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Variational integrators for fractional Birkhoffian systems

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Abstract

In this paper, we generalize the Pfaff–Birkhoff principle to the case of containing fractional derivatives and obtain the so-called fractional Pfaff–Birkhoff–d’Alembert principle. The fractional Birkhoff equations in the sense of Riemann–Liouville fractional derivative are derived. Under the framework of variational integrators, we develop the discrete fractional Birkhoff equations by approximating the Riemann–Liouville fractional derivative with the shifted Grünwald–Letnikov fractional derivative. The resulting algebraic equations can be served as an algorithm to numerically solve the fractional Birkhoff equations. A numerical example is demonstrated to show the validity and applicability of the presented methodology.

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References

  1. Birkhoff, G.D.: Dynamical Systems. AMS College Publ, Providence (1927)

    Book  MATH  Google Scholar 

  2. Santilli, R.M.: Foundations of Theoretical Mechanics II. Springer, New York (1982)

    Google Scholar 

  3. Mei, F.X.: The Noether’s theory of Birkhoffian systems. Sci. China Ser. A 36(12), 1456–1467 (1993)

    MathSciNet  MATH  Google Scholar 

  4. Mei, F.X.: On the Birkhoffian mechanics. Int. J. Non-Linear Mech. 36(5), 817–834 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  5. Mei, F.X., Shi, R.C., Zhang, Y.F., Wu, H.B.: Dynamics of Birkhoffian System. Beijing Institute of Technology Press, Beijing (1996). (in Chinese)

    Google Scholar 

  6. Marsden, J.E., Ratiu, T.S.: Introduction to Mechanics and Symmetry. Springer, New York (1999)

    Book  MATH  Google Scholar 

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

    MATH  Google Scholar 

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

    MATH  Google Scholar 

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

    MATH  Google Scholar 

  10. Podlubny, I.: Fractional Differential Equations. Academic Press, San Diego (1998)

    MATH  Google Scholar 

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

    MATH  Google Scholar 

  12. Diethelm, K.: The Analysis of Fractional Differential Equations: An Application-Oriented Exposition Using Differential Operators of Caputo Type. Springer, Berlin (2010)

    Book  MATH  Google Scholar 

  13. Abbas, S., Benchohra, M., N’Guérékata, G.M.: Topics in Fractional Differential Equations, vol. 27. Springer Science & Business Media, New York (2012)

    MATH  Google Scholar 

  14. Zhou, Y.: Basic Theory of Fractional Differential Equations, vol. 6. World Scientific, Singapore (2014)

    Book  MATH  Google Scholar 

  15. Sabatier, J., Agrawal, O., Machado, J.T.: Advances in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineering. Springer, Dordrecht (2007)

    Book  MATH  Google Scholar 

  16. Monje, C.A., Chen, Y.Q., Vinagre, B.M., Xue, D.Y., Feliu-Batlle, V.: Fractional-Order Systems and Controls: Fundamentals and Applications. Springer, London (2010)

    Book  MATH  Google Scholar 

  17. Baleanu, D., Güvenç, Z.B., Machado, J.T.: New Trends in Nanotechnology and Fractional Calculus Applications. Springer, New York (2010)

    Book  MATH  Google Scholar 

  18. Mainardi, F.: Fractional Calculus and Waves in Linear Viscoelasticity: An Introduction to Mathematical Models. Imperial College Press, London (2010)

    Book  MATH  Google Scholar 

  19. Das, S.: Functional Fractional Calculus. Springer, Berlin (2011)

    Book  MATH  Google Scholar 

  20. Carpinteri, A., Mainardi, F.: Fractals and Fractional Calculus in Continuum Mechanics. Springer, New York (2014)

    MATH  Google Scholar 

  21. Herrmann, R.: Fractional Calculus: An Introduction for Physicists. World Scientific, Singapore (2014)

    Book  MATH  Google Scholar 

  22. Atanackovic, T.M., Pilipovic, S., Stankovic, B., Zorica, D.: Fractional Calculus with Applications in Mechanics: Wave Propagation. Impact and Variational Principles. Wiley, New York (2014)

    Book  MATH  Google Scholar 

  23. Meiss, J., Zaslavsky, G.M.: Hamiltonian Chaos and Fractional Dynamics. Oxford University Press, New York (2006)

    Google Scholar 

  24. Tarasov, V.E.: Fractional Dynamics: Applications of Fractional Calculus to Dynamics of Particles, Fields and Media. Higher Education Press, Beijing (2011)

