Nonlinear Dynamics

, Volume 69, Issue 3, pp 977–982

Fractional Euler–Lagrange equations revisited

Original Paper

Abstract

This paper presents the necessary and sufficient optimality conditions for the Euler–Lagrange fractional equations of fractional variational problems with determining in which spaces the functional must exist where the functional contains right and left fractional derivatives in the Riemann–Liouville sense and the upper bound of integration less than the upper bound of the interval of the fractional derivative. In order to illustrate our results, one example is presented.

Keywords

Fractional integral Fractional derivative Fractional calculus of variations Lipschitz spaces 

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References

  1. 1.
    Agrawal, O.P.: Formulation of Euler–Lagrange equations for fractional variational problems. J. Math. Anal. Appl. 272, 368–379 (2002) MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Agrawal, O.P.: Fractional variational calculus and the transversality conditions. J. Phys. A 39(33), 10375–10384 (2006) MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Agrawal, O.P.: Generalized Euler–Lagrange equations and transversality conditions for FVPs in terms of the Caputo derivative. J. Vib. Control 13(9–10), 1217–1237 (2007) MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Agrawal, O.P., Baleanu, D.: Hamiltonian formulation and a direct numerical scheme for Fractional Optimal Control Problems. J. Vib. Control 13, 1217–1237 (2007) MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Almeida, R., Torres, D.F.M.: Calculus of variations with fractional derivative and fractional integrals. Appl. Math. Lett. 22(12), 1816–1820 (2009) MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Almeida, R., Malinowska, A.B., Torres, D.F.M.: A fractional calculus of variations for multiple integrals with application to vibrating string. J. Math. Phys. 51(4), 033503 (2010) MathSciNetCrossRefGoogle Scholar
  7. 7.
    Almeida, R., Torres, D.F.M.: Leitmann’s direct method for fractional optimization problems. Appl. Math. Comput. 217, 956–962 (2010) MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    Almeida, R., Torres, D.F.M.: Necessary and sufficient conditions for the fractional calculus of variations with Caputo derivatives. Commun. Nonlinear Sci. Numer. Simul. 16, 1490–1500 (2011) MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    Atanackovic, T.M., Konjik, S., Pilipovic, S.: Variational problems with fractional derivatives: Euler–Lagrange equations. arxiv:1101.2961v1 [math.FA] 15 Jan (2011)
  10. 10.
    Baleanu, D., Agrawal, O.P.: Fractional Hamilton formalism within Caputo’s derivative. Czechoslov. J. Phys. 56(10–11), 1087–1092 (2006) MathSciNetCrossRefGoogle Scholar
  11. 11.
    Baleanu, D., Muslih, S.I., Rabei, E.M.: On fractional Euler–Lagrange and Hamilton equations and the fractional generalization of total time derivative. Nonlinear Dyn. 53(1–2), 67–74 (2008) MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    Chen, Y.Q., Vinagre, B.M.: A new IIR-type digital fractional order differentiator. Signal Process. 83(11), 2359–2365 (2003) MATHCrossRefGoogle Scholar
  13. 13.
    Debnath, L.: Recent applications of fractional calculus to science and engineering. Int. J. Math. Math. Sci. 54, 3413–3442 (2003) MathSciNetCrossRefGoogle Scholar
  14. 14.
    El-Nabulsi, R.A.: A fractional approach of nonconservative Lagrangian dynamics. Fizika A 14(4), 289–298 (2005) Google Scholar
  15. 15.
    Fonseca Ferreira, N.M., Duarte, F.B., Lima, M.F.M., Marcos, M.G., Tenreiro, Machado J.A.: Application of fractional calculus in the dynamical analysis and control of mechanical manipulators. Fract. Calc. Appl. Anal. 11(1), 91–113 (2008) MathSciNetMATHGoogle Scholar
  16. 16.
    Frederico, G.S.F., Torres, D.F.M.: Fractional conservation laws in optimal control theory. Nonlinear Dyn. 53(3), 215–222 (2008) MathSciNetMATHCrossRefGoogle Scholar
  17. 17.
    Herzallah, M.A.E., Baleanu, D.: Fractional-order Euler–Lagrange equations and formulation of Hamiltonian equations. Nonlinear Dyn. 58, 385–391 (2009) MathSciNetMATHCrossRefGoogle Scholar
