메뉴 건너뛰기




Volumn 64, Issue 10, 2012, Pages 3351-3366

Generalized fractional calculus with applications to the calculus of variations

Author keywords

Calculus of variations; Coherent embedding; Fractional operators; Generalized fractional calculus; Integration by parts; Necessary optimality conditions

Indexed keywords

CALCULUS OF VARIATIONS; COHERENT EMBEDDING; FRACTIONAL CALCULUS; FRACTIONAL OPERATORS; INTEGRATION BY PARTS; NECESSARY OPTIMALITY CONDITION;

EID: 84868208170     PISSN: 08981221     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.camwa.2012.01.073     Document Type: Article
Times cited : (62)

References (44)
  • 5
    • 34250648556 scopus 로고    scopus 로고
    • A formulation of Noether's theorem for fractional problems of the calculus of variations
    • G.S.F. Frederico, and D.F.M. Torres A formulation of Noether's theorem for fractional problems of the calculus of variations J. Math. Anal. Appl. 334 2 2007 834 846
    • (2007) J. Math. Anal. Appl. , vol.334 , Issue.2 , pp. 834-846
    • Frederico, G.S.F.1    Torres, D.F.M.2
  • 6
    • 77957267794 scopus 로고    scopus 로고
    • Leitmann's direct method for fractional optimization problems
    • R. Almeida, and D.F.M. Torres Leitmann's direct method for fractional optimization problems Appl. Math. Comput. 217 3 2010 956 962
    • (2010) Appl. Math. Comput. , vol.217 , Issue.3 , pp. 956-962
    • Almeida, R.1    Torres, D.F.M.2
  • 7
    • 46249125645 scopus 로고    scopus 로고
    • Fractional conservation laws in optimal control theory
    • G.S.F. Frederico, and D.F.M. Torres Fractional conservation laws in optimal control theory Nonlinear Dynam. 53 3 2008 215 222
    • (2008) Nonlinear Dynam. , vol.53 , Issue.3 , pp. 215-222
    • Frederico, G.S.F.1    Torres, D.F.M.2
  • 8
    • 78049334398 scopus 로고    scopus 로고
    • Modified optimal energy and initial memory of fractional continuous-time linear systems
    • D. Mozyrska, and D.F.M. Torres Modified optimal energy and initial memory of fractional continuous-time linear systems Signal Process. 91 3 2011 379 385
    • (2011) Signal Process. , vol.91 , Issue.3 , pp. 379-385
    • Mozyrska, D.1    Torres, D.F.M.2
  • 9
    • 53949093156 scopus 로고    scopus 로고
    • Fractional optimal control in the sense of Caputo and the fractional Noether's theorem
    • G.S.F. Frederico, and D.F.M. Torres Fractional optimal control in the sense of Caputo and the fractional Noether's theorem Int. Math. Forum 3 9-12 2008 479 493
    • (2008) Int. Math. Forum , vol.3 , Issue.912 , pp. 479-493
    • Frederico, G.S.F.1    Torres, D.F.M.2
  • 10
    • 77950866104 scopus 로고    scopus 로고
    • Generalized natural boundary conditions for fractional variational problems in terms of the Caputo derivative
    • A.B. Malinowska, and D.F.M. Torres Generalized natural boundary conditions for fractional variational problems in terms of the Caputo derivative Comput. Math. Appl. 59 9 2010 3110 3116
    • (2010) Comput. Math. Appl. , vol.59 , Issue.9 , pp. 3110-3116
    • Malinowska, A.B.1    Torres, D.F.M.2
  • 11
    • 78049320912 scopus 로고    scopus 로고
    • Minimal modified energy control for fractional linear control systems with the Caputo derivative
    • D. Mozyrska, and D.F.M. Torres Minimal modified energy control for fractional linear control systems with the Caputo derivative Carpathian J. Math. 26 2 2010 210 221
