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Volumn 71, Issue 5-6, 2009, Pages 1504-1517

Variational problems with fractional derivatives: Invariance conditions and Nöther's theorem

Author keywords

Approximations; Conservation laws; Infinitesimal criterion; N ther's theorem; Riemann Liouville fractional derivative; Variational problem; Variational symmetry

Indexed keywords

APPROXIMATIONS; CONSERVATION LAWS; INFINITESIMAL CRITERION; RIEMANN-LIOUVILLE FRACTIONAL DERIVATIVE; VARIATIONAL PROBLEM; VARIATIONAL SYMMETRY;

EID: 67349097718     PISSN: 0362546X     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.na.2008.12.043     Document Type: Article
Times cited : (144)

References (41)
  • 1
    • 33745870156 scopus 로고    scopus 로고
    • Fractional Hamiltonian analysis of irregular systems
    • Baleanu D. Fractional Hamiltonian analysis of irregular systems. Signal Process. 86 (2006) 2632-2636
    • (2006) Signal Process. , vol.86 , pp. 2632-2636
    • Baleanu, D.1
  • 2
    • 4043139312 scopus 로고    scopus 로고
    • Lagrangians with linear velocities within Riemann-Liouville fractional derivatives
    • Baleanu D., and Avkar T. Lagrangians with linear velocities within Riemann-Liouville fractional derivatives. Il Nuovo Cim. B 119 (2004) 73-79
    • (2004) Il Nuovo Cim. B , vol.119 , pp. 73-79
    • Baleanu, D.1    Avkar, T.2
  • 3
    • 14844283120 scopus 로고    scopus 로고
    • Hamiltonian formulation of systems with linear velocities within Riemann-Liouville fractional derivatives
    • Baleanu D., and Muslih S.I. Hamiltonian formulation of systems with linear velocities within Riemann-Liouville fractional derivatives. J. Math. Anal. Appl. 304 (2005) 599-606
    • (2005) J. Math. Anal. Appl. , vol.304 , pp. 599-606
    • Baleanu, D.1    Muslih, S.I.2
  • 4
    • 1342310800 scopus 로고    scopus 로고
    • The Lagrange formalism with fractional derivatives in problems of mechanics
    • Rekhviashvili S.Sh. The Lagrange formalism with fractional derivatives in problems of mechanics. Tech. Phys. Lett. 30 (2004) 33-37
    • (2004) Tech. Phys. Lett. , vol.30 , pp. 33-37
    • Rekhviashvili, S.Sh.1
  • 6
    • 0039849267 scopus 로고    scopus 로고
    • Inferring conservation laws in particle physics: A case study in the problem of induction
    • Schulte O. Inferring conservation laws in particle physics: A case study in the problem of induction. Brit. J. Philos. Sci 51 (2000) 771-806
    • (2000) Brit. J. Philos. Sci , vol.51 , pp. 771-806
    • Schulte, O.1
  • 9
    • 34247636164 scopus 로고    scopus 로고
    • Diffusion wave equation with two fractional derivatives of different order
    • Atanacković T.M., Pilipović S., and Zorica D. Diffusion wave equation with two fractional derivatives of different order. J. Phys. A 40 (2007) 5319-5333
    • (2007) J. Phys. A , vol.40 , pp. 5319-5333
    • Atanacković, T.M.1    Pilipović, S.2    Zorica, D.3
  • 10
    • 34248159960 scopus 로고    scopus 로고
    • Convergence of the Grünwald-Letnikov scheme for time-fractional diffusion
    • Gorenflo R., and Abdel-Rehim E.A. Convergence of the Grünwald-Letnikov scheme for time-fractional diffusion. J. Comput. Appl. Math. 205 (2007) 871-881
    • (2007) J. Comput. Appl. Math. , vol.205 , pp. 871-881
    • Gorenflo, R.1    Abdel-Rehim, E.A.2
  • 11
    • 30644458726 scopus 로고    scopus 로고
    • Wright functions as scale-invariant solutions of the diffusion-wave equation
