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Combined Galerkin spectral/finite difference method over graded meshes for the generalized nonlinear fractional Schrödinger equation

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Abstract

The main aim of this paper is to construct an efficient Galerkin–Legendre spectral approximation combined with a finite difference formula of L1 type to numerically solve the generalized nonlinear fractional Schrödinger equation with both space- and time-fractional derivatives. We discretize the Riesz space-fractional derivative using the Legendre–Galerkin spectral method and the time-fractional derivative using the L1 scheme on nonuniform meshes. The stability and convergence analyses of the numerical scheme are studied in detail. The scheme is unconditionally stable and convergent of \(\min \{\kappa \theta ,2-\theta \}\) order convergence in time and of spectral accuracy in space, where \(\theta \) is the order of fractional derivative and \(\kappa \) is the grading mesh parameter. To verify the efficiency of the proposed algorithm, two numerical test problems are performed with convergence and error analysis.

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References

  1. Aboelenen, T.: A high-order nodal discontinuous Galerkin method for nonlinear fractional Schrödinger type equations. Commun. Nonlinear Sci. Numer. Simul. 54, 428–452 (2018)

    MathSciNet  MATH  Google Scholar 

  2. Aboelenen, T.: Local discontinuous Galerkin method for distributed-order time and space-fractional convection–diffusion and Schrödinger-type equations. Nonlinear Dyn. 92(2), 395–413 (2018)

    MATH  Google Scholar 

  3. Alikhanov, A.: A priori estimates for solutions of boundary value problems for fractional-order equations. Differ. Equ. 46(5), 660–666 (2010)

    MathSciNet  MATH  Google Scholar 

  4. Amore, P., Fernández, F.M., Hofmann, C.P., Sáenz, R.A.: Collocation method for fractional quantum mechanics. J. Math. Phys. 51(12), 122101 (2010)

    MathSciNet  MATH  Google Scholar 

  5. Antoine, X., Tang, Q., Zhang, J.: On the numerical solution and dynamical laws of nonlinear fractional Schrödinger/Gross-Pitaevskii equations. Int. J. Comput. Math. 95(6–7), 1423–1443 (2018)

    MathSciNet  Google Scholar 

  6. Bhrawy, A., Doha, E., Ezz-Eldien, S., Van Gorder, R.A.: A new Jacobi spectral collocation method for solving 1+ 1 fractional Schrödinger equations and fractional coupled Schrödinger systems. Eur. Phys. J. Plus 129(12), 260 (2014)

    Google Scholar 

  7. Bhrawy, A., Zaky, M.: An improved collocation method for multi-dimensional space-time variable-order fractional Schrödinger equations. Appl. Numer. Math. 111, 197–218 (2017)

    MathSciNet  MATH  Google Scholar 

  8. Bhrawy, A., Zaky, M.: Numerical simulation of multi-dimensional distributed-order generalized Schrödinger equations. Nonlinear Dyn. 89(2), 1415–1432 (2017)

    MATH  Google Scholar 

  9. Bhrawy, A.H., Abdelkawy, M.A.: A fully spectral collocation approximation for multi-dimensional fractional Schrödinger equations. J. Comput. Phys. 294, 462–483 (2015)

    MathSciNet  MATH  Google Scholar 

  10. Bhrawy, A.H., Alzaidy, J.F., Abdelkawy, M.A., Biswas, A.: Jacobi spectral collocation approximation for multi-dimensional time-fractional Schrödinger equations. Nonlinear Dyn. 84(3), 1553–1567 (2016)

    Google Scholar 

  11. Bhrawy, A.H., Zaky, M.A.: Highly accurate numerical schemes for multi-dimensional space variable-order fractional Schrödinger equations. Comput. Math. Appl. 73(6), 1100–1117 (2017)

    MathSciNet  MATH  Google Scholar 

  12. Castillo, P., Gómez, S.: On the conservation of fractional nonlinear Schrödinger equation’s invariants by the local discontinuous Galerkin method. J. Sci. Comput. 77(3), 1444–1467 (2018)

    MathSciNet  MATH  Google Scholar 

  13. Chen, X., Di, Y., Duan, J., Li, D.: Linearized compact ADI schemes for nonlinear time-fractional Schrödinger equations. Appl. Math. Lett. 84, 160–167 (2018)

    MathSciNet  MATH  Google Scholar 

  14. Cheng, B., Guo, Z., Wang, D.: Dissipativity of semilinear time fractional subdiffusion equations and numerical approximations. Appl. Math. Lett. 86, 276–283 (2018)

    MathSciNet  MATH  Google Scholar 

  15. Cho, Y., Hwang, G., Kwon, S., Lee, S.: Well-posedness and ill-posedness for the cubic fractional Schrödinger equations. Discrete Contin. Dyn. Syst. A 35(7), 2863 (2015)

    MathSciNet  MATH  Google Scholar 

  16. Dong, J., Xu, M.: Space-time fractional Schrödinger equation with time-independent potentials. J. Math. Anal. Appl. 344(2), 1005–1017 (2008)

    MathSciNet  MATH  Google Scholar 

  17. Ervin, V.J., Roop, J.P.: Variational solution of fractional advection dispersion equations on bounded domains in \({\mathbb{R}}^d\). Numer. Methods Part. Differ. Equ. Int. J. 23(2), 256–281 (2007)