    • (2010) Carpathian J. Math. , vol.26 , Issue.2 , pp. 210-221
    • Mozyrska, D.1    Torres, D.F.M.2
  • 12
    • 34250210234 scopus 로고    scopus 로고
    • Fractional variational calculus in terms of Riesz fractional derivatives
    • O.P. Agrawal Fractional variational calculus in terms of Riesz fractional derivatives J. Phys. A 40 24 2007 6287 6303
    • (2007) J. Phys. A , vol.40 , Issue.24 , pp. 6287-6303
    • Agrawal, O.P.1
  • 13
    • 77957259811 scopus 로고    scopus 로고
    • Fractional Noether's theorem in the Riesz-Caputo sense
    • G.S.F. Frederico, and D.F.M. Torres Fractional Noether's theorem in the Riesz-Caputo sense Appl. Math. Comput. 217 3 2010 1023 1033
    • (2010) Appl. Math. Comput. , vol.217 , Issue.3 , pp. 1023-1033
    • Frederico, G.S.F.1    Torres, D.F.M.2
  • 14
    • 79953081892 scopus 로고    scopus 로고
    • Hamilton-Jacobi formulation for systems in terms of Riesz's fractional derivatives
    • E. Rabei, I. Rawashdeh, S. Muslih, and D. Baleanu Hamilton-Jacobi formulation for systems in terms of Riesz's fractional derivatives Internat. J. Theoret. Phys. 50 5 2011 1569 1576
    • (2011) Internat. J. Theoret. Phys. , vol.50 , Issue.5 , pp. 1569-1576
    • Rabei, E.1    Rawashdeh, I.2    Muslih, S.3    Baleanu, D.4
  • 15
    • 0037047631 scopus 로고    scopus 로고
    • Stationarity-conservation laws for fractional differential equations with variable coefficients
    • DOI 10.1088/0305-4470/35/31/311, PII S0305447002353848
    • M. Klimek Stationarity-conservation laws for fractional differential equations with variable coefficients J. Phys. A 35 31 2002 6675 6693 (Pubitemid 37049354)
    • (2002) Journal of Physics A: Mathematical and General , vol.35 , Issue.31 , pp. 6675-6693
    • Klimek, M.1
  • 16
    • 30544445345 scopus 로고    scopus 로고
    • Lagrangian fractional mechanics - A noncommutative approach
    • DOI 10.1007/s10582-006-0024-7
    • M. Klimek Lagrangian fractional mechanics - a noncommutative approach Czech. J. Phys. 55 11 2005 1447 1453 (Pubitemid 43083201)
    • (2005) Czechoslovak Journal of Physics , vol.55 , Issue.11 , pp. 1447-1453
    • Klimek, M.1
  • 17
    • 84856342754 scopus 로고    scopus 로고
    • Fractional calculus of variations for a combined Caputo derivative
    • A.B. Malinowska, and D.F.M. Torres Fractional calculus of variations for a combined Caputo derivative Fract. Calc. Appl. Anal. 14 4 2011 523 537
    • (2011) Fract. Calc. Appl. Anal. , vol.14 , Issue.4 , pp. 523-537
    • Malinowska, A.B.1    Torres, D.F.M.2
  • 18
    • 77950867099 scopus 로고    scopus 로고
    • A fractional calculus of variations for multiple integrals with application to vibrating string
    • 12 pp
    • R. Almeida, A.B. Malinowska, and D.F.M. Torres A fractional calculus of variations for multiple integrals with application to vibrating string J. Math. Phys. 51 3 2010 033503 12 pp
    • (2010) J. Math. Phys. , vol.51 , Issue.3 , pp. 033503
    • Almeida, R.1    Malinowska, A.B.2    Torres, D.F.M.3
  • 19
    • 34047097242 scopus 로고    scopus 로고
    • Fractional embedding of differential operators and Lagrangian systems
    • 34 pp
    • J. Cresson Fractional embedding of differential operators and Lagrangian systems J. Math. Phys. 48 3 2007 033504 34 pp
    • (2007) J. Math. Phys. , vol.48 , Issue.3 , pp. 033504
    • Cresson, J.1