    • Gorenflo R., Luchko Y., and Mainardi F. Wright functions as scale-invariant solutions of the diffusion-wave equation. J. Math. Sci. (NY) 132 (2006) 614-628
    • (2006) J. Math. Sci. (NY) , vol.132 , pp. 614-628
    • Gorenflo, R.1    Luchko, Y.2    Mainardi, F.3
  • 12
    • 34047155650 scopus 로고    scopus 로고
    • Continuous-time random walk and parametric subordination in fractional diffusion
    • Gorenflo R., Mainardi F., and Vivoli A. Continuous-time random walk and parametric subordination in fractional diffusion. Chaos Solitons Fractals 34 (2007) 87-103
    • (2007) Chaos Solitons Fractals , vol.34 , pp. 87-103
    • Gorenflo, R.1    Mainardi, F.2    Vivoli, A.3
  • 13
    • 0000103589 scopus 로고    scopus 로고
    • Discrete and continuous random walk models for space-time fractional diffusion
    • Gorenflo R., Vivoli A., and Mainardi F. Discrete and continuous random walk models for space-time fractional diffusion. J. Comput. Appl. Math. 118 (2000) 175-191
    • (2000) J. Comput. Appl. Math. , vol.118 , pp. 175-191
    • Gorenflo, R.1    Vivoli, A.2    Mainardi, F.3
  • 14
    • 0035480489 scopus 로고    scopus 로고
    • Fractional master equation: Non-standard analysis and Liouville-Riemann derivative
    • Jumarie G. Fractional master equation: Non-standard analysis and Liouville-Riemann derivative. Chaos Solitons Fractals 12 (2001) 2577-2587
    • (2001) Chaos Solitons Fractals , vol.12 , pp. 2577-2587
    • Jumarie, G.1
  • 15
    • 29144520230 scopus 로고    scopus 로고
    • Fractionalization of the complex-valued Brownian motion of order n using Riemann-Liouville derivative, Applications to mathematical finance and stochastic mechanics
    • Jumarie G. Fractionalization of the complex-valued Brownian motion of order n using Riemann-Liouville derivative, Applications to mathematical finance and stochastic mechanics. Chaos Solitons Fractals 28 (2006) 1285-1305
    • (2006) Chaos Solitons Fractals , vol.28 , pp. 1285-1305
    • Jumarie, G.1
  • 16
    • 33745839870 scopus 로고    scopus 로고
    • Non-local continuum mechanics and fractional calculus
    • Lazopoulos K.A. Non-local continuum mechanics and fractional calculus. Mech. Res. Commun. 33 (2006) 753-757
    • (2006) Mech. Res. Commun. , vol.33 , pp. 753-757
    • Lazopoulos, K.A.1
  • 17
    • 34247367101 scopus 로고    scopus 로고
    • Some aspects of fractional diffusion equations of single and distributed order
    • Mainardi F., Pagnini G., and Gorenflo R. Some aspects of fractional diffusion equations of single and distributed order. Appl. Math. Comput. 187 (2007) 295-305
    • (2007) Appl. Math. Comput. , vol.187 , pp. 295-305
    • Mainardi, F.1    Pagnini, G.2    Gorenflo, R.3
  • 18
    • 0035421619 scopus 로고    scopus 로고
    • Relaxation properties of fractional derivative viscoelasticity models
    • Sakakibara S. Relaxation properties of fractional derivative viscoelasticity models. Nonlinear Anal. 47 (2001) 5449-5454
    • (2001) Nonlinear Anal. , vol.47 , pp. 5449-5454
    • Sakakibara, S.1
  • 19
    • 34047124843 scopus 로고    scopus 로고
    • The stochastic nature of complexity evolution in the fractional systems
    • Stanislavsky A.A. The stochastic nature of complexity evolution in the fractional systems. Chaos Solitons Fractals 34 (2007) 51-61
    • (2007) Chaos Solitons Fractals , vol.34 , pp. 51-61
    • Stanislavsky, A.A.1
  • 20
