369
Views
2
CrossRef citations to date
0
Altmetric
Research Article

Spectral approximation for nonlinear time fractional Schrödinger equation on graded meshes

&
Pages 2524-2541 | Received 01 Oct 2021, Accepted 21 Apr 2022, Published online: 04 May 2022

References

  • R.A. Adams, Sobolev Space, Academic Press, New York, 1975.
  • 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.
  • L. Chen, S.J. Lü, and T. Xu, Fourier spectral approximation for time fractional Burgers equation with nonsmooth solutions, Appl. Numer. Math. 169 (2021), pp. 164–178.
  • H. Chen and M. Stynes, Error analysis of a second-order method on fitted meshes for a time-fractional diffusion problem, J. Sci. Comput. 79 (2019), pp. 624–647.
  • Q.X. Ding and P.J.Y. Wong, Quintic non-polynomial spline for time-fractional nonlinear Schrödinger equation, Adv. Differ. Equ. 577 (2020), pp. 1–27.
  • M.F. Fei, N. Wang, C.M. Huang, and X.H. Ma, A second-order implicit difference scheme for the nonlinear time-space fractional Schrödinger equation, Appl. Numer. Math. 153 (2020), pp. 399–411.
  • A.S. Hendy and M.A. Zaky, Global consistency analysis of L1-Galerkin spectral schemes for coupled nonlinear space-time fractional Schrödinger equations, Appl. Numer. Math. 156 (2020), pp. 276–302.
  • A.S. Hendy and M.A. Zaky, Combined Galerkin spectral finite difference method over graded meshes for the generalized nonlinear fractional Schrödinger equation, Nonlinear Dyn. 103 (2021), pp. 2493–2507.
  • A. Iomin, Fractional-time Schrödinger equation: Fractional dynamics on a comb, Chaos Soliton. Fract. 44 (2011), pp. 348–352.
  • J.Q. Jia, X.Y. Jiang, and H. Zhang, An L1 Legendre-Galerkin spectral method with fast algorithm for the two-dimensional nonlinear coupled time fractional Schrödinger equation and its parameter estimation, Comput. Math. Appl. 82 (2021), pp. 13–35.
  • N. Kopteva, Error analysis of the L1 method on graded and uniform meshes for a fractional-derivative problem in two and three dimensions, Math. Comput. 88 (2019), pp. 2135–2155.
  • N. Laskin, Fractional quantum mechanics and lévy path integrals, Phys. Lett. A. 268 (2000), pp. 298–305.
  • 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.
  • H.L. Liao, W. Mclean, and J.W. Zhang, A discrete Grönwall inequality with application to numerical schemes for subdiffusion problems, SIAM J. Numer. Anal. 57 (2019), pp. 218–237.
  • H.L. Liao, W. Mclean, and J.W. Zhang, A second-order scheme with nonuniform time steps for a linear reaction-subdiffusion problem, Commun. Comput. Phys. 30 (2021), pp. 567–601.
  • W. McLean and K. Mustapha, A second-order accurate numerical method for a fractional wave equation, Numer. Math. 105 (2007), pp. 481–510.
  • M. Naber, Time fractional Schrödinger equation, J. Math. Phys. 45 (2004), pp. 3339–3352.
  • B.N. Narahari Achar, B.T. Yale, and J.W. Hanneken, Time fractional Schrödinger equation revisited, Adv. Math. Phys. 2013 (2013), pp. 1–11.
  • C.T. Pham, C. Nore, and M.E. Brachet, Boundary layers and emitted excitations in nonlinear Schrödinger superflow past a disk, Physica D. 210 (2005), pp. 203–226.
  • J.I. Ramos and F.R. Villatoro, The nonlinear schrödinger equation in the finite line, Math. Comput. Model. 20 (1994), pp. 31–59.
  • K. Sakamoto and M. Yamamoto, Initial value/boundary value problems for fractional diffusion-wave equations and applications to some inverse problems, J. Math. Anal. Appl. 382 (2011), pp. 426–447.
  • M. Stynes, E. O'Riordan, and J.L. Gracia, Error analysis of a finite difference method on graded meshes for a time-fractional diffusion equation, SIAM J. Numer. Anal. 55 (2017), pp. 1057–1079.
  • R. Temam, Navier-Stokes Equations: Theory and Numerical Analysis, North-Holland Publishing Company, Amsterdam, New York, Oxford, 1977.
  • A. Tofighi, Probability structure of time fractional Schrödinger equation, Acta Phys. Polon. A. 116 (2009), pp. 114–118.
  • L.L. Wang and B.Y. Guo, Interpolation approximations based on Gauss-Lobatto-Legendre-Birkhoff quadrature, J. Approx. Theory. 161 (2009), pp. 142–173.
  • Y. Wang, G. Wang, L.L. Bu, and L.Q. Mei, Two second-order and linear numerical schemes for the multi dimensional nonlinear time-fractional Schrödinger equation, Numer. Algor. 88 (2021), pp. 419–451.
  • Y. Yang, J.D. Wang, S.Y. Zhang, and E. Tohidi, Convergence analysis of space-time Jacobi spectral collocation method for solving time-fractional Schrödinger equations, Appl. Math. Comput. 387 (2020), pp. 1–17.

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.