A numerically efficient variational algorithm to solve a fractional nonlinear elastic string equation

Abstract

In this work, we propose a fractional extension of the one-dimensional nonlinear vibration problem on an elastic string. The fractional problem is governed by a hyperbolic partial differential equation that considers a nonlinear function of spatial derivatives of the Riesz type and constant damping. Initial and homogeneous Dirichlet boundary conditions are imposed on a bounded interval of the real line. We show that the problem can be expressed in variational form and propose a Hamiltonian function associated to the system. We prove that the total energy of the system is constant in the absence of damping, and it is non-increasing otherwise. Some boundedness properties of the solutions are established mathematically. Motivated by these facts, we design a finite-difference discretization of the continuous model based on the use of fractional-order centered differences. The discrete scheme has also a variational structure, and we propose a discrete form of the Hamiltonian function. As the continuous counterpart, we prove rigorously that the discrete total energy is conserved in the absence of damping, and dissipated when the damping coefficient is positive. The scheme is a second-order consistent discretization of the continuous model. Moreover, we prove the stability and quadratic convergence of the numerical model using a discrete form of the energy method. We provide some computer simulations using an implementation of our scheme to illustrate the validity of the conservative/dissipative properties.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5

References

  1. 1.

    Alikhanov, A.A.: A new difference scheme for the time fractional diffusion equation. J. Comput. Phys. 280, 424–438 (2015)

    MathSciNet  MATH  Article  Google Scholar 

  2. 2.

    Ben-Yu, G., Pascual, P.J., Rodriguez, M.J., Vázquez, L.: Numerical solution of the sine-Gordon equation. Appl. Math. Comput. 18(1), 1–14 (1986)

    MathSciNet  MATH  Google Scholar 

  3. 3.

    Carrier, G.: On the non-linear vibration problem of the elastic string. Q. Appl. Math. 3(2), 157–165 (1945)

    MathSciNet  MATH  Article  Google Scholar 

  4. 4.

    Cui, M.: Convergence analysis of high-order compact alternating direction implicit schemes for the two-dimensional time fractional diffusion equation. Numer. Algor. 62(3), 383–409 (2013)

    MathSciNet  MATH  Article  Google Scholar 

  5. 5.

    Dehghan, M., Abbaszadeh, M., Mohebbi, A.: Analysis of a meshless method for the time fractional diffusion-wave equation. Numer. Algor. 73(2), 445–476 (2016)

    MathSciNet  MATH  Article  Google Scholar 

  6. 6.

    Fei, Z., Vázquez, L.: Two energy conserving numerical schemes for the sine-Gordon equation. Appl. Math. Comput. 45(1), 17–30 (1991)

    MathSciNet  MATH  Google Scholar 

  7. 7.

    Friedman, A.: Foundations of Modern Analysis. Courier Corporation, New York (1970)

    Google Scholar 

  8. 8.

    Furihata, D.: Finite-difference schemes for nonlinear wave equation that inherit energy conservation property. J. Comput. Appl. Math. 134(1), 37–57 (2001)

    MathSciNet  MATH  Article  Google Scholar 

  9. 9.

    Furihata, D., Matsuo, T.: Discrete Variational Derivative Method: A Structure-Preserving Numerical Method for Partial Differential Equations. CRC Press, New York (2010)

    Google Scholar 

  10. 10.

    Furihata, D., Sato, S., Matsuo, T.: A novel discrete variational derivative method using“average-difference methods”. JSIAM Lett. 8, 81–84 (2016)

    MathSciNet  MATH  Article  Google Scholar 

  11. 11.

    Garrappa, R.: Trapezoidal methods for fractional differential equations: Theoretical and computational aspects. Math. Comput. Simul. 110, 96–112 (2015)

    MathSciNet  Article  Google Scholar 

  12. 12.

    Garrappa, R., Moret, I., Popolizio, M.: Solving the time-fractional Schrödinger equation by Krylov projection methods. J. Comput. Phys. 293, 115–134 (2015)

    MathSciNet  MATH  Article  Google Scholar 

  13. 13.

