194
Views
2
CrossRef citations to date
0
Altmetric
Original Articles

High order WSGL difference operators combined with Sinc-Galerkin method for time fractional Schrödinger equation

, , &
Pages 2259-2286 | Received 23 Aug 2019, Accepted 07 Nov 2019, Published online: 20 Nov 2019

References

  • M. Abdrabou, On the Sinc-Galerkin method for triharmonic boundary-value problems, Comput. Math. Appl. 76 (2018), pp. 520–533. doi: 10.1016/j.camwa.2018.04.034
  • J.R. Anglin and W. Ketterle, Bose-Einstein condensation of atomic gases, Nature 416 (2002), pp. 211–218. doi: 10.1038/416211a
  • X. Antoine, Q.L. Tang, J.W. Zhang, On the numerical solution and dynamical laws of nonlinear fractional Schrödinger/Gross-Pitaevskii equations. Int. J. Comput. Math. 95 (2018), pp. 1–25. doi: 10.1080/00207160.2018.1437911
  • W.Z. Bao, Mathematical models and numerical methods for Bose-Einstein condensation, Proc. Int. Congr. Math. IV (2014), pp. 971–996.
  • A.H. Bhrawy and M.A. Abdelkawy, A fully spectral collocation approximation for multi-dimensional fractional Schrödinger equations, J. Comput. Phys. 294 (2015), pp. 462–483. doi: 10.1016/j.jcp.2015.03.063
  • A.H. Bhrawy, J.F. Alzaidy, M.A. Abdelkawy, and A. Biswas, Jacobi spectral collocation approximation for multi-dimensional time-fractional Schrödinger equations, Nonlinear Dyn. 84 (2016), pp. 1553–1567. doi: 10.1007/s11071-015-2588-x
  • X.L. Chen, Y.N. Di, J.Q. Duan, and D.F. 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
  • M. Dehghan and F. Emami-Naeini, The Sinc-collocation and Sinc-Galerkin methods for solving the two-dimensional Schrödinger equation with nonhomogeneous boundary conditions, Appl. Math. Comput. 37 (2013), pp. 9379–9397.
  • M. Dehghan and A. Taleei, A compact split-step finite difference method for solving the nonlinear Schrödinger equations with constant and variable coefficients, Comput. Phys. Comm. 181 (2010), pp. 43–51. doi: 10.1016/j.cpc.2009.08.015
  • R.P. Feynman and A.R. Hibbs, Quantum Mechanics and Path Integrals, McGraw-Hill, New York, 1965.
  • L. Galeone and R. Garrappa, Explicit methods for fractional differential equations and their stability properties, J. Comput. Appl. Math. 228 (2009), pp. 548–560. doi: 10.1016/j.cam.2008.03.025
  • 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
  • J.S. Hu and C.X. Zheng, Fast and stable evaluation of the exact absorbing boundary condition for the semi-discrete linear Schrödinger equation in unbounded domains, J. Comput. Appl. Math. 326 (2017), pp. 116–125. doi: 10.1016/j.cam.2017.05.018
  • A. Iomin, Fractional-time Schrödinger equation: fractional dynamics on a comb, Chaos Solitons Fractals 44 (2011), pp. 348–352. doi: 10.1016/j.chaos.2011.03.005
  • C.C. Ji and Z.Z. Sun, A high-order compact finite difference scheme for the fractional sub-diffusion equation, J. Sci. Comput. 64 (2015), pp. 959–985. doi: 10.1007/s10915-014-9956-4
  • N.A. Khan, M. Jamil, A. Ara, Approximate solutions to time-fractional Schrödinger equation via homotopy analysis method. ISRN Math. Phys. 2012 (2012), pp. 1–11. doi: 10.5402/2012/197068
  • L.L. Li and D.F. Li, Exact solutions and numerical study of time fractional Burgers' equations, Appl. Math. Lett. 100 (2020), pp. 106011.
  • D.F. Li, J.L. Wang, and J.W. Zhang, Unconditionally convergent L1-Galerkin FEMs for nonlinear time-fractional Schrödinger equations, SIAM J. Sci. Comput. 39 (2017), pp. A3067–A3088. doi: 10.1137/16M1105700
  • Y.M. Lin and C.J. Xu, Finite difference/spectral approximations for the time-fractional diffusion equation, J. Comput. Phys. 225 (2007), pp. 1533–1552. doi: 10.1016/j.jcp.2007.02.001
