References
- C. Canuto, M.Y. Hussaini, A. Quarteroni, and T.A. Zang, Spectral Methods, Scientific Computation, Springer-Verlag, Berlin, 2006, Fundamentals in single domains.
- 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.
- H. Chen, X. Hu, J. Ren, T. Sun, and Y. Tang, L1 scheme on graded mesh for the linearized time fractional Kdv equation with initial singularity, Int. J. Mod. Sim. Sci. Comp. 10(1) (2019), p. 1941006.
- M. Cheng, Bound state for the fractional Schrödinger equation with unbounded potential, J. Math. Phys. 53(4) (2012), p. 043507.
- H. Dai, L. Wei, and X. Zhang, Numerical algorithm based on an implicit fully discrete local discontinuous Galerkin method for the fractional diffusion-wave equation, Numer. Algor. 67(4) (2014), pp. 845–862.
- P. Felmer and A.Q.J. Tan, Positive solutions of the nonlinear Schrödinger equation with the fractional Laplacian, Proc. Roy. Soc. Edinb. 142(6) (2012), pp. 1237–1262.
- J.L. Gracia, E. O'Riordan, and M. Stynes, A fitted scheme for a caputo initial-boundary value problem, J. Sci. Comput. 10 (2018), pp. 1–27.
- Y.J. Jiang and J.T. Ma, Moving finite element methods for time fractional partial differential equations, Sci. China Math. 56(6) (2013), pp. 1287–1300.
- B. Jin, R. Lazarov, and Z. Zhou, An analysis of the L1 scheme for the subdiffusion equation with nonsmooth data, IMA J. Numer. Anal. 36(1) (2015), pp. 197–221.
- N. Kopteva, Error analysis of the L1 method on graded and uniform meshes for a fractional-derivative problem in two and three dimensions, Math. Comp. 88 (2019), pp. 2135–2155.
- N. Laskin, Fractional quantum mechanics, Phys. Rev. E 62(3) (2000), p. 3135.
- N. Laskin, Fractional quantum mechanics and lévy path integrals, Phys. Lett. A 268(4–6) (2000), pp. 298–305.
- 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.
- M. Li, C. Huang, and P. Wang, Galerkin finite element method for nonlinear fractional Schrödinger equations, Numer. Algor. 74(2) (2017), pp. 499–525.
- D. Li, J. Wang, and J. Zhang, Unconditionally convergent L1-Galerkin FEMs for nonlinear time-fractional Schrödinger equations, SIAM J. Sci. Comput. 39(6) (2017), pp. A3067–A3088.
- M. Li, X.M. 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 W. Ming, A relaxation-type Galerkin fem for nonlinear fractional Schrödinger equations, Numer. Algor. (2019), in press.
- H.L. Liao, D. Li, and J. Zhang, Sharp error estimate of the nonuniform L1 formula for linear reaction-subdiffusion equations, SIAM J. Numer. Anal. 56(2) (2018), pp. 1112–1133.
- Y. Lin and C. Xu, Finite difference/spectral approximations for the time-fractional diffusion equation, J. Comput. Phys. 225(2) (2007), pp. 1533–1552.
- M. Naber, Time fractional Schrödinger equation, J. Math. Phys. 45(8) (2004), pp. 3339–3352.
- J. Shen, Z. Sun, and R. Du, Fast finite difference schemes for time-fractional diffusion equations with a weak singularity at initial time, East. Asia. J. Appl. Math. 8 (2018), pp. 834–858.
- 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(2) (2017), pp. 1057–1079.
- Z. Sun and X. Wu, A fully discrete difference scheme for a diffusion-wave system, Appl. Numer. Math. 56(2) (2006), pp. 193–209.
- S. Wang and M. Xu, Generalized fractional Schrödinger equation with space-time fractional derivatives, J. Math. Phys. 48(4) (2007), p. 041502.
- J.R. Wang, Y. Zhou, and W. Wei, Fractional Schrödinger equations with potential and optimal controls, Nonlinear Anal. Real World Appl. 13(6) (2012), pp. 2755–2766.
- L. Wei, Y. He, X. Zhang, and S. Wang, A numerical study based on an implicit fully discrete local discontinuous Galerkin method for the time-fractional coupled Schrödinger equation, Finite Elem. Anal. Des. 59 (2012), pp. 28–34.
- 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(8) (2012), pp. 2603–2615.
- M. Wyns, Convergence analysis of the modified Craig–Sneyd scheme for two-dimensional convection–diffusion equations with nonsmooth initial data, IMA J. Numer. Anal. 37(2) (2018), pp. 798–831.
- X. Zhu, Z. Yuan, J. Wang, Y. Nie, and Z. Yang, Finite element method for time-space-fractional Schrödinger equation, Electr. J. Differ. Equ. 2017(166) (2017), pp. 1–18.