    Garrappa, R., Moret, I., Popolizio, M.: On the time-fractional Schrodinger̈ equation: Theoretical analysis and numerical solution by matrix Mittag-Leffler functions. Comput. Math. Appl. 74(5), 977–992 (2017)

    MathSciNet  MATH  Article  Google Scholar 

  14. 14.

    Glöckle, W.G., Nonnenmacher, T.F.: A fractional calculus approach to self-similar protein dynamics. Biophys. J. 68(1), 46–53 (1995)

    Article  Google Scholar 

  15. 15.

    Huang, J., Tang, Y., Vázquez, L., Yang, J.: Two finite difference schemes for time fractional diffusion-wave equation. Numer. Algor. 64(4), 707–720 (2013)

    MathSciNet  MATH  Article  Google Scholar 

  16. 16.

    Ibrahimbegovic, A., Mamouri, S.: Energy conserving/decaying implicit time-stepping scheme for nonlinear dynamics of three-dimensional beams undergoing finite rotations. Comput. Methods Appl. Mech. Eng. 191(37-38), 4241–4258 (2002)

    MATH  Article  Google Scholar 

  17. 17.

    Ide, T.: Some energy preserving finite element schemes based on the discrete variational derivative method. Appl. Math. Comput. 175(1), 277–296 (2006)

    MathSciNet  MATH  Google Scholar 

  18. 18.

    Ide, T., Okada, M.: Numerical simulation for a nonlinear partial differential equation with variable coefficients by means of the discrete variational derivative method. J. Comput. Appl. Math. 194(2), 425–459 (2006)

    MathSciNet  MATH  Article  Google Scholar 

  19. 19.

    Ishikawa, A., Yaguchi, T.: Application of the variational principle to deriving energy-preserving schemes for the Hamilton equation. JSIAM Lett. 8, 53–56 (2016)

    MathSciNet  MATH  Article  Google Scholar 

  20. 20.

    Koeller, R.: Applications of fractional calculus to the theory of viscoelasticity. ASME, Transactions. J.f Appl. Mech. (ISSN 0021-8936)(51), 299–307 (1984)

    MathSciNet  MATH  Article  Google Scholar 

  21. 21.

    Kuramae, H., Matsuo, T.: An alternating discrete variational derivative method for coupled partial differential equations. JSIAM Lett. 4, 29–32 (2012)

    MathSciNet  MATH  Article  Google Scholar 

  22. 22.

    Laskin, N.: Fractional Schrödinger equation. Phys. Rev. E 66(5), 056108 (2002)

    MathSciNet  Article  Google Scholar 

  23. 23.

    Laursen, T., Chawla, V.: Design of energy conserving algorithms for frictionless dynamic contact problems. Int. J. Numer. Methods Eng. 40(5), 863–886 (1997)

    MathSciNet  MATH  Article  Google Scholar 

  24. 24.

    Liu, F., Zhuang, P., Turner, I., Anh, V., Burrage, K.: A semi-alternating direction method for a 2-D fractional FitzHugh–Nagumo monodomain model on an approximate irregular domain. J. Comput. Phys. 293, 252–263 (2015)

    MathSciNet  MATH  Article  Google Scholar 

  25. 25.

    Macías-Díaz, J.E.: A structure-preserving method for a class of nonlinear dissipative wave equations with Riesz space-fractional derivatives. J. Comput. Phys. 351, 40–58 (2017)

    MathSciNet  MATH  Article  Google Scholar 

  26. 26.

    Macías-Díaz, J.E.: An explicit dissipation-preserving method for Riesz space-fractional nonlinear wave equations in multiple dimensions. Commun. Nonlinear Sci. Numer. Simul. 59, 67–87 (2018)

    MathSciNet  MATH  Article  Google Scholar 

  27. 27.

    Macías-Díaz, J.E.: Numerical simulation of the nonlinear dynamics of harmonically driven Riesz-fractional extensions of the Fermi–Pasta–Ulam chains. Commun. Nonlinear Sci. Numer. Simul. 55, 248–264 (2018)

    MathSciNet  MATH  Article  Google Scholar 

  28. 28.

