173
Views
2
CrossRef citations to date
0
Altmetric
Original Articles

Error analysis of the unstructured mesh finite element method for the two-dimensional time-space fractional Schrödinger equation with a time-independent potential

&
Pages 1663-1682 | Received 19 Jul 2018, Accepted 28 Sep 2020, Published online: 24 Nov 2020

References

  • A. Bhrawy, M. Abdelkawy, A fully spectral collocation approximation for multi-dimensional fractional Schrödinger equations, J. Comput. Phys. 294 (2015), pp. 462–483.
  • A. Bhrawy, J. Alzaidy, M. Abdelkawy, and A. Biswas, Jacobi spectral collocation approximation for multi-dimensional time-fractional Schrödinger equations, Nonlinear Dyn. 84(3) (2016), pp. 1553–1567.
  • A. Bhrawy and M. Zaky, Numerical simulation of multi-dimensional distributed-order generalized Schrödinger equationsNonlinear Dyn. 89(2) (2017), pp. 1415–1432.
  • A.H. Bhrawy and M.A. Zaky, Highly accurate numerical schemes for multi-dimensional space variable-order fractional Schrödinger equations, Comput. Math. Appl. 73(6) (2017), pp. 1100–1117.
  • A.H. Bhrawy and M.A. Zaky, An improved collocation method for multi-dimensional space-time variable-order fractional Schrödinger equations, Appl. Numer. Math. 111 (2017), pp. 197–218.
  • W. Bu, X. Liu, Y. Tang, and J. Yang, Finite element multigrid method for multiterm time fractional advection diffusion equations, Int. J. Model. Simul. Sci. Comput. 6(1) (2015), pp. 1540001.
  • 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.
  • E.C. de Oliveira, F.S. Costa, and J. Vaz Jr, The fractional Schrödinger equation for delta potentials, J. Math. Phys. 51(12) (2010), p. 123517.
  • V.J. Ervin and J.P. Roop, Variational formulation for the stationary fractional advection dispersion equation, Numer. Methods Partial Differential Equations 22(3) (2006), pp. 558–576.
  • V.J. Ervin and J.P. Roop, Variational solution of fractional advection dispersion equations on bounded domains in Rd, Numer. Methods Partial Differential Equations 23(2) (2007), pp. 256–281.
  • W. Fan, X. Jiang, F. Liu, and V. Anh, The unstructured mesh finite element method for the two-dimensional multi-term time-space fractional diffusion-wave equation on an irregular convex domain, J. Sci. Comput. 77 (2018), pp. 27–52.
  • W. Fan, F. Liu, X. Jiang, and T. Ian, A novel unstructured mesh finite element method for solving the time-space fractional wave equation on a two-dimensional irregular convex domain, Fract. Calc. Appl. Anal. 20(2) (2017), pp. 954–383.
  • W. Fan, F. Liu, X. Jiang, and I. Turner, Some novel numerical techniques for an inverse problem of the multi-term time fractional partial differential equation, J. Comput. Appl. Math. 336 (2018), pp. 114–126.
  • B. Feng and H. Zhang, Stability of standing waves for the fractional Schrödinger-Choquard equation, Comput. Math. Appl. 75 (2018), pp. 2499–2507.
  • R.P. Feynman, A.R. Hibbs, D.F. Styer, Quantum Mechanics and Path Integrals, Dover publications, Inc. Mineola, New York, 2010.
  • M.A. Herzallah and K.A. Gepreel, Approximate solution to the time-space fractional cubic nonlinear Schrödinger equation, Appl. Math. Model. 36(11) (2012), pp. 5678–5685.
  • Y. Huang, X. Li, and A. Xiao, Fourier pseudospectral method on generalized sparse grids for the space-fractional Schrödinger equation, Comput. Math. Appl. 75(12) (2018), pp. 4241–4255.
  • X. Jiang, Time-space fractional Schrödinger like equation with a nonlocal term, Eur. Phys. J. Spec. Top. 193(1) (2011), pp. 61–70.