    Google Scholar 

  25. Petras, I.: Fractional-Order Nonlinear Systems: Modeling, Analysis and Simulation. Higher Education Press, Beijing (2011)

    Book  MATH  Google Scholar 

  26. Klafter, J., Lim, S., Metzler, R.: Fractional Dynamics: Recent Advances. World Scientific, Singapore (2012)

    MATH  Google Scholar 

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

    Book  MATH  Google Scholar 

  28. Malinowska, A.B., Odzijewicc, T., Torres, D.F.: Advanced Methods in the Fractional Calculus of Variations. Springer, Cham (2015)

    Book  Google Scholar 

  29. Baleanu, D., Diethelm, K., Scalas, E., Trujillo, J.J.: Fractional Calculus: Models and Numerical Methods, vol. 3. World Scientific, Singapore (2012)

    MATH  Google Scholar 

  30. Li, C.P., Zeng, F.H.: Numerical Methods for Fractional Calculus. CRC Press, Boca Raton (2015)

    MATH  Google Scholar 

  31. Guo, B.L., Pu, X.K., Huang, F.H.: Fractional Partial Differential Equations and Their Numerical Solutions. Science Press, Beijing (2015)

    Book  MATH  Google Scholar 

  32. Riewe, F.: Nonconservative Lagrangian and Hamiltonian mechanics. Phys. Rev. E 53(2), 1890–1899 (1996)

    Article  MathSciNet  Google Scholar 

  33. Riewe, F.: Mechanics with fractional derivatives. Phys. Rev. E 55(3), 3581–3592 (1997)

    Article  MathSciNet  Google Scholar 

  34. Agrawal, O.P.: Formulation of Euler–Lagrange equations for fractional variational problems. J. Math. Anal. Appl. 272(1), 368–379 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  35. 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)

    Article  MathSciNet  MATH  Google Scholar 

  36. Agrawal, O.P.: Fractional variational calculus in terms of Riesz fractional derivatives. J. Phys. A Math. Theor. 40(24), 6287–6303 (2007)

    Article  MathSciNet  MATH  Google Scholar 

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

    Google Scholar 

  38. Muslih, S.I., Baleanu, D.: Formulation of Hamiltonian equations for fractional variational problems. Czechoslov. J. Phys. 55(6), 633–642 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  39. Muslih, S.I., Baleanu, D.: Hamiltonian formulation of systems with linear velocities within Riemann–Liouville fractional derivatives. J. Math. Anal. Appl. 304(2), 599–606 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  40. Baleanu, D., Trujillo, J.J.: On exact solutions of a class of fractional Euler–Lagrange equations. Nonlinear Dyn. 52(4), 331–335 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  41. Baleanu, D., Muslih, S.I., Taş, K.: Fractional Hamiltonian analysis of higher order derivatives systems. J. Math. Phys. 47(10), 103,503 (8pp) (2006)

    Article  MathSciNet  Google Scholar 

  42. Baleanu, D.: Fractional Hamiltonian analysis of irregular systems. Signal Process. 86(10), 2632–2636 (2006)

    Article  MATH  Google Scholar 

  43. Baleanu, D.: About fractional quantization and fractional variational principles. Commun. Nonlinear Sci. Numer. Simul. 14(6), 2520–2523 (2009)

    Article  Google Scholar 

  44. Rekhviashvili, S.S.: The Lagrange formalism with fractional derivatives in problems of mechanics. Tech. Phys. Lett. 30(1), 55–57 (2004)

    Article  Google Scholar 

  45. Cresson, J.: Fractional embedding of differential operators and Lagrangian systems. J. Math. Phys. 48(3), 033,504(34pp) (2007)

    Article  MathSciNet  Google Scholar 

  46. Atanacković, T., Konjik, S., Pilipović, S.: Variational problems with fractional derivatives: Euler–Lagrange equations. J. Phys. A 41(9), 095,201(12pp) (2008)

    MathSciNet  MATH  Google Scholar 

  47. Herzallah, M.A., Baleanu, D.: Fractional Euler–Lagrange equations revisited. Nonlinear Dyn. 69(3), 977–982 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  48. Tarasov, V.E.: Fractional generalization of gradient and Hamiltonian systems. J. Phys. A Math. Gen. 38(26), 5929–5943 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  49. Stanislavsky, A.A.: Hamiltonian formalism of fractional systems. Eur. Phys. J. B 49(1), 93–101 (2006)

