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Numerical solution of the fractional Euler-Lagrange’s equations of a thin elastica model

Abstract

In this manuscript, we investigated the fractional thin elastic system. We studied the obtained fractional Euler-Lagrange’s equations of the system numerically. The numerical study is based on Grünwald–Letnikov approach, which is power series expansion of the generating function. We present an illustrative example of the proposed numerical model of the system.

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References

  1. 1.

    Oldham, K.B., Spanier, J.: The Fractional Calculus. Academic Press, New York (1974)

    MATH  Google Scholar 

  2. 2.

    Samko, S.G., Kilbas, A.A., Marichev, O.I.: Fractional Integrals and Derivatives: Theory and Applications. Gordon and Breach, Amsterdam (1993)

    MATH  Google Scholar 

  3. 3.

    Machado, J.A.T., Kiryakova, V., Mainardi, F.: Recent history of fractional calculus. Commun. Nonlinear Sci. 16, 1140–1153 (2011)

    MATH  MathSciNet  Article  Google Scholar 

  4. 4.

    Li, C., Chen, Y.Q., Kurths, J.: Fractional calculus and its applications. Philos. Trans. R. Soc. A. 371, 20130037 (2013)

    MathSciNet  Article  Google Scholar 

  5. 5.

    Debnath, L.: Recent applications of fractional calculus to science and engineering. Int. J. Math. Math. Sci. 54, 3413–3442 (2003)

    MathSciNet  Article  Google Scholar 

  6. 6.

    Montgomery-Smith, S., Huang, W.: A numerical method to model dynamic behavior of thin inextensible elastic rods in three dimensions. J. Comput. Nonlinear Dyn. 9(1), 011015 (2014)

    Google Scholar 

  7. 7.

    David, S.A., Linares, J.L., Pallone, E.M.J.A.: Fractional order calculus: historical apologia, basic concepts and some applications. Rev. Bras. Ensino Fis. 33(4), 4302 (2011)

    Article  Google Scholar 

  8. 8.

    Kilbas, A.A., Srivastava, H.M., Trujiilo, J.J.: Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam (2006)

    MATH  Google Scholar 

  9. 9.

    Miller, K.S., Ross, B.: An Introduction to the Fractional Calculus and Fractional Differential Equations. Wiley, New York (1993)

    MATH  Google Scholar 

  10. 10.

    Podlubny, I.: Fractional Differential Equations. Academic Press, San Diego (1999)

    MATH  Google Scholar 

  11. 11.

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

    MathSciNet  Article  Google Scholar 

  12. 12.

    Riewe, F.: Mechanics with fractional derivatives. Phys. Rev. E 55, 3581–3592 (1997)

    MathSciNet  Article  Google Scholar 

  13. 13.

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

    MATH  MathSciNet  Article  Google Scholar 

  14. 14.

    Klimek, M.: Fractional sequential mechanics—models with symmetric fractional derivative. Czechoslov. J. Phys. 51(12), 1348–1354 (2001)

    MATH  MathSciNet  Article  Google Scholar 

  15. 15.

    Baleanu, D., Avkar, T.: Lagrangians with linear velocities within Riemann–Liouville fractional derivatives. Nuovo Cimento, B 119, 73–79 (2004)

    Google Scholar 

  16. 16.

    Klimek, M.: On Solutions of Linear Fractional Differential Equations of a Variational Type. Czestochowa University of Technology, Czestochowa (2009)

    Google Scholar 

  17. 17.

    Diethelm, K., Ford, N.J.: Numerical solution of the Bagley-Torvik equation. BIT Numer. Math. 42(3), 490 (2002)

    MATH  MathSciNet  Google Scholar 

  18. 18.

    Ray, S.S., Bera, R.K.: Analytical solution of the Bagley Torvik equation by Adomian decomposition method. Appl. Math. Comput. 168(1), 398 (2005)

    MATH  MathSciNet  Article  Google Scholar 

  19. 19.

    Momani, S., Al-Khaled, K.: Numerical solutions for systems of fractional differential equations by the decomposition method. Appl. Math. Comput. 162(3), 1351 (2005)

    MATH  MathSciNet  Article  Google Scholar 

  20. 20.

