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Fractional Calculus of Variations: A Novel Way to Look At It

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Abstract

In this work we look at the original fractional calculus of variations problem in a somewhat different way. As a simple consequence, we show that a fractional generalization of a classical problem has a solution without any restrictions on the derivative-order α.

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References

  1. O.P. Agrawal, Formulation of Euler-Lagrange equations for fractional variational problems. J. Math. Anal. Appl. 272, No 1 (2002), 368–379.; DOI: 10.1016/S0022-247X(02)00180-4.

    Article  MathSciNet  MATH  Google Scholar 

  2. H.M. Ali, F. Lobo Pereira, S.M.A. Gama, A new approach to the Pontryagin maximum principle for nonlinear fractional optimal control problems. Math. Methods Appl. Sci. 39, No 13 (2016), 3640–3649.; DOI: 10.1002/mma.3811.

    Article  MathSciNet  MATH  Google Scholar 

  3. R. Almeida, R.A.C. Ferreira, D.F.M. Torres, Isoperimetric problems of the calculus of variations with fractional derivatives. Acta Math. Sci. Ser. B (Engl. Ed.) 32, No 2 (2012), 619–630; DOI: 10.1016/S0252-9602(12)60043-5.

    Article  MathSciNet  MATH  Google Scholar 

  4. T.M. Atanackovic, S. Konjik, S. Pilipovic, Variational problems with fractional derivatives: Euler-Lagrange equations. J. Phys. A 41, No 9 (2008), Art. 095201, 12pp; DOI: 10.1088/1751-8113/41/9/095201.

    Google Scholar 

  5. L. Bourdin, D. Idczak, A fractional fundamental lemma and a fractional integration by parts formula—Applications to critical points of Bolza functionals and to linear boundary value problems. Adv. Differential Equations 20, No 3-4 (2015), 213–232; https://projecteuclid.org/euclid.ade/1423055200.

    MathSciNet  MATH  Google Scholar 

  6. J. Cresson, Fractional embedding of differential operators and Lagrangian systems. J. Math. Phys. 48, No 3 (2007), Art. 033504, 34 pp; DOI: 10.1063/1.2483292.

    Google Scholar 

  7. A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and Applications of Fractional Differential Equations. North-Holland Mathematics Studies, 204, Elsevier Science B.V., Amsterdam, (2006).

    MATH  Google Scholar 

  8. M.J. Lazo, D.F.M. Torres, The DuBois-Reymond fundamental lemma of the fractional calculus of variations and an Euler-Lagrange equation involving only derivatives of Caputo. J. Optim. Theory Appl. 156, No 1 (2013), 56–67.; DOI: 10.1007/s10957-012-0203-6.

    Article  MathSciNet  MATH  Google Scholar 

  9. A. Malinowska, D.F.M. Torres, Fractional calculus of variations for a combined Caputo derivative. Fract. Calc. Appl. Anal. 14, No 4 (2011), 523–537.; DOI: 10.2478/s13540-011-0032-6; https://www.degruyter.com/view/j/fca.2011.14.issue-4/issue-files/fca.2011.14.issue-4.xml.

    Article  MathSciNet  MATH  Google Scholar 

  10. J. Mawhin, M. Willem, Critical Point Theory and Hamiltonian Systems. Applied Mathematical Sciences, 74, Springer-Verlag, New York, (1989).

  11. F. Riewe, Nonconservative Lagrangian and Hamiltonian mechanics. Phys. Rev. E 3 53, No 2 (1996), 1890–1899; DOI: 10.1103/PhysRevE.53.1890.

    Article  MathSciNet  Google Scholar 

  12. S.G. Samko, A.A. Kilbas, O.I. Marichev, Fractional Integrals and Derivatives (Transl. from the 1987 Russian original) Gordon and Breach Science Publishers, Yverdon, (1993).

    MATH  Google Scholar 

  13. B. van Brunt, The Calculus of Variations. Universitext, Springer-Verlag, New York, (2004).

    Book  MATH  Google Scholar 

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Correspondence to Rui A. C. Ferreira.

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Ferreira, R.A.C. Fractional Calculus of Variations: A Novel Way to Look At It. FCAA 22, 1133–1144 (2019). https://doi.org/10.1515/fca-2019-0059

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  • DOI: https://doi.org/10.1515/fca-2019-0059

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