    Macías-Díaz, J.E.: On the solution of a Riesz space-fractional nonlinear wave equation through an efficient and energy-invariant scheme. Int. J. Comput. Math. 96 (2), 337–361 (2019)

    MathSciNet  Article  Google Scholar 

  29. 29.

    Macías-Díaz, J.E., Hendy, A.S., De Staelen, R.H.: A compact fourth-order in space energy-preserving method for Riesz space-fractional nonlinear wave equations. Appl. Math. Comput. 325, 1–14 (2018)

    MathSciNet  MATH  Google Scholar 

  30. 30.

    Macías-Díaz, J.E., Hendy, A.S., De Staelen, R.H.: A pseudo energy-invariant method for relativistic wave equations with Riesz space-fractional derivatives. Comput. Phys. Commun. 224, 98–107 (2018)

    MathSciNet  Article  Google Scholar 

  31. 31.

    Matsuo, T.: Dissipative/conservative Galerkin method using discrete partial derivatives for nonlinear evolution equations. J. Comput. Appl. Math. 218(2), 506–521 (2008)

    MathSciNet  MATH  Article  Google Scholar 

  32. 32.

    Matsuo, T., Furihata, D.: Dissipative or conservative finite-difference schemes for complex-valued nonlinear partial differential equations. J. Comput. Phys. 171(2), 425–447 (2001)

    MathSciNet  MATH  Article  Google Scholar 

  33. 33.

    Namias, V.: The fractional order Fourier transform and its application to quantum mechanics. IMA J. Appl. Math. 25(3), 241–265 (1980)

    MathSciNet  MATH  Article  Google Scholar 

  34. 34.

    Narasimha, R.: Non-linear vibration of an elastic string. J. Sound Vib. 8(1), 134–146 (1968)

    MATH  Article  Google Scholar 

  35. 35.

    Ortigueira, M.D.: Riesz potential operators and inverses via fractional centred derivatives. Int. J. Math. Math. Sci., 2006 (2006)

  36. 36.

    Pen-Yu, K.: Numerical methods for incompressible viscous flow. Sci. Sin. 20, 287–304 (1977)

    MathSciNet  Google Scholar 

  37. 37.

    Podlubny, I.: Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications, vol. 198. Elsevier (1998)

  38. 38.

    Povstenko, Y.: Theory of thermoelasticity based on the space-time-fractional heat conduction equation. Physica Scripta 2009(T136), 014017 (2009)

    Article  Google Scholar 

  39. 39.

    Romero, I., Armero, F.: An objective finite element approximation of the kinematics of geometrically exact rods and its use in the formulation of an energy–momentum conserving scheme in dynamics. Int. J. Numer. Methods Eng. 54 (12), 1683–1716 (2002)

    MathSciNet  MATH  Article  Google Scholar 

  40. 40.

    Scalas, E., Gorenflo, R., Mainardi, F.: Fractional calculus and continuous-time finance. Physica A: Stat. Mech. Appl. 284(1), 376–384 (2000)

    MathSciNet  Article  Google Scholar 

  41. 41.

    Strauss, W., Vazquez, L.: Numerical solution of a nonlinear Klein-Gordon equation. J. Comput. Phys. 28(2), 271–278 (1978)

    MathSciNet  MATH  Article  Google Scholar 

  42. 42.

    Suzuki, Y., Ohnawa, M.: Generic formalism and discrete variational derivative method for the two-dimensional vorticity equation. J. Comput. Appl. Math. 296, 690–708 (2016)

    MathSciNet  MATH  Article  Google Scholar 

  43. 43.

    Tang, Y.F., Vázquez, L., Zhang, F., Pérez-García, V.: Symplectic methods for the nonlinear Schrödinger equation. Comput. Math. Appl. 32(5), 73–83 (1996)

    MathSciNet  MATH  Article  Google Scholar 

  44. 44.

    Tarasov, V.E.: Continuous limit of discrete systems with long-range interaction. J. Phys. A Math. Gen. 39(48), 14895 (2006)

    MathSciNet  MATH  Article  Google Scholar 

  45. 45.

