Advertisement

Numerical Algorithms

, Volume 78, Issue 2, pp 485–511 | Cite as

A linearized second-order scheme for nonlinear time fractional Klein-Gordon type equations

  • Pin Lyu
  • Seakweng Vong
Original Paper

Abstract

We consider difference schemes for nonlinear time fractional Klein-Gordon type equations in this paper. A linearized scheme is proposed to solve the problem. As a result, iterative method need not be employed. One of the main difficulties for the analysis is that certain weight averages of the approximated solutions are considered in the discretization and standard energy estimates cannot be applied directly. By introducing a new grid function, which further approximates the solution, and using ideas in some recent studies, we show that the method converges with second-order accuracy in time.

Keywords

Linearized scheme Time fractional differential equations Nonlinear Klein-Gordon equations Convergence 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Notes

Acknowledgment

The authors would like to thank the referees for their comments which improve the paper significantly. We also want to thank Prof. Honglin Liao for the helpful discussion on the estimates of Lemmas 2.2 and 2.3.

References

  1. 1.
    Barkai, E., Metzler, R., Klafter, J.: From continuous time random walks to the fractional Fokker-Planck equation. Phys. Rev. E 61, 132–138 (2000)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Fitt, A.D., Goodwin, A.R.H., Ronaldson, K.A., Wakeham, W.A.: A fractional differential equation for a MEMS viscometer used in the oil industry. J. Comput. Appl. Math. 229, 373–381 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Kilbas, A., Srivastava, H., Trujillo, J.: Theory and Applications of Fractional Differential Equations. Elsevier Science and Technology, Amsterdam (2006)zbMATHGoogle Scholar
  4. 4.
    Meerschaert, M.M., Benson, D.A., Baeumer, B.: Operator lévy motion and multiscaling anomalous diffusion. Phys. Rev. E 63, 1112–1117 (2001)CrossRefGoogle Scholar
  5. 5.
    Meerschaert, M.M., Benson, D.A., Scheffler, H.-P., Baeumer, B.: Stochastic solution of space-time fractional diffusion equations. Phys. Rev. E 65, 1103–1106 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    West, B.: Fractional calculus in bioengineering. J. Stat. Phys. 126, 1285–1286 (2007)CrossRefGoogle Scholar
  7. 7.
    Cui, M.: Compact finite difference method for the fractional diffusion equation. J. Comput. Phys. 228, 7792–7804 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Deng, W.H.: Finite element method for the space and time fractional Fokker-Planck equation. SIAM J. Numer. Anal. 47, 204–226 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Deng, W.H.: Numerical algorithm for the time fractional Fokker-Planck equation. J. Comput. Phys. 227, 1510–1522 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Gao, G.H., Sun, Z.Z., Zhang, H.W.: A new fractional numerical differentiation formula to approximate the Caputo fractional derivative and its applications. J. Comput. Phys. 259, 33–50 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Jin, B., Lazarov, R., Zhou, Z.: Error estimates for a semidiscrete finite element method for fractional order parabolic equations. SIAM J. Numer. Anal. 51, 445–466 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Jin, B., Lazarov, R., Zhou, Z.: Two fully discrete schemes for fractional diffusion and diffusion-wave equations with nonsmooth data. SIAM J. Numer. Anal. 38, A146–A170 (2016)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Lei, S.L., Sun, H.W.: A circulant preconditioner for fractional diffusion equations. J. Comput. Phys. 242, 715–725 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Lin, Y., Xu, C.: Finite difference/spectral approximations for the time-fractional diffusion equation. J. Comput. Phys. 225, 1533–1552 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Liu, F., Meerschaert, M.M., McGough, R.J., Zhuang, P., Liu, Q.: Numerical methods for solving the multi-term time-fractional wave-diffusion equation. Fractional Calc. Appl. Anal. 16, 9–25 (2013)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Vong, S., Wang, Z.: A high order compact finite difference scheme for time fractional Fokker-Planck equations. Appl. Math. Lett. 43, 38–43 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Wang, Z., Vong, S.: A high-order exponential ADI scheme for two dimensional time fractional convection-diffusion equations. Comput. Math. Appl. 68, 185–196 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Yuste, S.B., Acedo, L.: An explicit finite difference method and a new Von Neumann-type stability analysis for fractional diffusion equations. SIAM J. Numer. Anal. 42, 1862–1874 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Zhao, Z., Jin, X.Q., Lin, M.M.: Preconditioned iterative methods for space-time fractional advection-diffusion equations. J. Comput. Phys. 319, 266–279 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Zhuang, P., Liu, F., Anh, V., Turner, I.: New solution and analytical techniques of the implicit numerical method for the anomalous subdiffusion equation. SIAM J. Numer. Anal. 46, 1079–1095 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    El-Sayed, S.M.: The decomposition method for studying the Klein-Gordon equation. Chaos, Solitons Fractals 18, 1025–1030 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Kaya, D., El-Sayed, S.M.: A numerical solution of the Klein-Gordon equation and convergence of the decomposition method. Appl. Math. Comput. 156, 341–353 (2004)MathSciNetzbMATHGoogle Scholar