  • 20
    • 70449526839 scopus 로고    scopus 로고
    • An approach via fractional analysis to non-linearity induced by coarse-graining in space
    • G. Jumarie An approach via fractional analysis to non-linearity induced by coarse-graining in space Nonlinear Anal. RWA 11 1 2010 535 546
    • (2010) Nonlinear Anal. RWA , vol.11 , Issue.1 , pp. 535-546
    • Jumarie, G.1
  • 21
    • 79953211976 scopus 로고    scopus 로고
    • Composition functionals in fractional calculus of variations
    • A.B. Malinowska, M.R. Sidi Ammi, and D.F.M. Torres Composition functionals in fractional calculus of variations Commun. Frac. Calc. 1 2010 32 40
    • (2010) Commun. Frac. Calc. , vol.1 , pp. 32-40
    • Malinowska, A.B.1    Ammi, M.R.S.2    Torres, D.F.M.3
  • 22
    • 79961009317 scopus 로고    scopus 로고
    • Fractional variational calculus with classical and combined Caputo derivatives
    • T. Odzijewicz, A.B. Malinowska, and D.F.M. Torres Fractional variational calculus with classical and combined Caputo derivatives Nonlinear Anal. 75 3 2012 1507 1515
    • (2012) Nonlinear Anal. , vol.75 , Issue.3 , pp. 1507-1515
    • Odzijewicz, T.1    Malinowska, A.B.2    Torres, D.F.M.3
  • 23
    • 76449118673 scopus 로고    scopus 로고
    • Generalized variational problems and Euler-Lagrange equations
    • O.P. Agrawal Generalized variational problems and Euler-Lagrange equations Comput. Math. Appl. 59 5 2010 1852 1864
    • (2010) Comput. Math. Appl. , vol.59 , Issue.5 , pp. 1852-1864
    • Agrawal, O.P.1
  • 26
    • 0001655260 scopus 로고
    • Some properties of fractional integrals i
    • G.H. Hardy, and J.E. Littlewood Some properties of fractional integrals I Math. Z. 28 1928 565 606
    • (1928) Math. Z. , vol.28 , pp. 565-606
    • Hardy, G.H.1    Littlewood, J.E.2
  • 27
    • 0006972557 scopus 로고
    • Functions that have no first order derivative might have fractional derivatives of all orders less than one
    • B. Ross, S.G. Samko, and E.R. Love Functions that have no first order derivative might have fractional derivatives of all orders less than one Real Anal. Exchange 20 1 1994-1995 140 157
    • (1994) Real Anal. Exchange , vol.20 , Issue.1 , pp. 140-157
    • Ross, B.1    Samko, S.G.2    Love, E.R.3
  • 28
    • 82155197606 scopus 로고    scopus 로고
    • Fractional variational problems depending on indefinite integrals
    • R. Almeida, S. Pooseh, and D.F.M. Torres Fractional variational problems depending on indefinite integrals Nonlinear Anal. 75 3 2012 1009 1025
    • (2012) Nonlinear Anal. , vol.75 , Issue.3 , pp. 1009-1025
    • Almeida, R.1    Pooseh, S.2    Torres, D.F.M.3
  • 29
    • 77957362548 scopus 로고    scopus 로고
    • Necessary and sufficient conditions for the fractional calculus of variations with Caputo derivatives
    • R. Almeida, and D.F.M. Torres Necessary and sufficient conditions for the fractional calculus of variations with Caputo derivatives Commun. Nonlinear Sci. Numer. Simul. 16 3 2011 1490 1500
    • (2011) Commun. Nonlinear Sci. Numer. Simul. , vol.16 , Issue.3 , pp. 1490-1500
    • Almeida, R.1    Torres, D.F.M.2
  • 32
    • 21144433519 scopus 로고    scopus 로고
    • The isoperimetric problem
    • V. Blåsjö The isoperimetric problem Amer. Math. Monthly 112 6 2005 526 566 (Pubitemid 40882597)
    • (2005) American Mathematical Monthly , vol.112 , Issue.6 , pp. 526-566
