0043.18001