zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
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:
49J20Optimal control problems with PDE (existence)
49S05Variational principles of physics
26A33Fractional derivatives and integrals (real functions)
References:
[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