    • Blasjo, V.1
  • 33
    • 33845663453 scopus 로고    scopus 로고
    • Complementary extremum principles for isoperimetric optimization problems
    • J.P. Curtis Complementary extremum principles for isoperimetric optimization problems Optim. Eng. 5 4 2004 417 430
    • (2004) Optim. Eng. , vol.5 , Issue.4 , pp. 417-430
    • Curtis, J.P.1
  • 34
    • 67650544428 scopus 로고    scopus 로고
    • Hölderian variational problems subject to integral constraints
    • R. Almeida, and D.F.M. Torres Hölderian variational problems subject to integral constraints J. Math. Anal. Appl. 359 2 2009 674 681
    • (2009) J. Math. Anal. Appl. , vol.359 , Issue.2 , pp. 674-681
    • Almeida, R.1    Torres, D.F.M.2
  • 35
    • 66249145058 scopus 로고    scopus 로고
    • Isoperimetric problems on time scales with nabla derivatives
    • R. Almeida, and D.F.M. Torres Isoperimetric problems on time scales with nabla derivatives J. Vib. Control 15 6 2009 951 958
    • (2009) J. Vib. Control , vol.15 , Issue.6 , pp. 951-958
    • Almeida, R.1    Torres, D.F.M.2
  • 36
    • 77957608394 scopus 로고    scopus 로고
    • Isoperimetric problems of the calculus of variations on time scales
    • A. Leizarowitz, B.S. Mordukhovich, I. Shafrir, A.J. Zaslavski, Contemp. Math. Amer. Math. Soc. Providence, RI
    • R.A.C. Ferreira, and D.F.M. Torres Isoperimetric problems of the calculus of variations on time scales Nonlinear Analysis and Optimization II A. Leizarowitz, B.S. Mordukhovich, I. Shafrir, A.J. Zaslavski, Contemp. Math. vol. 514 2010 Amer. Math. Soc. Providence, RI 123 131
    • (2010) Nonlinear Analysis and Optimization II , vol.514 , pp. 123-131
    • Ferreira, R.A.C.1    Torres, D.F.M.2
  • 37
    • 34547556183 scopus 로고    scopus 로고
    • Stochastic embedding of dynamical systems
    • 54 pp
    • J. Cresson, and S. Darses Stochastic embedding of dynamical systems J. Math. Phys. 48 7 2007 072703 54 pp
    • (2007) J. Math. Phys. , vol.48 , Issue.7 , pp. 072703
    • Cresson, J.1    Darses, S.2
  • 38
    • 67650547434 scopus 로고    scopus 로고
    • Constants of motion for non-differentiable quantum variational problems
    • J. Cresson, G.S.F. Frederico, and D.F.M. Torres Constants of motion for non-differentiable quantum variational problems Topol. Methods Nonlinear Anal. 33 2 2009 217 231
    • (2009) Topol. Methods Nonlinear Anal. , vol.33 , Issue.2 , pp. 217-231
    • Cresson, J.1    Frederico, G.S.F.2    Torres, D.F.M.3
  • 42
    • 67650479716 scopus 로고    scopus 로고
    • Solution of the fractional Langevin equation and the Mittag-Leffler functions
    • 8 pp
    • R.F. Camargo, A.O. Chiacchio, R. Charnet, and E. Capelas de Oliveira Solution of the fractional Langevin equation and the Mittag-Leffler functions J. Math. Phys. 50 6 2009 063507 8 pp
    • (2009) J. Math. Phys. , vol.50 , Issue.6 , pp. 063507
    • Camargo, R.F.1    Chiacchio, A.O.2    Charnet, R.3    De Oliveira, E.C.4
  • 43
    • 70349490327 scopus 로고    scopus 로고
    • Calculus of variations with fractional derivatives and fractional integrals
    • R. Almeida, and D.F.M. Torres Calculus of variations with fractional derivatives and fractional integrals Appl. Math. Lett. 22 12 2009 1816 1820
    • (2009) Appl. Math. Lett. , vol.22 , Issue.12 , pp. 1816-1820
    • Almeida, R.1    Torres, D.F.M.2


* 이 정보는 Elsevier사의 SCOPUS DB에서 KISTI가 분석하여 추출한 것입니다.