    Podlubny, I.: Matrix approach to discrete fractional calculus. Fract. Calc. Appl. Anal. 3(4), 359 (2010)

    MathSciNet  Google Scholar 

  21. 21.

    Podlubny, I., Chechkin, A.V., Skovranek, T., Chen, Y.Q., Vinagre, B.: Matrix approach to discrete fractional calculus II: partial fractional differential equations. J. Comput. Phys. 228(8), 3137 (2009)

    MATH  MathSciNet  Article  Google Scholar 

  22. 22.

    Baleanu, D., Petras, I., Asad, J.H., Pilar, M.: Velasco, fractional Pais–Uhlenbeck oscillator. Int. J. Theor. Phys. 51(4), 1253–1258 (2012)

    MATH  Article  Google Scholar 

  23. 23.

    Baleanu, D., Asad, J.H., Petras, I.: Fractional- order two- electric pendulum. Rom. Rep. Phys. 64(4), 907–914 (2012)

    Google Scholar 

  24. 24.

    Baleanu, D., Asad, J.H., Petras, I., Elagan, S., Bilgen, A.: Fractional Euler-Lagrange equation of Caldirola-Kanai oscillator. Rom. Rep. Phys. 64, 1171–1177 (2012)

    Google Scholar 

  25. 25.

    Baleanu, D., Asad, J.H., Petras, I.: Fractional Bateman–Feshbach Tikochinsky oscillator. Commun. Theo. Phys. 61(2), 221–225 (2014)

    Article  Google Scholar 

  26. 26.

    Euler, L.: Methodus inveniendi lineas curvas maximi minimive proprietate gaudentes, sivesolutio problematis isoperimetrici lattissimo sensu accepti, chapter Additamentum 1. eulerarchive.org, E065 (1744)

  27. 27.

    Levien, R.: The elastica: a mathematical history (2008). http://levien.com/phd/elastica_hist.pdf

  28. 28.

    Burden, R.L., Faires, J.D.: Numerical Analysis, 3rd edn. Prindle, Weber and Schmidt, Belmont, Boston (1985)

    Google Scholar 

  29. 29.

    Quarteroni, A., Sacco, R., Saleri, F.: Numerical Mathematics. Springer, New York (2000)

    Google Scholar 

  30. 30.

    Huang, W.: Numerical Recipes in C. University Press, Cambridge (2002)

    Google Scholar 

  31. 31.

    Allgower, E.L., Georg, K.: Introduction to Numerical Continuation Methods. SIAM, Philadelphia (2003)

    MATH  Book  Google Scholar 

  32. 32.

    Santillan, S.A.: Analysis of the Elastica with Applications to Vibration Isolation. In: Ph.D. Thesis, Duke University (2007). http://dukespace.lib.duke.edu/dspace/handle/10161/180

  33. 33.

    Cusumano, J.P.: Low-dimensional, Chaotic, Non-planar Motions of the Elastic. In: Ph.D. Thesis, Cornell University, NY (1990)

  34. 34.

    Pak, C.H., Rand, R.H., Moon, F.C.: Free vibrations of a thin elastica by normal modes. Nonlinear Dyn. 3, 347–364 (1992)

    Article  Google Scholar 

  35. 35.

    Rabei, E.M., Nawafleh, K.I., Hijjawi, R.S., Muslih, S.I., Baleanu, D.: The Hamilton formalism with fractional derivatives. J. Math. Appl. Anal. 327, 891–897 (2007)

    MATH  MathSciNet  Article  Google Scholar 

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Acknowledgments

The work of Ivo Petras was supported in part by Grants VEGA: 1/0552/14, 1/0908/15, and APVV-0482-11.

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Baleanu, D., Asad, J.H. & Petras, I. Numerical solution of the fractional Euler-Lagrange’s equations of a thin elastica model. Nonlinear Dyn 81, 97–102 (2015). https://doi.org/10.1007/s11071-015-1975-7

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Keywords

  • Riemann–Liouville derivatives
  • Vibration
  • Thin elastica
  • Fractional Euler-Lagrange equations
  • Grünwald–Letnikov approach