    Article  MathSciNet  Google Scholar 

  50. 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(2), 891–897 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  51. Rabei, E.M., Almayteh, I., Muslih, S.I., Baleanu, D.: Hamilton–Jacobi formulation of systems within Caputo’s fractional derivative. Phys. Scr. 77(1), 015,101(6pp) (2008)

    Article  MATH  Google Scholar 

  52. Zhang, Y., Zhou, Y.: Symmetries and conserved quantities for fractional action-like Pfaffian variational problems. Nonlinear Dyn. 73(1–2), 783–793 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  53. Yan, Z., Zhang, Y.: Noether’s theorems of a fractional Birkhoffian system within Riemann–Liouville derivatives. Chin. Phys. B 23(12), 124,502(8pp) (2014)

    Article  Google Scholar 

  54. Zhai, X.H., Zhang, Y.: Noether symmetries and conserved quantities for Birkhoffian systems with time delay. Nonlinear Dyn. 77(1–2), 73–86 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  55. Zhang, Y., Zhai, X.H.: Noether symmetries and conserved quantities for fractional Birkhoffian systems. Nonlinear Dyn. 81(1–2), 469–480 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  56. Jia, Q.L., Wu, H.B., Mei, F.X.: Noether symmetries and conserved quantities for fractional forced Birkhoffian systems. J. Math. Anal. Appl. 442(2), 782–795 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  57. Zhou, Y., Zhang, Y.: Fractional Pfaff-Birkhoff principle and Birkhoff’s equations in terms of Riesz fractional derivatives. Trans. Nanjing Univ. Aeronaut. Astronaut. 31(1), 63–69 (2014)

    Google Scholar 

  58. Luo, S.K., Xu, Y.L.: Fractional Birkhoffian mechanics. Acta Mech. 226(3), 1–16 (2014)

    MathSciNet  Google Scholar 

  59. Marsden, J.E., West, M.: Discrete mechanics and variational integrators. Acta Numer. 2001(10), 357–514 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  60. Lew, A., Marsden, J.E., Ortiz, M., West, M.: An Overview of Variational Integrators. Finite Element Methods: 1970s and Beyond. CIMNE, Barcelona (2004)

    Google Scholar 

  61. Bourdin, L., Cresson, J., Greff, I., Inizan, P.: Variational integrator for fractional Euler–Lagrange equations. Appl. Numer. Math. 71, 14–23 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  62. Wang, D.L., Xiao, A.G.: Fractional variational integrators for fractional variational problems. Commun. Nonlinear Sci. Numer. Simul. 17(2), 602–610 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  63. Wang, D.L., Xiao, A.G.: Fractional variational integrators for fractional Euler–Lagrange equations with holonomic constraints. Commun. Nonlinear Sci. Numer. Simul. 18(4), 905–914 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  64. Kong, X.L., Wu, H.B., Mei, F.X.: Structure-preserving algorithms for Birkhoffian systems. J. Geom. Phys. 62(5), 1157–1166 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  65. Kong, X.L., Wu, H.B., Mei, F.X.: Discrete optimal control for Birkhoffian systems. Nonlinear Dyn. 74(3), 711–719 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  66. Kong, X.L., Wu, H.B., Mei, F.X.: Variational integrators for forced Birkhoffian systems. Appl. Math. Comput. 225, 326–332 (2013)

    MathSciNet  MATH  Google Scholar 

  67. Meerschaert, M.M., Tadjeran, C.: Finite difference approximations for fractional advection–dispersion flow equations. J. Comput. Appl. Math. 172(1), 65–77 (2004)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

This project was supported by the National Natural Science Foundation of China (Grant Nos. 10932002, 10972031 and 11672032).

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Authors and Affiliations

  1. School of Mathematics and Statistics, Beijing Institute of Technology, Beijing, 100081, China

    Lin He & Huibin Wu

  2. Beijing Key Laboratory on MCAACI, Beijing Institute of Technology, Beijing, 100081, China

    Lin He & Huibin Wu

  3. School of Aerospace Engineering, Beijing Institute of Technology, Beijing, 100081, China

    Fengxiang Mei

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  1. Lin He
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Correspondence to Lin He.

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He, L., Wu, H. & Mei, F. Variational integrators for fractional Birkhoffian systems. Nonlinear Dyn 87, 2325–2334 (2017). https://doi.org/10.1007/s11071-016-3192-4

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