×

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.

MSC:

49J20 Existence theories for optimal control problems involving partial differential equations
49S05 Variational principles of physics
26A33 Fractional derivatives and integrals
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Tenreiro Machado, A. J.; Kiryakova, V.; Mainardi, F., Recent history of fractional calculus, Commun. Nonlinear Sci. Numer. Simul., 16, 1140-1153 (2011) · Zbl 1221.26002
[2] Riewe, F., Nonconservative Lagrangian and Hamiltonian mechanics, Phys. Rev. E, 53, 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
[4] Kobelev, V. V., Linear non-conservative systems with fractional damping and the derivatives of critical load parameter, GAMM-Mitt., 30, 2, 287-299 (2007) · Zbl 1156.74017
[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, 9-10, 1217-1237 (2007) · Zbl 1158.49006
[7] Malinowska, A. B.; Torres, D. F.M., Multiobjective fractional variational calculus in terms of a combined Caputo derivative, Appl. Math. Comput., 218, 9, 5099-5111 (2012) · Zbl 1238.49029
[8] Odzijewicz, T.; Malinowska, A. B.; Torres, D. F.M., Fractional variational calculus with classical and combined Caputo derivatives, Nonlinear Anal., 75, 3, 1507-1515 (2012) · Zbl 1236.49043
[9] Almeida, R., Fractional variational problems with the Riesz-Caputo derivative, Appl. Math. Lett., 25, 142-148 (2012) · Zbl 1236.49044
[10] Klimek, M., Fractional sequential mechanics-models with symmetric fractional derivatives, Czech. J. Phys., 52, 1247-1253 (2002) · Zbl 1064.70013
[11] Malinowska, A. B.; Sidi Ammi, M. R.; Torres, D. F.M., Composition functionals in fractional calculus of variations, Commun. Frac. Calc., 1, 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 \((\alpha, \beta)\), Math. Methods Appl. Sci., 30, 15, 1931-1939 (2007) · Zbl 1177.49036
[14] El-Nabulsi, A. R., The fractional calculus of variations from extended Erdelyi-Kober operator, Int. J. Mod. Phys. B, 23, 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) · Zbl 1325.37036
[16] El-Nabulsi, A. R., Fractional variational problems from extended exponentially fractional integral, Appl. Math. Comput., 217, 22, 9492-9496 (2011) · Zbl 1220.26004
[17] El-Nabulsi, A. R., Universal fractional Euler-Lagrange equation from a generalized fractional derivate operator, Cent. Eur. J. Phys., 9, 1, 250-256 (2011)
[18] El-Nabulsi, R. A.; Torres, D. F.M., Fractional actionlike variational problems, J. Math. Phys., 49, 5, 053521 (2008), 7 pp · Zbl 1152.81422
[19] El-Nabulsi, A. R., Fractional field theories from multi-dimensional fractional variational problems, J. Mod. Geom. Meth. Mod. Phys., 5, 6, 863-892 (2008) · Zbl 1172.26305
[20] Cresson, J., Fractional embedding of differential operators and Lagrangian systems, J. Math. Phys., 48, 3, 033504 (2007), 34 pp · Zbl 1137.37322
[21] Cresson, J., Inverse problem of fractional calculus of variations for partial differential equations, Commun. Nonlinear Sci. Numer. Simul., 15, 4, 987-996 (2010) · Zbl 1221.35447
[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, 3, 033503 (2010), 12 pp · Zbl 1309.49003
[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, 2, 834-846 (2007) · Zbl 1119.49035
[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, 5-6, 1504-1517 (2009) · Zbl 1163.49022
[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, 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, 3, 1023-1033 (2010) · Zbl 1200.49019
[27] Kilbas, A. A.; Srivastava, H. M.; Trujillo, J. J., Theory and Applications of Fractional Differential Equations (2006), Elsevier: Elsevier Amsterdam · Zbl 1092.45003
[28] Podlubny, I., Fractional Differential Equations (1999), Academic Press: Academic Press San Diego, CA · Zbl 0918.34010
[29] Klimek, M., On Solutions of Linear Fractional Differential Equations of a Variational Type (2009), The Publishing Office of Czestochowa University of Technology: The Publishing Office of Czestochowa University of Technology Czestochowa
[30] Love, E. R.; Young, L. C., On fractional integration by parts, Proc. Lond. Math. Soc., 44, 1-35 (1938) · Zbl 0019.01006
[31] Das, S., Functional Fractional Calculus for System Identification and Controls (2008), Springer: Springer Berlin Heidelberg · Zbl 1154.26007
[32] Giaquinta, M.; Hildebrandt, S., Calculus of Variations. I (1996), Springer: Springer Berlin
[33] Blackledge, J. M., A generalized nonlinear model for the evolution of low frequency freak waves, Int. J. Appl. Math., 41, 1, 06 (2011)
[34] Modes, C. D.; Bhattacharya, K.; Warner, M., Gaussian curvature from flat elastica sheets, Proc. R. Soc. A, 467, 2128, 1121-1140 (2011) · Zbl 1219.74026
[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, 2, 021014 (2011), 6p.
[37] Schneider, W. R.; Wyss, W., Fractional difussion and wave equations, J. Math. Phys., 30, 134-144 (1989) · Zbl 0692.45004
[38] Parsian, H., Time fractional wave equation: Caputo sense, Adv. Stud. Theor. Phys., 6, 2, 95-100 (2012) · Zbl 1247.35191
[39] Goldstein, H., Classical Mechanics (1951), Addison-Wesley Press, Inc.: Addison-Wesley Press, Inc. Cambridge, MA · Zbl 0043.18001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.