    • 0000735791 scopus 로고    scopus 로고
    • Nonconservative Lagrangian and Hamiltonian mechanics
    • Riewe F. Nonconservative Lagrangian and Hamiltonian mechanics. Phys. Rev. E 53 (1996) 1890-1899
    • (1996) Phys. Rev. E , vol.53 , pp. 1890-1899
    • Riewe, F.1
  • 21
    • 4243530410 scopus 로고    scopus 로고
    • Mechanics with fractional derivatives
    • Riewe F. Mechanics with fractional derivatives. Phys. Rev. E 55 (1997) 3581-3592
    • (1997) Phys. Rev. E , vol.55 , pp. 3581-3592
    • Riewe, F.1
  • 22
    • 0036701004 scopus 로고    scopus 로고
    • Formulation of Euler-Lagrange equations for fractional variational problems
    • Agrawal O.P. Formulation of Euler-Lagrange equations for fractional variational problems. J. Math. Anal. Appl. 272 (2002) 368-379
    • (2002) J. Math. Anal. Appl. , vol.272 , pp. 368-379
    • Agrawal, O.P.1
  • 23
    • 33746876366 scopus 로고    scopus 로고
    • Fractional variational calculus and the transversality conditions
    • Agrawal O.P. Fractional variational calculus and the transversality conditions. J. Phys. A 39 (2006) 10375-10384
    • (2006) J. Phys. A , vol.39 , pp. 10375-10384
    • Agrawal, O.P.1
  • 24
    • 34250210234 scopus 로고    scopus 로고
    • Fractional variational calculus in terms of Riesz fractional derivatives
    • Agrawal O.P. Fractional variational calculus in terms of Riesz fractional derivatives. J. Phys. A 40 (2007) 6287-6303
    • (2007) J. Phys. A , vol.40 , pp. 6287-6303
    • Agrawal, O.P.1
  • 25
    • 34250648556 scopus 로고    scopus 로고
    • A formulation of Nöther's theorem for fractional problems of the calculus of variations
    • Frederico G.S.F., and Torres D.F.M. A formulation of Nöther's theorem for fractional problems of the calculus of variations. J. Math. Anal. Appl. 334 (2007) 834-846
    • (2007) J. Math. Anal. Appl. , vol.334 , pp. 834-846
    • Frederico, G.S.F.1    Torres, D.F.M.2
  • 26
    • 41149152515 scopus 로고    scopus 로고
    • Conservation laws and Hamilton's equations for systems with long-range interaction and memory
    • Tarasov V.E., and Zaslavsky G.M. Conservation laws and Hamilton's equations for systems with long-range interaction and memory. Commun. Nonlinear Sci. Numer. Simul. 13 (2008) 1860-1878
    • (2008) Commun. Nonlinear Sci. Numer. Simul. , vol.13 , pp. 1860-1878
    • Tarasov, V.E.1    Zaslavsky, G.M.2
  • 27
    • 33845944224 scopus 로고    scopus 로고
    • On a nonlinear distributed order fractional differential equation
    • Atanacković T.M., Oparnica Lj., and Pilipović S. On a nonlinear distributed order fractional differential equation. J. Math. Anal. Appl. 328 (2007) 590-608
    • (2007) J. Math. Anal. Appl. , vol.328 , pp. 590-608
    • Atanacković, T.M.1    Oparnica, Lj.2    Pilipović, S.3
  • 28
    • 34547619943 scopus 로고    scopus 로고
    • On a class of differential equations with left and right fractional derivatives
    • Atanacković T.M., and Stanković B. On a class of differential equations with left and right fractional derivatives. Z. Angew. Math. Mech. 87 (2007) 537-546
    • (2007) Z. Angew. Math. Mech. , vol.87 , pp. 537-546
    • Atanacković, T.M.1    Stanković, B.2
  • 29
    • 1842530950 scopus 로고    scopus 로고
    • Extending Bauer's corollary to fractional derivatives
    • Dreisigmeyer D.W., and Young P.M. Extending Bauer's corollary to fractional derivatives. J. Phys. A 37 (2003) 117-121