  • X. Jiang, H. Qi, and M. Xu, Exact solutions of fractional Schrödinger-like equation with a nonlocal term, J. Math. Phys. 52(4) (2011), pp. 042105.
  • N. Laskin, Fractional quantum mechanics and Lévy path integrals, Phys. Lett. Appl. 268(4) (2000), pp. 298–305.
  • M. Li, Xian-Ming Gu, C. Huang, M. Fei, and G. Zhang, A fast linearized conservative finite element method for the strongly coupled nonlinear fractional Schrödinger equations, J. Comput. Phys. 358 (2018), pp. 256–282.
  • M. Li, C. Huang, and P. Wang, Galerkin finite element method for nonlinear fractional Schrödinger equations, Numer. Algor. 74(2) (2017), pp. 499–525.
  • M. Li, C. Huang, and Z. Zhang, Unconditional error analysis of Galerkin FEMs for nonlinear fractional Schrödinger equation, Appl. Anal. 97(2) (2018), pp. 295–315.
  • Y. Li, F. Liu, I. Turner, and T. Li, Time-fractional diffusion equation for signal smoothing, Appl. Math. Comput. 326 (2018), pp. 108–116.
  • M. Li and Y. Zhao, A fast energy conserving finite element method for the nonlinear fractional Schrödinger equation with wave operator, Appl. Math. Comput. 338 (2018), pp. 758–773.
  • Q. Liu, F.H. Zeng, and C.P. Li, Finite difference method for time-space-fractional Schrödinger equation, Int. J. Comput. Math. 92 (2015), pp. 1439–1451.
  • Y. Luchko, Fractional Schrödinger equation for a particle moving in a potential well, J. Math. Phys. 54(1) (2013), pp. 012111.
  • B.B. Mandelbrot, The Fractal Geometry of Nature, Freeman, New York, 1982.
  • M. Naber, Time fractional Schrödinger equation, J. Math. Phys. 45(8) (2004), pp. 3339–3352.
  • I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, 1998.
  • J.P. Roop, Variational solution of the fractional advection dispersion equation, Ph.D. thesis, 2004.
  • N.H. Sweilam and M.A. Hasan, Numerical solutions for 2-D fractional Schrödinger equation with the Riesz–Feller derivative, Math. Comput. Simulat. 140 (2017), pp. 53–68.
  • P. Wang and C. Huang, Split-step alternating direction implicit difference scheme for the fractional Schrödinger equation in two dimensions, Comput. Math. Appl. 71(5) (2016), pp. 1114–1128.
  • P. Wang and C. Huang, Structure-preserving numerical methods for the fractional Schrödinger equation, Appl. Numer. Math. 129 (2018), pp. 137–158.
  • 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.
  • S. Wang and M. Xu, Generalized fractional Schrödinger equation with space-time fractional derivatives, J. Math. Phys. 48(4) (2007), pp. 043502.
  • 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.
  • G. Zhang, C. Huang, and M. Li, A mass-energy preserving Galerkin FEM for the coupled nonlinear fractional Schrödinger equations, Eur. Phys. J. Plus 133(4) (2018), pp. 514.
  • H. Zhang, X. Jiang, C. Wang, and S. Chen, Crank-Nicolson Fourier spectral methods for the space fractional nonlinear Schrödinger equation and its parameter estimation, Int. J. Comput. Math. 96 (2) (2018), pp. 1–29.
  • H. Zhang, F. Liu, and V. Anh, Galerkin finite element approximation of symmetric space-fractional partial differential equations, Appl. Math. Comput. 217(6) (2010), pp. 2534–2545.
  • X. Zhao, Z. Sun, and Z. Hao, A fourth-order compact ADI scheme for two-dimensional nonlinear space fractional Schrödinger equation, SIAM J. Sci. Comput 36(6) (2014), pp. A2865–A2886.
  • X. Zhu, Z. Yuan, J. Wang, Y. Nie, and Z. Yang, Finite element method for time-space-fractional Schrödinger equation, Electron. J. Differ. Eq. 2017(166) (2017), pp. 1–18.

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.