  • Y. Liu, Y.W. Du, H. Li, and J.F. Wang, A two-grid finite element approximation for a nonlinear time-fractional Cable equation, Nonlinear Dyn. 85 (2016), pp. 2535–2548. doi: 10.1007/s11071-016-2843-9
  • Y. Liu, M. Zhang, and J.C. Li, High-order local discontinuous Galerkin method combined with WSGD-approximation for a fractional subdiffusion equation, Comput. Math. Appl. 73 (2017), pp. 1298–1314. doi: 10.1016/j.camwa.2016.08.015
  • F. Liu, P.H. Zhuang, and Q.X. Liu, The Applications and Numerical Methods of Fractional Differential Equations, Science Press, Beijing, 2015.
  • C. Lubich, Discretized fractional calculus, SIAM J. Math. Anal. 17 (1986), pp. 704–719. doi: 10.1137/0517050
  • N. Mark, Time fractional Schrödinger equation, J. Math. 1159 (2014), pp. 3339–3352.
  • M.M. Meerschaert and C. Tadjeran, Finite difference approximations for fractional advection-dispersion flow equations, J. Comput. Appl. Math. 172 (2004), pp. 65–77. doi: 10.1016/j.cam.2004.01.033
  • 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
  • K. Parand, M. Dehghan, and A. Pirkhedri, The use of Sinc-collocation method for solving Falkner-Skan boundary-layer equation, Int. J. Numer. Methods Fluids 68 (2012), pp. 36–47. doi: 10.1002/fld.2493
  • I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, 1999.
  • F. Stenger, Numerical methods based on Whittaker cardinal, or Sinc functions, SIAM Rev. 23 (1981), pp. 165–224. doi: 10.1137/1023037
  • M. Sugihara and T. Matsuo, Recent developments of the Sinc numerical methods, J. Comput. Appl. Math. 164 (2004), pp. 673–689. doi: 10.1016/j.cam.2003.09.016
  • Z.Z. Sun and X.N. Wu, A fully discrete difference scheme for a diffusion-wave system, Appl. Numer. Math. 56 (2006), pp. 193–209. doi: 10.1016/j.apnum.2005.03.003
  • J. Tang and D. Xu, The global behavior of finite difference-spatial spectral collocation methods for a partial integro-differential equation with a weakly singular kernel, Numer. Math. Theory Methods Appl. 6 (2013), pp. 556–570. doi: 10.4208/nmtma.2013.1111nm
  • W.Y. Tian, H. Zhou, and W.H. Deng, A class of second order difference approximations for solving space fractional diffusion equations, Math. Comput. 84 (2015), pp. 1703–1727. doi: 10.1090/S0025-5718-2015-02917-2
  • A. Tofighi, Probability structure of time fractional Schrödinger equation, Acta Phys. Polon. 116 (2009), pp. 114–118. doi: 10.12693/APhysPolA.116.114
  • A.F. Vargas, N. Morales-Durán, and P. Bargueño, A Bohmian approach to the non-Markovian non-linear Schrödinger-Langevin equation, Ann. Phys. 356 (2015), pp. 498–504. doi: 10.1016/j.aop.2015.03.021
  • Z.B. Wang and S. Vong, Compact difference schemes for the modified anomalous fractional sub-diffusion equation and the fractional diffusion-wave equation, J. Comput. Phys. 277 (2014), pp. 1–15. doi: 10.1016/j.jcp.2014.08.012
  • L.L. Wei, Y.N. He, X.D. Zhang, and S.L. 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
  • M. Zarebnia and M. Sajjadian, The Sinc-Galerkin method for solving Troesch's problem, Math. Comput. Modell. 56 (2012), pp. 218–228. doi: 10.1016/j.mcm.2011.11.071
  • M. Zarebnia and M. Sajjadian, Convergence of the Sinc-Galerkin method for the Bratu equation, Chiang Mai J. Sci. 41 (2014), pp. 714–723.
  • C.X. Zheng, J.S. Hu, Q. Du, and J.W. Zhang, Numerical solution of the nonlocal diffusion equation on the real line, SIAM J. Sci. Comput. 39 (2017), pp. A1951–A1968. doi: 10.1137/16M1090107
  • Y. Zhou, R.W. Jin, and Z. Lu, Basic Theory of Fractional Differential Equations, World Scientific, Singapore, 2016.
  • H. Zhou, W.Y. Tian, and W.H. Deng, Quasi-compact finite difference schemes for space fractional diffusion equations, J. Sci. Comput. 56 (2013), pp. 45–66. doi: 10.1007/s10915-012-9661-0

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.