    • (2003) J. Phys. A , vol.37 , pp. 117-121
    • Dreisigmeyer, D.W.1    Young, P.M.2
  • 30
    • 0043132218 scopus 로고    scopus 로고
    • Nonconservative Lagrangian mechanics: A generalized function approach
    • Dreisigmeyer D.W., and Young P.M. Nonconservative Lagrangian mechanics: A generalized function approach. J. Phys. A: Math. Gen. 36 (2003) 8297-8310
    • (2003) J. Phys. A: Math. Gen. , vol.36 , pp. 8297-8310
    • Dreisigmeyer, D.W.1    Young, P.M.2
  • 31
    • 3142686670 scopus 로고    scopus 로고
    • Fractional variational principle in macroscopic picture
    • Gaies A., and El-Akrmi A. Fractional variational principle in macroscopic picture. Phys. Scripta 70 (2004) 7-10
    • (2004) Phys. Scripta , vol.70 , pp. 7-10
    • Gaies, A.1    El-Akrmi, A.2
  • 32
    • 0035737230 scopus 로고    scopus 로고
    • Fractional sequential mechanics - model with symmetric fractional derivative
    • Klimek M. Fractional sequential mechanics - model with symmetric fractional derivative. Czech. J. Phys. 51 (2001) 1348-1354
    • (2001) Czech. J. Phys. , vol.51 , pp. 1348-1354
    • Klimek, M.1
  • 33
    • 33748425302 scopus 로고    scopus 로고
    • Numerical comparison of methods for solving linear differential equations of fractional order
    • Momani S., and Odibat Z. Numerical comparison of methods for solving linear differential equations of fractional order. Chaos Solitons Fractals 31 (2007) 1248-1255
    • (2007) Chaos Solitons Fractals , vol.31 , pp. 1248-1255
    • Momani, S.1    Odibat, Z.2
  • 34
    • 30344464250 scopus 로고    scopus 로고
    • Application of variational iteration method to nonlinear differential equations of fractional order
    • Odibat Z., and Momani S. Application of variational iteration method to nonlinear differential equations of fractional order. Int. J. Nonlinear Sci. Numer. Simul. 7 (2006) 27-34
    • (2006) Int. J. Nonlinear Sci. Numer. Simul. , vol.7 , pp. 27-34
    • Odibat, Z.1    Momani, S.2
  • 36
    • 46249125645 scopus 로고    scopus 로고
    • Fractional conservation laws in optimal control theory
    • Frederico G.S.F., and Torres D.F.M. Fractional conservation laws in optimal control theory. Nonlinear Dyn. 53 (2008) 215-222
    • (2008) Nonlinear Dyn. , vol.53 , pp. 215-222
    • Frederico, G.S.F.1    Torres, D.F.M.2
  • 37
    • 53949093156 scopus 로고    scopus 로고
    • Fractional optimal control in the sense of Caputo and the fractional Nöther's theorem
    • Frederico G.S.F., and Torres D.F.M. Fractional optimal control in the sense of Caputo and the fractional Nöther's theorem. Int. Math. Forum 3 (2008) 479-493
    • (2008) Int. Math. Forum , vol.3 , pp. 479-493
    • Frederico, G.S.F.1    Torres, D.F.M.2
  • 38
    • 41849118424 scopus 로고    scopus 로고
    • Variational problems with fractional derivatives: Euler-Lagrange equations
    • Atanacković T.M., Konjik S., and Pilipović S. Variational problems with fractional derivatives: Euler-Lagrange equations. J. Phys. A 41 (2008) 095201
    • (2008) J. Phys. A , vol.41 , pp. 095201
    • Atanacković, T.M.1    Konjik, S.2    Pilipović, S.3
  • 39
    • 0942287926 scopus 로고
    • Springer, New York
    • Olver P.J. Applications of Lie Groups to Differential Equations. second ed. Graduate Texts in Mathematics vol. 107 (1993), Springer, New York
    • (1993) Graduate Texts in Mathematics , vol.107
    • Olver, P.J.1


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