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Journal of Mathematical Chemistry

, Volume 57, Issue 5, pp 1394–1412 | Cite as

An efficient Hamiltonian numerical model for a fractional Klein–Gordon equation through weighted-shifted Grünwald differences

  • Ahmed S. Hendy
  • Jorge E. Macías-DíazEmail author
Original Paper

Abstract

In this work, we investigate numerically a nonlinear wave equation with fractional derivatives of the Riesz type in space. As opposed to previously published papers which employed fractional centered differences, the present approach is based on the use of weighted and shifted Grünwald difference operators. The mathematical model has an associated energy function which is preserved under suitable parameter conditions. In this manuscript, we propose a discrete energy function that estimates the continuous counterpart and which is preserved under the same conditions. As some of the main result of this work, we show that the method is stable and second-order convergent. Moreover, we establish that the technique is quadratically consistent, and we prove the existence and uniqueness of solutions of the numerical model for arbitrary initial conditions. Some numerical results are provided in order to confirm the quadratic order of convergence of the method.

Keywords

Fractional wave equation Riesz space-fractional equations Weighted and shifted Grünwald differences Hamiltonian numerical model Convergence and stability 

Notes

Acknowledgements

For the first author, this work was supported by Government of the Russian Federation Resolution 211 of March 16, 2013. The authors wish to thank the anonymous reviewers and the editor in charge of handling this paper for their suggestions and criticisms. Their comments helped substantially in improving the quality of this manuscript.

