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2013 | 42 | nr 2 | 443--458
Tytuł artykułu

A Generalized Fractional Calculus of Variations

Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
We study incommensurate fractional variational problems in terms of a generalized fractional integral with Lagrangians depending on classical derivatives and generalized fractional integrals and derivatives. We obtain necessary optimality conditions for the basic and isoperimetric problems, transversality conditions for free boundary value problems, and a generalized Noether type theorem. (original abstract)
Rocznik
Tom
42
Numer
Strony
443--458
Opis fizyczny
Twórcy
  • University of Aveiro, Portugal
  • University of Aveiro, Portugal
Bibliografia
  • Agrawal, O. P. (2010) Generalized variational problems and Euler-Lagrange equations. Comput. Math. Appl. 59 (5), 1852-1864.
  • Almeida, R., Malinowska, A. B. and Torres, D. F. M. (2010) A fractional calculus of variations for multiple integrals with application to vibrating string. J. Math. Phys. 51 (3), 033503.
  • Almeida, R., Malinowska, A. B. and Torres, D. F. M. (2012) Fractional Euler-Lagrange differential equations via Caputo derivatives. Fractional Dynamics and Control, Springer New York, Part 2, 109-118.
  • Almeida R. and Torres D. F. M. (2009) Calculus of variations with fractional derivatives and fractional integrals. Appl. Math. Lett. 22 (12), 1816-1820.
  • Bastos N. R. O., Ferreira R. A. C. and Torres, D. F. M. (2011a) Necessary optimality conditions for fractional difference problems of the calculus of variations. Discrete Contin. Dyn. Syst. 29 (2), 417-437.
  • Bastos N. R. O., Ferreira R. A. C. and Torres, D. F. M. (2011b) Discrete-time fractional variational problems. Signal Process. 91 (3), 513-524.
  • Cieśliński, J. L. And Nikiciuk, T. (2010) A direct approach to the construction of standard and non-standard Lagrangians for dissipative-like dynamical systems with variable coefficients. J. Phys. A 43 (17), 175205.
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  • Cresson, J. (2007) Fractional embedding of differential operators and Lagrangian systems. J. Math. Phys. 48 (3), 033504.
  • Frederico, G. S. F. and Torres, D. F. M. (2007a) Nonconservative Noether's theorem in optimal control. Int. J. Tomogr. Stat. 5 (W07), 109-114.
  • Frederico, G. S. F. and Torres, D. F. M. (2007b) A formulation of Noether's theorem for fractional problems of the calculus of variations. J. Math. Anal. Appl. 334 (2), 834-846.
  • Frederico, G. S. F. and Torres, D. F. M. (2008) Fractional optimal control in the sense of Caputo and the fractional Noether's theorem. Int. Math. Forum 3 (10), 479-493.
  • Ghosh, S., Choudhuri, A. and Talukdar, B. (2009) On the quantization of damped harmonic oscillator. Acta Phys. Polon. B 40 (1), 49-57.
  • Giaquinta, M. and Hildebrandt, S. (1996) Calculus of Variations. I. Springer, Berlin.
  • Gouveia, P. D. F., Torres, D. F. M. and Rocha, E. A. M. (2006) Symbolic computation of variational symmetries in optimal control. Control Cybernet. 35 (4), 831-849.
  • Helemskii, A. Ya. (2006) Lectures and Excercises on Functional Analysis. American Mathematical Society.
  • Herrera, L., Nunez, L., Patino, A. and Rago, H. (1986) A variational principle and the classical and quantum mechanics of the damped harmonic oscillator. Am. J. Phys. 54 (3), 273-277.
  • Katugampola, U. N. (2010) New approach to a generalized fractional integral. Appl. Math. Comput. 218 (3), 860-865.
  • Kilbas, A. A. and Saigo, M. (2004) GeneralizedMittag-Leffler function and generalized fractional calculus operators. Integral Transform. Spec. Func. 15 (1), 31-49.
  • Kilbas, A. A., Sruvastava, H. M. and Trujillo, J. J. (2006) Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam.
  • Klimek, M. (2009) On solutions of linear fractional differential equations of a variational type. The Publishing Office of Czestochowa University of Technology, Czestochowa.
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  • Lanezos, C. (1970) The Variational Principles of Mechanics. 4th edition, Dover, New York.
  • Malinowska, A. B. and Torres, D. F. M. (2012) Introduction to the Fractional Calculus of Variations. Imp. Coll. Press, London.
  • Menon, V. J., Chanana, N. and Singh, Y. (1997) A Fresh Look at the BCK Frictional Lagrangian. Prog. Theor. Phys. 98 (2), 321-329.
  • Mozyrska, D. and Torres, D. F. M. (2011) Modified optimal energy and initial memory of fractional continuous-time linear systems. Signal Process. 91 (3), 379-385.
  • Musielak, Z. E. (2008) Standard and non-standard Lagrangians for dissipative dynamical systems with variable coefficients. J. Phys. A 41 (5), 055205
  • Neuenschwander, D.E. (2011) Emmy Noether's Wonderful Theorem. Johns Hopkins University Press, Baltimore.
  • Odzijewicz, T., Malinowska, A.B. and Torres, D. F. M. (2012a) Fractional variational calculus with classical and combined Caputo derivatives. Nonlinear Anal. 75 (3), 1507-1515.
  • Odzijewicz, T., Malinowska, A.B. and Torres, D. F. M. (2012b) Fractional calculus of variations in terms of a generalized fractional integraf with applications to Physics. Abstr. Appl. Anal. 2012, Art. ID 871912.
  • Odzijewicz, T., Malinowska, A.B. and Torres, D. F. M. (2012c) Generalized fractional calculus with applications to the calculus of variations. Comput. Math. Appl. 64 (10), 3351-3366.
  • Odzijewicz, T., Malinowska, A.B. and Torres, D. F. M. (2013) Fractional variational calculus of variable order. In: A. Almeida, L. Castro, F.-O. Speck, Advances in Harmonic Analysis and Operator Theory, The Stefan Samko Anniversary Volume. Operator Theory: Advances and Applications 229, 291-301.
  • Odzijewicz, T. and Torres, D.F.M. (2011) Fractional calculus of variations for double integrals. Balkan J. Geom. Appl. 16 (2), 102-113.
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Typ dokumentu
Bibliografia
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