  23. 23.
    Batiha, B., Noorani, M.S.M., Hashim, I.: Numerical solution of sine-Gordon equation by variational iteration method. Phys. Lett. A 370, 437–440 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Yusufoğlu, E.: The variational iteration method for studying the Klein-Gordon equation. Appl. Math. Lett. 21, 669–674 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Jafari, H., Saeidy, M., Arab Firoozjaee, M.: Solving nonlinear Klein-Gordon equation with a quadratic nonlinear term using homotopy analysis method. Iran. J. Optim. 1, 162–172 (2009)Google Scholar
  26. 26.
    Golmankhaneh, A.K., Golmankhaneh, A.K., Baleanu, D.: On nonlinear fractional Klein-Gordon equation. Sig. Process. 91, 446–451 (2011)CrossRefzbMATHGoogle Scholar
  27. 27.
    Jafari, H., Tajadodi, H., Kadkhoda, N., Baleanu, D.: Fractional subequation method for Cahn-Hilliard and Klein-Gordon equations. Abstr. Appl. Anal. (2013). doi: 10.1155/2013/587179
  28. 28.
    Cui, M.: Fourth-order compact scheme for the one-dimensional sine-Gordon equation. Numer. Meth. Part. Differ. Equ. 25, 685–711 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Vong, S., Wang, Z.: A compact difference scheme for a two dimensional fractional Klein-Gordon equation with Neumann boundary conditions. J. Comput. Phys. 274, 268–282 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Vong, S., Wang, Z.: A high-order compact scheme for the nonlinear fractional Klein-Gordon equation. Numer. Meth. Part. Differ. Equ. 31, 706–722 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Chen, H., Lu, S., Chen, W.: A fully discrete spectral method for the nonlinear time fractional Klein-Gordon equation. Taiwan. J. Math. 21, 231–251 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Dehghan, M., Abbaszadeh, M., Mohebbi, A.: An implicit RBF meshless approach for solving the time fractional nonlinear sine-Gordon and Klein-Gordon equations. Eng. Anal. Bound. Elem. 50, 412–434 (2015)MathSciNetCrossRefGoogle Scholar
  33. 33.
    Alikhanov, A.A.: A new difference scheme for the time fractional diffusion equation. J. Comput. Phys. 280, 424–438 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Hao, Z.P., Sun, Z.Z.: A linearized high-order difference scheme for the fractional Ginzburg-Landau equation. Numer. Meth. Part. Differ. Equ. (2016). doi: 10.1002/num.22076
  35. 35.
    Ran, M., Zhang, C.: A conservative difference scheme for solving the strongly coupled nonlinear fractional schrödinger equations. Commun. Nonlin. Sci. Numer. Simulat. 41, 64–83 (2016)CrossRefGoogle Scholar
  36. 36.
    Wang, D., Xiao, A., Yang, W.: A linearly implicit conservative difference scheme for the space fractional coupled nonlinear Schrödinger equations. J. Comput. Phys. 272, 644–655 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Zhao, X., Sun, Z.Z., Hao, Z.P.: A fourth-order compact ADI scheme for two-dimensional nonlinear space fractional Schrödinger equation. SIAM J. Sci. Comput. 36, A2865–A2886 (2014)CrossRefzbMATHGoogle Scholar
  38. 38.
    Liao, H.L., Zhao, Y., Teng, X.H.: A weighted ADI scheme for subdiffusion equations. J. Sci. Comput. (2016). doi: 10.1007/s10915-016-0230-9
  39. 39.
    Liao, H.L., Zhao, Y., Teng, X.H.: Convergence of a weighted compact ADI scheme for fractional diffusion-wave equations. submittedGoogle Scholar
  40. 40.
    Quarteroni, A., Valli, A.: Numerical Approximation of Partial Diferential Equations. Springer, Berlin (1997)Google Scholar
  41. 41.
    Sun, Z.Z.: Numerical Methods of Partial Differential Equations, 2nd edn. Science Press, Beijing (2012)Google Scholar
  42. 42.
    Vong, S., Lyu, P., Chen, X., Lei, S.L.: High order finite difference method for time-space fractional differential equations with Caputo and Riemann-Liouville derivatives. Numer. Algor. 72, 195–210 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  43. 43.
    Sun, Z.Z., Wu, X.: A fully discrete difference scheme for a diffusion-wave system. Appl. Numer. Math. 56, 193–209 (2006)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2017

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of MacauMacauChina

Personalised recommendations