293
Views
13
CrossRef citations to date
0
Altmetric
Articles

A class of linearized energy-conserved finite difference schemes for nonlinear space-fractional Schrödinger equations

Pages 609-626 | Received 02 Aug 2014, Accepted 01 Jan 2015, Published online: 17 Mar 2015

References

  • P. Amore, F.M. Fernández, C.P. Hofmann, and R.A. Sáenz, Collocation method for fractional quantum mechanics, J. Math. Phys. 51 (2010), p. 16. (Art. ID 122101).
  • K. Aruna and A.S.V. Ravi Kanth, Approximate solutions of non-linear fractional Schrödinger equation via differential transform method and modified differential transform method, Natl. Acad. Sci. Lett. 36 (2013), pp. 201–213. doi: 10.1007/s40009-013-0119-1
  • A. Ashyralyev and B. Hicdurmaz, On the numerical solution of fractional Schrödinger differential equations with the Dirichlet condition, Int. J. Comput. Math. 89 (2012), pp. 1927–1936. doi: 10.1080/00207160.2012.698841
  • A. Atangana and A.H. Cloot, Stability and convergence of the space fractional variable-order Schrödinger equation, Adv. Differ. Equ. 2013 (2013), pp. 1–10. doi: 10.1186/1687-1847-2013-1
  • C. Çelik and M. Duman, Crank–Nicolson method for the fractional diffusion equation with the Riesz fractional derivative, J. Comput. Phys. 231 (2012), pp. 1743–1750. doi: 10.1016/j.jcp.2011.11.008
  • M.H. Chen and W.H. Deng, Fourth order accurate scheme for the space fractional diffusion equations, SIAM J. Numer. Anal. 52 (2014), pp. 1418–1438. doi: 10.1137/130933447
  • J.P. Dong and M.Y. Xu, Space-time fractional Schrödinger equation with time-independent potentials, J. Math. Anal. Appl. 344 (2008), pp. 1005–1017. doi: 10.1016/j.jmaa.2008.03.061
  • X.Y. Guo and M.Y. Xu, Some physical applications of fractional Schrödinger equation, J. Math. Phys. 47 (2006), p. 9. (Art. ID 082104).
  • B. Guo, Y. Han, and J. Xin, Existence of the global smooth solution to the period boundary value problem of fractional nonlinear Schrödinger equation, Appl. Math. Comput. 204 (2008), pp. 468–477. doi: 10.1016/j.amc.2008.07.003
  • M.A.E. Herzallaha and K.A. Gepreel, Approximate solution to the time-space fractional cubic nonlinear Schrodinger equation, Appl. Math. Model. 36 (2012), pp. 5678–5685. doi: 10.1016/j.apm.2012.01.012
  • J. Hu, J. Xin, and H. Lu, The global solution for a class of systems of fractional nonlinear Schrödinger equations with periodic boundary condition, Comput. Math. Appl. 62 (2011), pp. 1510–1521. doi: 10.1016/j.camwa.2011.05.039
  • X.Y. Jiang, H.T. Qi, and M.Y. Xu, Exact solutions of fractional Schrödinger-like equation with a nonlocal term, J. Math. Phys. 52 (2011), p. 10. (Art. ID 042105). doi: 10.1063/1.3576189
  • N. Laskin, Fractional quantum mechanics, Phys. Rev. E 62 (2000), pp. 3135–3145. doi: 10.1103/PhysRevE.62.3135
  • N. Laskin, Fractional quantum mechanics and Lévy path integrals, Phys. Lett. A 268 (2000), pp. 298–305. doi: 10.1016/S0375-9601(00)00201-2
  • N. Laskin, Fractional Schrödinger equation, Phys. Rev. E 66 (2002), p. 056108. doi: 10.1103/PhysRevE.66.056108
  • E.K. Lenzi, H.V. Ribeiro, H. Mukai, and R.S. Mendes, Continuous-time random walk as a guide to fractional Schrödinger equation, J. Math. Phys. 51 (2010), p. 7. (Art. ID 092102). doi: 10.1063/1.3491333
  • M.M. Meerschaert and C. Tadjeran, Finite difference approximations for two-sided space-fractional partial differential equations, Appl. Numer. Math. 56 (2006), pp. 80–90. doi: 10.1016/j.apnum.2005.02.008
  • A. Mohebbi, M. Abbaszadeh, and M. Dehghan, The use of a meshless technique based on collocation and radial basis functions for solving the time fractional nonlinear Schrödinger equation arising in quantum mechanics, Eng. Anal. Bound. Elem. 37 (2013), pp. 475–485. doi: 10.1016/j.enganabound.2012.12.002
  • S.I. Muslih, O.P. Agrawal, and D. Baleanu, A fractional Schrödinger equation and its solution, Int. J. Theor. Phys. 49 (2010), pp. 1746–1752. doi: 10.1007/s10773-010-0354-x
  • M. Naber, Time fractional Schrödinger equation, J. Math. Phys. 45 (2004), pp. 3339–3352. doi: 10.1063/1.1769611
  • W.Y. Tian, H. Zhou, and W.H. Deng, A class of second order difference approximations for solving space fractional diffusion equations, (2012), p. 24. available at arXiv:1201.5949.
  • P.D. Wang and C.M. Huang, An energy conservative difference scheme for the nonlinear fractional Schrödinger equations, J. Comput. Phys. (2014), in press.
  • D.L. Wang, A.G. Xiao, and W. Yang, Crank–Nicolson difference scheme for the coupled nonlinear Schrödinger equations with the Riesz space fractional derivative, J. Comput. Phys. 242 (2013), pp. 670–681. doi: 10.1016/j.jcp.2013.02.037
  • D.L. Wang, A.G. Xiao, and W. Yang, A linearly implicit conservative difference scheme for the space fractional coupled nonlinear Schrödinger equations, J. Comput. Phys. 272 (2014), pp. 644–655. doi: 10.1016/j.jcp.2014.04.047
  • L. Wei, Y. He, X. Zhang, and S. Wang, Analysis of an implicit fully discrete local discontinuous Galerkin method for the time-fractional Schrödinger equation, Finite Elem. Anal. Des. 59 (2012), pp. 28–34. doi: 10.1016/j.finel.2012.03.008
  • L. Wei, X. Zhang, S. Kumar, and A. Yildirim, A numerical study based on an implicit fully discrete local discontinuous Galerkin method for the time-fractional coupled Schrödinger system, Comput. Math. Appl. 64 (2012), pp. 2603–2615. doi: 10.1016/j.camwa.2012.07.004
  • H. Wu, H.P. Ma, and H.Y. Li, Optimal error estimates of the Chebyshev–Legendre spectral method for solving the generalized Burgers equation, SIAM J. Numer. Anal. 41 (2003), pp. 659–672. doi: 10.1137/S0036142901399781

Reprints and Corporate Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

To request a reprint or corporate permissions for this article, please click on the relevant link below:

Academic Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

Obtain permissions instantly via Rightslink by clicking on the button below:

If you are unable to obtain permissions via Rightslink, please complete and submit this Permissions form. For more information, please visit our Permissions help page.