275
Views
2
CrossRef citations to date
0
Altmetric
Original Articles

Finite difference schemes for time-fractional Schrödinger equations via fractional linear multistep method

Pages 1561-1573 | Received 31 Jan 2020, Accepted 25 Sep 2020, Published online: 02 Nov 2020

References

  • 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.H. Bhrawy and M.A. Abdelkawy, A fully spectral collocation approximation for multidimensional fractional Schrödinger equations, J. Comput. Phys. 294 (2015), pp. 462–483. doi: 10.1016/j.jcp.2015.03.063
  • X. Chen, Y. Di, J. Duan, and D. Li, Linearized compact ADI schemes for nonlinear time-fractional Schrödinger equations, Appl. Math. Lett. 84 (2018), pp. 160–167. doi: 10.1016/j.aml.2018.05.007
  • N.J. Ford, M.M. Rodrigues, and N. Vieira, A numerical method for the fractional Schrödinger type equation of spatial dimension two, Fract. Calc. Appl. Anal. 16(2) (2013), pp. 454–468. doi: 10.2478/s13540-013-0028-5
  • L. Galeone and R. Garrappa, Fractional Adams–Moulton methods, Math. Comput. Simul. 79(4) (2008), pp. 1358–1367. doi: 10.1016/j.matcom.2008.03.008
  • R. Garrappa, On some explicit Adams multistep methods for fractional differential equations, J. Comput. Appl. Math. 229 (2009), pp. 392–399. doi: 10.1016/j.cam.2008.04.004
  • R. Garrappa, I. Moret, and M. Popolizio, Solving the time-fractional Schrödinger equation by Krylov projection methods, J. Comput. Phys. 293 (2015), pp. 115–134. doi: 10.1016/j.jcp.2014.09.023
  • B.L. Guo, Y.Q. 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.
  • L. Guo, F. Zeng, I. Turner, Efficient multistep methods for tempered fractional calculus: Algorithms and simulations. SIAM J. Sci. Comput. 41 (4) (2019), pp. A2510–A2535. doi: 10.1137/18M1230153
  • D. He and K. Pan, A linearly implicit conservative difference scheme for the generalized Rosenau-Kawahara-RLW equation, Appl. Math. Comput. 271 (2015), pp. 323–336.
  • B. Hicdurmaz and A. Ashyralyev, A stable numerical method for multidimensional time fractional Schrödinger equations, Comput. Math. Appl. 72 (2016), pp. 1703–1713. doi: 10.1016/j.camwa.2016.07.036
  • B. Jin, B. Li, and Z. Zhou, Subdiffusion with a time-dependent coefficient: Analysis and numerical solution, Math. Comput. 88 (2019), pp. 2157–2186. doi: 10.1090/mcom/3413
  • E.I. Jury, Theory and Application of the z-Transform Method, Robert E Krieger Publishing Co., Huntington, NY, 1973.
  • M. Kirane and A. Nabti, Life span of solutions to a nonlocal in time nonlinear fractional Schrödinger equation, Z. Angew. Math. Phys. 66 (2015), pp. 1473–1482. doi: 10.1007/s00033-014-0473-y
  • N. Laskin, Fractional quantum mechanics and Lèvy integrals, Phys. Lett. A 268 (2000), pp. 298–305. doi: 10.1016/S0375-9601(00)00201-2
  • C.T. Ledesma, Existence and symmetric result for Liouville–Weyl fractional nonlinear Schrödinger equation, Commun. Nonlinear Sci. Numer. Simul. 27 (2015), pp. 314–327. doi: 10.1016/j.cnsns.2015.02.019
  • Q. Li and X. Wu, Soliton solutions for fractional Schrödinger equations, Appl. Math. Lett. 53 (2016), pp. 119–124. doi: 10.1016/j.aml.2015.10.006
  • D. Li, C. Wu, and Z. Zhang, Linearized Galerkin FEMs for nonlinear time fractional parabolic problems with non-smooth solutions in time direction, J. Sci. Comput. 80 (2019), pp. 403–419. doi: 10.1007/s10915-019-00943-0
