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Variational Problems Involving a Caputo-Type Fractional Derivative

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Abstract

The aim of this paper is to study certain problems of calculus of variations that are dependent upon a Lagrange function on a Caputo-type fractional derivative. This type of fractional operator is a generalization of the Caputo and the Caputo–Hadamard fractional derivatives that are dependent on a real parameter \(\rho \). Sufficient and necessary conditions of the first and second order are presented. The cases of integral and holonomic constraints are also considered.

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References

  1. Riewe, F.: Nonconservative Lagrangian and Hamiltonian mechanics. Phys. Rev. E 53, 1890–1899 (1996)

    Article  MathSciNet  Google Scholar 

  2. Almeida, R., Pooseh, S., Torres, D.F.M.: Computational Methods in the Fractional Calculus of Variations. Imperial College Press, London (2015)

    Book  MATH  Google Scholar 

  3. Malinowska, A.B., Torres, D.F.M.: Introduction to the Fractional Calculus of Variations. Imperial College Press, London (2012)

    Book  MATH  Google Scholar 

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

  5. Gambo, Y.Y., Jarad, F., Baleanu, D., Abdeljawad, T.: On Caputo modification of the Hadamard fractional derivatives. Adv. Differ. Equ. 2014, Article 10 (2014). doi:10.1186/1687-1847-2014-10

  6. Jarad, F., Abdeljawad, T., Baleanu, D.: Caputo-type modification of the Hadamard fractional derivatives. Adv. Differ. Equ. 2012, Article 142 (2012). doi:10.1186/1687-1847-2012-142

  7. Katugampola, U.N.: New approach to a generalized fractional integral. Appl. Math. Comput. 218, 860–865 (2011)

    MathSciNet  MATH  Google Scholar 

  8. Katugampola, U.N.: New approach to generalized fractional derivatives. Bull. Math. Anal. Appl. 6(4), 1–15 (2014)

    MathSciNet  MATH  Google Scholar 

  9. Agrawal, O.P.: Formulation of Euler–Lagrange equations for fractional variational problems. J. Math. Anal. Appl. 272, 368–379 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  10. Atanacković, T.M., Konjik, S., Pilipović, S.: Variational problems with fractional derivatives: Euler–Lagrange equations. J. Phys. A 41, 095201 (2008)

    MathSciNet  MATH  Google Scholar 

  11. Baleanu, D., Maaraba, T., Jarad, F.: Fractional variational principles with delay. J. Phys. A 41, 315403 (2008)

    MathSciNet  MATH  Google Scholar 

  12. Malinowska, A.B., Torres, D.F.M.: Generalized natural boundary conditions for fractional variational problems in terms of the Caputo derivative. Comput. Math. Appl. 59, 3110–3116 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  13. Malinowska, A.B., Torres, D.F.M.: Multiobjective fractional variational calculus in terms of a combined Caputo derivative. Appl. Math. Comput. 218, 5099–5111 (2012)

    MathSciNet  MATH  Google Scholar 

  14. Odzijewicz, T., Malinowska, A.B., Torres, D.F.M.: Fractional variational calculus with classical and combined Caputo derivatives. Nonlinear Anal. 75, 1507–1515 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  15. Almeida, R., Torres, D.F.M.: Calculus of variations with fractional derivatives and fractional integrals. Appl. Math. Lett. 22, 1816–1820 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  16. Agrawal, O.P., Xu, Y.: Generalized vector calculus on convex domain. Commun. Nonlinear Sci. Numer. Simulat. 23, 129140 (2015)

    Article  MathSciNet  Google Scholar 

  17. Xu, Y., Agrawal, O.P.: Models and numerical solutions of generalized oscillator equations. J. Vib. Acoust. 136, 050903-1 (2014)

    Article  Google Scholar 

  18. Lazo, M.J., Torres, D.F.M.: The Legendre condition of the fractional calculus of variations. Optimization 63, 1157–1165 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  19. Anosov, D.V., Aranson, S.K., Arnold, V.I., Bronshtein, I.U., Grines, V.Z., Ilyashenko, Y.S.: Ordinary Differential Equations and Smooth Dynamical Systems, vol. 1. Encyclopaedia of Mathematical Sciences. Springer-Verlag, Berlin, Heidelberg (1988)

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Acknowledgments

The author is very grateful to Tatiana Odzijewicz, Udita Katugampola, and Paulo Almeida, for a careful and thoughtful reading of the manuscript, and to the anonymous referee and to the Editor-in-Chief, for valuable remarks and comments. This work was supported by Portuguese funds through the CIDMA—Center for Research and Development in Mathematics and Applications, and the Portuguese Foundation for Science and Technology (FCT-Fundação para a Ciência e a Tecnologia), within project UID/MAT/04106/2013.

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  1. Center for Research and Development in Mathematics and Applications (CIDMA), Department of Mathematics, University of Aveiro, 3810-193, Aveiro, Portugal

    Ricardo Almeida

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  1. Ricardo Almeida
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Correspondence to Ricardo Almeida.

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Almeida, R. Variational Problems Involving a Caputo-Type Fractional Derivative. J Optim Theory Appl 174, 276–294 (2017). https://doi.org/10.1007/s10957-016-0883-4

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  • DOI: https://doi.org/10.1007/s10957-016-0883-4

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