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Applicable Analysis
An International Journal
Volume 97, 2018 - Issue 2
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Original Articles

Unconditional error analysis of Galerkin FEMs for nonlinear fractional Schrödinger equation

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Pages 295-315 | Received 24 Jul 2016, Accepted 15 Nov 2016, Published online: 28 Nov 2016

References

  • Guo X, Xu M. Some physical applications of fractional Schrödinger equation. J Math Phys. 2006;47:082104.
  • Lenzi E, Ribeiro H, Mukai H, et al. Continuous-time random walk as a guide to fractional Schrödinger equation. J Math Phys. 2010;51:092102.
  • Naber M. Time fractional Schrödinger equation. J Math Phys. 2004;45:3339–3352.
  • Laskin N. Fractional quantum mechanics. Phys Rev E. 2000;62:3135–3145.
  • Laskin N. Fractional quantum mechanics and Lévy path integrals. Phys Lett A. 2000;268:298–305.
  • Laskin N. Fractional Schrödinger equation. Phys Rev E. 2002;66:056108.
  • Kirkpatrick K, Lenzmann E, Staffilani G. On the continuum limit for discrete NLS with long-range lattice interactions. Commun Math Phys. 2013;317:563–591.
  • Aruna K, Kanth AR. Approximate solutions of non-linear fractional Schrodinger equation via differential transform method and modified differential transform method. Natl Acad Sci Lett. 2013;36:201–213.
  • Guo B, Han Y, Xin J. Existence of the global smooth solution to the period boundary value problem of fractional nonlinear Schrödinger equation. Appl Math Comput. 2008;204:468–477.
  • Hu J, Xin J, Lu H. The global solution for a class of systems of fractional nonlinear Schrödinger equations with periodic boundary condition. Comput Math Appl. 2011;62:1510–1521.
  • Jiang X, Qi H, Xu M. Exact solutions of fractional Schrödinger-like equation with a nonlocal term. J Math Phys. 2011;52:042105.
  • Muslih SI, Agrawal OP, Baleanu D. A fractional Schrödinger equation and its solution. Int J Theor Phys. 2010;49:1746–1752.
  • Hajaiej H. Existence of minimizers of functionals involving the fractional gradient in the absence of compactness, symmetry and monotonicity. J Math Anal Appl. 2012;399:17–26.
  • Ciaurri Ö, Roncal L, Stinga PR, et al. Fractional discrete Laplacian versus discretized fractional Laplacian. Mathematics. 2015. arXiv:1507.04986.
  • Mohebbi A, Abbaszadeh M, Dehghan M. 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. 2013;37:475–485.
  • Khan NA, Jamil M, Ara A. Approximate solutions to time-fractional Schrödinger equation via homotopy analysis method. ISRN Math Phys. 2012;2012:1–11.
  • Wei L, Zhang X, Kumar S, et al. A numerical study based on an implicit fully discrete local discontinuous Galerkin method for the time-fractional coupled Schrödinger system. Comput Math Appl. 2012;64:2603–2615.
  • Wei L, He Y, Zhang X, et al. Analysis of an implicit fully discrete local discontinuous Galerkin method for the time-fractional Schrödinger equation. Finite Elem Anal Des. 2012;59:28–34.
  • Amore P, Fernández FM, Hofmann CP, et al. Collocation method for fractional quantum mechanics. J Math Phys. 2010;51:12210101–12210116.
  • Wang D, Xiao A, Yang W. Crank--Nicolson difference scheme for the coupled nonlinear Schrödinger equations with the Riesz space fractional derivative. J Comp Phys. 2013;242:670–681.
  • Zhao X, Sun Z, Hao Z. A fourth-order compact adi scheme for two-dimensional nonlinear space fractional Schrödinger equation. SIAM J Sci Comput. 2014;36:A2865–A2886.
  • Kirkpatrick K, Zhang Y. Fractional Schrödinger dynamics and decoherence. Phys D Nonlinear Phenom. 2016:41–54. doi:10.1016/j.physd.2016.05.015 (Online).
  • Ran Y, Wang J, Wang D. On HSS-like iteration method for the space fractional coupled nonlinear Schrödinger equations. Appl Math Comput. 2015;271:482–488.
  • Wang P, Huang C. An energy conservative difference scheme for the nonlinear fractional Schrödinger equations. J Comp Phys. 2015;293:238–251.
  • Wang P, Huang C. Split-step alternating direction implicit difference scheme for the fractional Schrödinger equation in two dimensions. Comput Math Appl. 2016;71:1114–1128.
  • Wang P, Huang C, Zhao L. Point-wise error estimate of a conservative difference scheme for the fractional Schrödinger equation. J Comput Appl Math. 2016;306:231–247.
  • Wang P, Huang C. A conservative linearized difference scheme for the nonlinear fractional Schrödinger equation[J]. Numer Alg. 2015;69:625–641.
  • Podlubny I. Fractional differential equations: an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications. 1st ed. Vol. 198, Mathematics in science and engineering. New York (NY): Academic Press; 1999.
  • Caffarelli L, Silvestre L. An extension problem related to the fractional laplacian. Comm PDE. 2007;32:1245–1260.
  • Akrivis GD, Dougalis VA, Karakashian OA. On fully discrete Galerkin methods of second-order temporal accuracy for the nonlinear Schrödinger equation. Numer Math. 1991;59:31–53.
  • Li B. Mathematical modelling, analysis and computation of some complex and nonlinear flow problems [PhD thesis]. Hong Kong: City University of Hong Kong; 2012.
  • Li B, Sun W. Unconditional convergence and optimal error estimates of a Galerkin-mixed FEM for incompressible miscible flow in porous media. SIAM J Numer Anal. 2013;51:1959–1977.
  • Li B, Sun W. Error analysis of linearized semi-implicit Galerkin finite element methods for nonlinear parabolic equations. Int J Numer Anal Model. 2013;10:622–633.
  • Wang J. A new error analysis of Crank-Nicolson Galerkin FEMs for a generalized nonlinear Schrödinger equation. J Sci Comput. 2014;60:390–407.
  • Li M, Huang C, Wang P. Galerkin finite element method for nonlinear fractional Schrödinger equations. Numer Alg. Forthcoming 2016. Available from: http://dx.doi.org/10.1007/s11075-016-0160-5.
  • Ervin VJ, Roop JP. Variational formulation for the stationary fractional advection dispersion equation. Numer Methods PDE. 2006;22:558–576.
  • Roop JP. Variational solution of the fractional advection dispersion equation [PhD thesis]. South Carolina: Clemson University; 2004.
  • Li C, Zhao Z, Chen Y. Numerical approximation of nonlinear fractional differential equations with subdiffusion and superdiffusion. Comput Math Appl. 2011;62:855–875.
  • Zhang H, Liu F, Anh V. Galerkin finite element approximation of symmetric space-fractional partial differential equations. Appl Math Comput. 2010;217:2534–2545.
  • Brenner SC, Scott R. The mathematical theory of finite element methods. 3rd ed. Vol. 15, Texts in applied mathematics. New York (NY): Springer-Verlag; 2008.
  • Bu W, Liu X, Tang Y, et al. Finite element multigrid method for multi-term time fractional advection diffusion equations. Int J Mod Sim Sci Comp. 2015;6:154000101–154000125.

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