  • D. Li, J. Wang, J. Zhang, Unconditionally convergent L1-Galerkin FEMs for nonlinear time-Fractional Schrödinger equations. SIAM J. Sci. Comput. 39 (6) (2017), pp. A3067–A3088. doi:10.1137/16M1105700.
  • C. Lubich, Discretized fractional calculus, SIAM J. Math. Anal. 17(3) (1986), pp. 704–719. doi: 10.1137/0517050
  • 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
  • M. Naber, Time fractional Schrödinger equation, J. Math. Phys. 45 (2004), pp. 3339–3352. doi: 10.1063/1.1769611
  • I. Podlubny, Fractional Differential Equations, Mathematics in Science and Engineering, Vol. 198, Academic Press, San Diego, CA, 1999.
  • E. Scalas, D. Baleanu, F. Mainardi, and A. Mura, Fractional calculus and the Schrödinger equation, IFAC Proceedings Volumes (Part 1), 2006, pp. 234–237.
  • P.E. Sobolevskii, Difference Methods for the Approximate Solution of Differential Equations, Izdat. Voronezh Gusud. Univ., Voronezh, 1975 (Russian).
  • J.C. Strikwerda, Finite Difference Schemes and Partial Differential Equations. 2nd ed.SIAMPhiladelphia2004.
  • J. Tang and D. Xu, The global behaviour of finite difference-spatial spectral collocation methods for a partial integro-differential equation with a weakly singular kernel, Numer. Math. Theory Methods Appl. 6(3) (2013), pp. 556–570. doi: 10.4208/nmtma.2013.1111nm
  • P. Wang and C. Huang, An energy conservative difference scheme for the nonlinear fractional Schrödinger equations, J. Comput. Phys. 293 (2015), pp. 238–251. doi: 10.1016/j.jcp.2014.03.037
  • D. Wang, A. 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
  • 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
  • J. Xu, Z. Wei, and W. Dong, Existence of weak solutions for a fractional Schrödinger equation, Commun. Nonlinear Sci. Numer. Simul. 22 (2015), pp. 1215–1222. doi: 10.1016/j.cnsns.2014.06.051
  • S. Yan, F. Zhao, C. Li, High order WSGL difference operators combined with Sinc-Galerkin method for time fractional Schrödinger equation. Int. J. Comput. Math. 97(11) (2019), pp. 2259–2286. doi:10.1080/00207160.2019.1692200.
  • A. Yokus and D. Kaya, Numerical and exact solutions for time fractional Burgers' equation, J. Nonlinear Sci. Appl. 10 (2017), pp. 3419–3428. doi: 10.22436/jnsa.010.07.06
  • F. Zeng, Second-order stable finite difference schemes for the time-fractional diffusion-wave equation, J. Sci. Comput. 65 (2015), pp. 411–430. doi:10.1007/s10915-014-9966-2.
  • F. Zeng, C. Li, F. Liu, and I. Turner, The use of finite difference/element approaches for solving the time-fractional subdiffusion equation, SIAM J. Sci. Comput. 35(6) (2013), pp. A2976–A3000. doi: 10.1137/130910865
  • F. Zeng, C. Li, F. Liu, and I. Turner, Numerical algorithms for time-fractional subdiffusion equation with second-order accuracy, SIAM J. Sci. Comput. 37(1) (2015), pp. A55–A78. doi: 10.1137/14096390X
  • J. Zhang, D. Li, and X. Antoine, Efficient numerical computation of time-fractional nonlinear Schrödinger equations in unbounded domain, Commun. Comput. Phys. 25 (2019), pp. 218–243.
  • M. Zheng, F. Liu, and Z. Jin, The global analysis on the spectral collocation method for time fractional Schrödinger equation, Appl. Math. Comput. 365 (2020), p. 124689.

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.