References

  1. 1.
    M. Abbaszade, M. Mohebbi, Fourth-order numerical solution of a fractional PDE with the nonlinear source term in the electroanalytical chemistry. Iran. J. Math. Chem. 3(2), 195–220 (2012)Google Scholar
  2. 2.
    G.D. Akrivis, Finite difference discretization of the cubic Schrödinger equation. IMA J. Numer. Anal. 13(1), 115–124 (1993)CrossRefGoogle Scholar
  3. 3.
    G. Alfimov, T. Pierantozzi, L. Vázquez, Numerical study of a fractional sine-Gordon equation. Fract. Differ. Appl. 4, 153–162 (2004)Google Scholar
  4. 4.
    O. Belhamiti, B. Absar, A numerical study of fractional order reverse osmosis desalination model using legendre wavelet approximation. Iran. J. Math. Chem. 8(4), 345–364 (2017)Google Scholar
  5. 5.
    G. Ben-Yu, P.J. Pascual, M.J. Rodriguez, L. Vázquez, Numerical solution of the sine-Gordon equation. Appl. Math. Comput. 18(1), 1–14 (1986)Google Scholar
  6. 6.
    F.P. Benetti, A.C. Ribeiro-Teixeira, R. Pakter, Y. Levin, Nonequilibrium stationary states of 3d self-gravitating systems. Phys. Rev. Lett. 113(10), 100602 (2014)PubMedCrossRefGoogle Scholar
  7. 7.
    G.N.C. Beukam, J.P. Nguenang, A. Trombettoni, T. Dauxois, R. Khomeriki, S. Ruffo, Fermi-pasta-ulam chains with harmonic and anharmonic long-range interactions. Commun. Nonlinear Sci. Numer. Simul. 60(1), 115–127 (2018)Google Scholar
  8. 8.
    A. Bhrawy, M. Zaky, Highly accurate numerical schemes for multi-dimensional space variable-order fractional Schrödinger equations. Comput. Math. Appl. 73(6), 1100–1117 (2017)CrossRefGoogle Scholar
  9. 9.
    A. Campa, T. Dauxois, S. Ruffo, Statistical mechanics and dynamics of solvable models with long-range interactions. Phys. Rep. 480(3), 57–159 (2009)CrossRefGoogle Scholar
  10. 10.
    C. Çelik, M. Duman, Crank-Nicolson method for the fractional diffusion equation with the Riesz fractional derivative. J. Comput. Phys. 231(4), 1743–1750 (2012)CrossRefGoogle Scholar
  11. 11.
    H. Christodoulidi, T. Bountis, L. Drossos, Numerical integration of variational equations for Hamiltonian systems with long range interactions. Appl. Numer. Math. 104, 158–165 (2016)CrossRefGoogle Scholar
  12. 12.
    H. Christodoulidi, T. Bountis, C. Tsallis, L. Drossos, Dynamics and statistics of the Fermi–Pasta–Ulam \(\beta \)-model with different ranges of particle interactions. J. Stat. Mech. Theory Exp. 2016(12), 123206 (2016)CrossRefGoogle Scholar
  13. 13.
    H. Christodoulidi, C. Tsallis, T. Bountis, Fermi–Pasta–Ulam model with long-range interactions: dynamics and thermostatistics. Europhys. Lett. 108(4), 40006 (2014)CrossRefGoogle Scholar
  14. 14.
    A. Coronel-Escamilla, J. Gómez-Aguilar, E. Alvarado-Méndez, G. Guerrero-Ramírez, R. Escobar-Jiménez, Fractional dynamics of charged particles in magnetic fields. Int. J. Mod. Phys. C 27(08), 1650084 (2016)CrossRefGoogle Scholar
  15. 15.
    K. Diethelm, A.D. Freed, On the solution of nonlinear fractional-order differential equations used in the modeling of viscoplasticity, in Scientific Computing in Chemical Engineering II, ed. by F. Keil, W. Mackens, H. Voss, J. Werther (Springer, 1999), pp. 217–224Google Scholar
  16. 16.
    Z. Fei, L. Vázquez, Two energy conserving numerical schemes for the sine-Gordon equation. Appl. Math. Comput. 45(1), 17–30 (1991)Google Scholar
  17. 17.
    G.S. Frederico, D.F. Torres, Fractional conservation laws in optimal control theory. Nonlinear Dyn. 53(3), 215–222 (2008)CrossRefGoogle Scholar
  18. 18.
    A. Friedman, Foundations of Modern Analysis (Courier Corporation, New York, 1970)Google Scholar
  19. 19.
    D. Furihata, Finite-difference schemes for nonlinear wave equation that inherit energy conservation property. J. Comput. Appl. Math. 134(1), 37–57 (2001)CrossRefGoogle Scholar
  20. 20.
    D. Furihata, T. Matsuo, Discrete Variational Derivative Method: A Structure-Preserving Numerical Method for Partial Differential Equations (CRC Press, New York, 2010)CrossRefGoogle Scholar
  21. 21.
    V. Gafiychuk, B.Y. Datsko, Pattern formation in a fractional reaction-diffusion system. Physica A Stat. Mech. Appl. 365(2), 300–306 (2006)CrossRefGoogle Scholar
  22. 22.
    E. Kharazmi, M. Zayernouri, G.E. Karniadakis, A Petrov–Galerkin spectral element method for fractional elliptic problems. Comput. Methods Appl. Mech. Eng. 324, 512–536 (2017)CrossRefGoogle Scholar
  23. 23.
    N. Korabel, G.M. Zaslavsky, V.E. Tarasov, Coupled oscillators with power–law interaction and their fractional dynamics analogues. Commun. Nonlinear Sci. Numer. Simul. 12(8), 1405–1417 (2007)CrossRefGoogle Scholar
  24. 24.
    N. Laskin, Fractional Schrödinger equation. Phys. Rev. E 66(5), 056108 (2002)CrossRefGoogle Scholar
  25. 25.
    I. Latella, A. Pérez-Madrid, A. Campa, L. Casetti, S. Ruffo, Long-range interacting systems in the unconstrained ensemble. Phys. Rev. E 95(1), 012140 (2017)PubMedCrossRefGoogle Scholar
  26. 26.
    S. Li, L. Vu-Quoc, Finite difference calculus invariant structure of a class of algorithms for the nonlinear Klein–Gordon equation. SIAM J. Numer. Anal. 32(6), 1839–1875 (1995)CrossRefGoogle Scholar
  27. 27.