    Tarasov, V.E., Zaslavsky, G.M.: Conservation laws and Hamilton’s equations for systems with long-range interaction and memory. Commun. Nonlinear Sci. Numer. Simul. 13(9), 1860–1878 (2008)

    MathSciNet  MATH  Article  Google Scholar 

  46. 46.

    Toda, M.: Waves in nonlinear lattice. Prog. Theor. Phys. Suppl. 45, 174–200 (1970)

    Article  Google Scholar 

  47. 47.

    Välimäki, V., Pakarinen, J., Erkut, C., Karjalainen, M.: Discrete-time modelling of musical instruments. Reports Progress Phys. 69(1), 1 (2005)

    Article  Google Scholar 

  48. 48.

    Wang, P., Huang, C.: An energy conservative difference scheme for the nonlinear fractional Schrödinger equations. J. Comput. Phys. 293, 238–251 (2015)

    MathSciNet  MATH  Article  Google Scholar 

  49. 49.

    Wang, X., Liu, F., Chen, X.: Novel second-order accurate implicit numerical methods for the Riesz space distributed-order advection-dispersion equations. Adv. Math. Phys. 2015, 590435 (2015)

    MathSciNet  MATH  Google Scholar 

  50. 50.

    Yagdjian, K.: On the global solutions of the Higgs boson equation. Commun. Partial Diff. Equ. 37(3), 447–478 (2012)

    MathSciNet  MATH  Article  Google Scholar 

  51. 51.

    Yagdjian, K., Balogh, A.: The maximum principle and sign changing solutions of the hyperbolic equation with the Higgs potential. J. Math. Anal. Appl. 465(1), 403–422 (2018)

    MathSciNet  MATH  Article  Google Scholar 

  52. 52.

    Yaguchi, T., Matsuo, T., Sugihara, M.: The discrete variational derivative method based on discrete differential forms. J. Comput. Phys. 231(10), 3963–3986 (2012)

    MathSciNet  MATH  Article  Google Scholar 

Download references

Acknowledgments

The author wishes to thank the anonymous reviewers for their comments and criticisms. All of their comments were taken into account in the revised version of the paper, resulting in a substantial improvement with respect to the original submission. The present work reports on a set of final results of the research project “Conservative methods for fractional hyperbolic systems: analysis and applications”, funded by the National Council for Science and Technology of Mexico (CONACYT) through grant A1-S-45928.

Author information

Affiliations

Authors

Corresponding author

Correspondence to Jorge E. Macías-Díaz.

Ethics declarations

Conflict of interests

The author declares that there is no conflict of interest.

Additional information

Publisher’s note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Appendix: Matlab code

Appendix: Matlab code

The purpose of this appendix is to provide a Matlab computer implementation of the finite-difference scheme (28)–(29). Beforehand, it is worth pointing out that the implementation can be applied to solve systems like that described in Example 1 of Section 4, in which the initial conditions are given by the algebraic sum of the kink and the anti-kink of the Toda lattice, and F = 0. However, a suitable modification of this program may be used to solve different general scenarios.

The modifiable variables of the code are the following:

  • The value of L, which corresponds to the value of b in B = (0, L).

  • The value of T, which corresponds to the value of T.

  • The value of alpha, which corresponds to the value of of the fractional order of differentiation α.

  • The value of damp, which corresponds to the value of the damping coefficient γ.

  • The value of h, which corresponds to the value of h.

  • The value of tau, which corresponds to the value of τ.

As outcomes, the algorithm obtains the graphs of the numerical solutions for U and \(\mathcal {H}\) versus x and t, as well as the graph of \(\mathcal {E}\) versus t.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Macías-Díaz, J.E. A numerically efficient variational algorithm to solve a fractional nonlinear elastic string equation. Numer Algor 86, 75–102 (2021). https://doi.org/10.1007/s11075-020-00880-2

Download citation

Keywords

  • Nonlinear vibration problem
  • Fractional elastic string
  • Discrete variational derivative method
  • Numerical efficiency analysis

Mathematics Subject Classification (2010)

  • 65Mxx
  • 65Qxx