  18. 18.
    Herzallah, M.A.E., Baleanu, D.: Fractional-order variational calculus with generalized boundary conditions. Adv. Differ. Equ. (2011). doi:10.1155/2011/357580 MathSciNetGoogle Scholar
  19. 19.
    Herzallah, M.A.E., Muslih, S.I., Baleanu, D., Rabei, E.M.: Hamilton–Jacobi and fractional like action with time scaling. Nonlinear Dyn. (2011). doi:10.1007/s11071-010-9933-x MathSciNetGoogle Scholar
  20. 20.
    Hilfer, R.: Applications of Fractional Calculus in Physics. World Sci., River Edge (2000) MATHCrossRefGoogle Scholar
  21. 21.
    Jesus, I.S., Machado, J.A.T.: Fractional control of heat diffusion systems. Nonlinear Dyn. 54(3), 263–282 (2008) MathSciNetMATHCrossRefGoogle Scholar
  22. 22.
    Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. North-Holland Mathematical Studies, vol. 204. Elsevier, Amsterdam (2006) MATHCrossRefGoogle Scholar
  23. 23.
    Klimek, M.: Fractional sequential mechanics–models with symmetric fractional derivative. Czechoslov. J. Phys. 51, 1348–1354 (2001) MathSciNetMATHCrossRefGoogle Scholar
  24. 24.
    Kulish, V.V., Lage, J.L.: Application of fractional calculus to fluid mechanics. J. Fluids Eng. 124(3), 803–806 (2002) CrossRefGoogle Scholar
  25. 25.
    Lorenzo, C.F., Hartley, T.T.: Fractional trigonometry and the spiral functions. Nonlinear Dyn. 38(1–4), 23–60 (2004) MathSciNetMATHCrossRefGoogle Scholar
  26. 26.
    Magin, R.: Fractional calculus in bioengineering. Part 1–3. Crit. Rev. Bioeng. 32 (2004) Google Scholar
  27. 27.
    Miller, K.S., Ross, B.: An Introduction to the Fractional Calculus and Fractional Differential Equations. Wiley, New York (1993) MATHGoogle Scholar
  28. 28.
    Muslih, S.I., Baleanu, D.: Hamiltonian formulation of systems with linear velocities within Riemann–Liouville fractional derivatives. J. Math. Anal. Appl. 304, 599–606 (2005) MathSciNetMATHCrossRefGoogle Scholar
  29. 29.
    Muslih, S., Baleanu, D.: Formulation of Hamiltonian equations for fractional variational problems. Czechoslov. J. Phys. 55(6), 633–642 (2005) MathSciNetMATHCrossRefGoogle Scholar
  30. 30.
    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) MathSciNetMATHCrossRefGoogle Scholar
  31. 31.
    Odzijewicz, T., Torres, D.F.M.: Calculus of variations with fractional and classical derivatives. In: Podlubny, I., Vinagre Jara, B.M., Chen, Y.Q., Feliu Batlle, V., Tejado Balsera, I. (eds.) Proceedings of FDA’10, The 4th IFAC Workshop on Fractional Differentiation and its Applications, Badajoz, Spain, October 18–20, 2010. ISBN 9788055304878. Article no. FDA10-076, 5 pp Google Scholar
  32. 32.
    Muslih, S.I., Baleanu, D.: Fractional Euler–Lagrange equations of motion in fractional space. J. Vib. Control 13(9–10), 1209–1216 (2007) MathSciNetMATHCrossRefGoogle Scholar
  33. 33.
    Podlubny, I.: Fractional Differential Equations. Acad. Press, San Diego, New York, London (1999) MATHGoogle Scholar
  34. 34.
    Rabei, E.M., Altarazi, I.A., Muslih, S.I., Baleanu, D.: Fractional WKB approximation. Nonlinear Dyn. 57(1–2), 171–175 (2009) MathSciNetMATHCrossRefGoogle Scholar
  35. 35.
    Rabei, E.M., Ababenh, B.S.: Hamilton–Jacobi fractional mechanics. J. Math. Anal. Appl. 344, 799–805 (2008) MathSciNetMATHCrossRefGoogle Scholar
  36. 36.
    Rabei, E.M., Nawafleh, K.I., Hijjawi, R.S., Muslih, S.I., Baleanu, D.: The Hamilton formalism with fractional derivatives. J. Math. Anal. Appl. 327, 891–897 (2007) MathSciNetMATHCrossRefGoogle Scholar
  37. 37.
    Samko, S.G., Kilbas, A.A., Marichev, O.I.: Fractional Integrals and Derivatives: Theory and Applications. Gordon and Breach, New York (1993) MATHGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2012

Authors and Affiliations

  1. 1.College of Science in ZulfiMajmaah UniversityRiyadhSaudi Arabia
  2. 2.Faculty of ScienceZagazig UniversityZagazigEgypt
  3. 3.Department of Mathematics and Computer ScienceÇankaya UniversityAnkaraTurkey
  4. 4.Institute of Space SciencesMagurele-BucharestRomania

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