    F. Liu, V. Anh, I. Turner, Numerical solution of the space fractional Fokker–Planck equation. J. Comput. Appl. Math. 166(1), 209–219 (2004)CrossRefGoogle Scholar
  28. 28.
    F. Liu, M. Meerschaert, R. McGough, P. Zhuang, Q. Liu, Numerical methods for solving the multi-term time-fractional wave-diffusion equation. Fract. Calc. Appl. Anal. 16(1), 9–25 (2013)PubMedPubMedCentralCrossRefGoogle Scholar
  29. 29.
    J.E. Macías-Díaz, A structure-preserving method for a class of nonlinear dissipative wave equations with Riesz space-fractional derivatives. J. Comput. Phys. 351, 515–528 (2017)CrossRefGoogle Scholar
  30. 30.
    J.E. Macías-Díaz, An explicit dissipation-preserving method for Riesz space-fractional nonlinear wave equations in multiple dimensions. Commun. Nonlinear Sci. Numer. Simul. 59, 67–87 (2018)CrossRefGoogle Scholar
  31. 31.
    J.E. Macías-Díaz, A.S. Hendy, R.H. De Staelen, A pseudo energy-invariant method for relativistic wave equations with Riesz space-fractional derivatives. Comput. Phys. Commun. 224, 98–107 (2017)CrossRefGoogle Scholar
  32. 32.
    T. Matsuo, D. Furihata, Dissipative or conservative finite-difference schemes for complex-valued nonlinear partial differential equations. J. Comput. Phys. 171(2), 425–447 (2001)CrossRefGoogle Scholar
  33. 33.
    A. Miele, J. Dekker, Long-range chromosomal interactions and gene regulation. Mol. Biosyst. 4(11), 1046–1057 (2008)PubMedPubMedCentralCrossRefGoogle Scholar
  34. 34.
    G. Miloshevich, J.P. Nguenang, T. Dauxois, R. Khomeriki, S. Ruffo, Traveling solitons in long-range oscillator chains. J. Phys. A Math. Theor. 50(12), 12LT02 (2017)CrossRefGoogle Scholar
  35. 35.
    K.B. Oldham, Fractional differential equations in electrochemistry. Adv. Eng. Softw. 41(1), 9–12 (2010)CrossRefGoogle Scholar
  36. 36.
    K. Pen-Yu, Numerical methods for incompressible viscous flow. Sci. Sin. 20, 287–304 (1977)Google Scholar
  37. 37.
    S. Shen, F. Liu, V. Anh, I. Turner, The fundamental solution and numerical solution of the Riesz fractional advection–dispersion equation. IMA J. Appl. Math. 73(6), 850–872 (2008)CrossRefGoogle Scholar
  38. 38.
    W. Strauss, L. Vazquez, Numerical solution of a nonlinear Klein–Gordon equation. J. Comput. Phys. 28(2), 271–278 (1978)CrossRefGoogle Scholar
  39. 39.
    Y.F. Tang, L. Vázquez, F. Zhang, V. Pérez-García, Symplectic methods for the nonlinear Schrödinger equation. Comput. Math. Appl. 32(5), 73–83 (1996)CrossRefGoogle Scholar
  40. 40.
    V.E. Tarasov, Fractional generalization of gradient and Hamiltonian systems. J. Phys. A Math. General 38(26), 5929 (2005)CrossRefGoogle Scholar
  41. 41.
    V.E. Tarasov, Continuous limit of discrete systems with long-range interaction. J. Phys. A Math. General 39(48), 14895 (2006)CrossRefGoogle Scholar
  42. 42.
    V.E. Tarasov, G.M. Zaslavsky, Fractional dynamics of systems with long-range interaction. Commun. Nonlinear Sci. Numer. Simul. 11(8), 885–898 (2006)CrossRefGoogle Scholar
  43. 43.
    V.E. Tarasov, G.M. Zaslavsky, Conservation laws and Hamiltons equations for systems with long-range interaction and memory. Commun. Nonlinear Sci. Numer. Simul. 13(9), 1860–1878 (2008)CrossRefGoogle Scholar
  44. 44.
    W. Tian, H. Zhou, W. Deng, A class of second order difference approximations for solving space fractional diffusion equations. Math. Comput. 84(294), 1703–1727 (2015)CrossRefGoogle Scholar
  45. 45.
    S. Vong, P. Lyu, Z. Wang, A compact difference scheme for fractional sub-diffusion equations with the spatially variable coefficient under Neumann boundary conditions. J. Sci. Comput. 66(2), 725–739 (2016)CrossRefGoogle Scholar
  46. 46.
    P. Wang, C. Huang, An energy conservative difference scheme for the nonlinear fractional Schrödinger equations. J. Comput. Phys. 293, 238–251 (2015)CrossRefGoogle Scholar
  47. 47.
    P. Wang, C. Huang, L. Zhao, Point-wise error estimate of a conservative difference scheme for the fractional Schrödinger equation. J. Comput. Appl. Math. 306, 231–247 (2016)CrossRefGoogle Scholar
  48. 48.
    Q. Yang, F. Liu, I. Turner, Stability and convergence of an effective numerical method for the time-space fractional Fokker–Planck equation with a nonlinear source term. Int. J. Differ. Equ. 2010 (2010)Google Scholar
  49. 49.
    X. Zhao, Z.Z. Sun, Compact Crank–Nicolson schemes for a class of fractional Cattaneo equation in inhomogeneous medium. J. Sci. Comput. 62(3), 747–771 (2015)CrossRefGoogle Scholar
  50. 50.
    P. Zhuang, F. Liu, V. Anh, I. Turner, Numerical methods for the variable-order fractional advection-diffusion equation with a nonlinear source term. SIAM J. Numer. Anal. 47(3), 1760–1781 (2009)CrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.Department of Mathematics, Faculty of ScienceBenha UniversityBenhaEgypt
  2. 2.Department of Computational Mathematics and Computer Science, Institute of Natural Sciences and MathematicsUral Federal UniversityYekaterinburgRussia
  3. 3.Departamento de Matemáticas y FísicaUniversidad Autónoma de AguascalientesAguascalientesMexico

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