    MATH  Google Scholar 

  18. Fan, W., Qi, H.: An efficient finite element method for the two-dimensional nonlinear time-space fractional Schrödinger equation on an irregular convex domain. Appl. Math. Lett. 86, 103–110 (2018)

    MathSciNet  MATH  Google Scholar 

  19. Fei, M., Wang, N., Huang, C., Ma, X.: A second-order implicit difference scheme for the nonlinear time-space fractional Schrödinger equation. Appl. Numer. Math. 153, 399–411 (2020)

    MathSciNet  MATH  Google Scholar 

  20. Feng, B., et al.: On fractional nonlinear schrödinger equation with combined power-type nonlinearities. Discrete Contin. Dyn. Syst. A 39(8), 4565 (2019)

    MathSciNet  MATH  Google Scholar 

  21. 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. 204(1), 468–477 (2008)

    MathSciNet  MATH  Google Scholar 

  22. Guo, X., Xu, M.: Some physical applications of fractional Schrödinger equation. J. Math. Phys. 47(8), 082104 (2006)

    MathSciNet  MATH  Google Scholar 

  23. 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. 62(3), 1510–1521 (2011)

    MathSciNet  MATH  Google Scholar 

  24. Huang, J., Nie, N., Tang, Y.: A second order finite difference-spectral method for space fractional diffusion equations. Science China Math. 57(6), 1303–1317 (2014)

    MathSciNet  MATH  Google Scholar 

  25. Laskin, N.: Fractional quantum mechanics and Lévy path integrals. Phys. Lett. A 268(4–6), 298–305 (2000)

    MathSciNet  MATH  Google Scholar 

  26. Laskin, N.: Fractional Schrödinger equation. Phys. Rev. E 66(5), 056108 (2002)

    MathSciNet  Google Scholar 

  27. Li, D., Wang, J., Zhang, J.: Unconditionally convergent L1-Galerkin FEMs for nonlinear time-fractional schrodinger equations. SIAM J. Sci. Comput. 39(6), A3067–A3088 (2017)

    MathSciNet  MATH  Google Scholar 

  28. Li, M., Ding, X., Xu, Q.: Non-polynomial spline method for the time-fractional nonlinear Schrödinger equation. Adv. Differ. Equ. 2018(1), 318 (2018)

    MATH  Google Scholar 

  29. Li, M., Huang, C., Wang, P.: Galerkin finite element method for nonlinear fractional Schrödinger equations. Numer. Algorithms 74(2), 499–525 (2017)

    MathSciNet  MATH  Google Scholar 

  30. Li, D., Zhang, J., Zhang, Z.: The numerical computation of the time fractional Schrödinger equation on an unbounded domain. Comput. Methods Appl. Math. 18(1), 77–94 (2017)

    MATH  Google Scholar 

  31. Liao, H.L., Li, D., Zhang, J.: Sharp error estimate of the nonuniform L1 formula for linear reaction–subdiffusion equations. SIAM J. Numer. Anal. 56(2), 1112–1133 (2018)

    MathSciNet  MATH  Google Scholar 

  32. Liu, Q., Zeng, F., Li, C.: Finite difference method for time–space–fractional Schrödinger equation. Int. J. Comput. Math. 92(7), 1439–1451 (2015)

    MathSciNet  MATH  Google Scholar 

  33. Longhi, S.: Fractional Schrödinger equation in optics. Opt. Lett. 40(6), 1117–1120 (2015)

    Google Scholar 

  34. Machado, J.T., Kiryakova, V., Mainardi, F.: Recent history of fractional calculus. Commun. Nonlinear Sci. Numer. Simul. 16(3), 1140–1153 (2011)

    MathSciNet  MATH  Google Scholar 

  35. Mainardi, F., Luchko, Y., Pagnini, G.: The fundamental solution of the space–time fractional diffusion equation. Fract. Calc. Appl. Anal. 4(2), 153–192 (2001)

    MathSciNet  MATH  Google Scholar 

  36. Metzler, R., Klafter, J.: The random walk’s guide to anomalous diffusion: a fractional dynamics approach. Phys. Rep. 339(1), 1–77 (2000)

    MathSciNet  MATH  Google Scholar 

  37. Naber, M.: Time fractional Schrödinger equation. J. Math. Phys. 45(8), 3339–3352 (2004)

    MathSciNet  MATH  Google Scholar 

  38. Podlubny, I.: Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications, vol. 198. Elsevier, Amsterdam (1998)

    MATH  Google Scholar 

  39. Ran, M., Zhang, C.: Linearized Crank–Nicolson scheme for the nonlinear time–space fractional Schrödinger equations. J. Comput. Appl. Math. 355, 218–231 (2019)

    MathSciNet  MATH  Google Scholar 

  40. Samko, S.G., Kilbas, A.A., Marichev, O.I.: Fractional Integrals and Derivatives, vol. 1. Gordon and Breach, New York (1993)

    MATH  Google Scholar 

  41. Shen, J.: Efficient spectral-Galerkin method I. Direct solvers of second-and fourth-order equations using Legendre polynomials. SIAM J. Sci. Comput. 15(6), 1489–1505 (1994)

    MathSciNet  MATH  Google Scholar 

  42. Shen, J., Tang, T., Wang, L.L.: Spectral Methods: Algorithms, Analysis and Applications, vol. 41. Springer, Berlin (2011)

    MATH  Google Scholar 

  43. Wang, D., Xiao, A., Yang, W.: Crank–Nicolson difference scheme for the coupled nonlinear Schrödinger equations with the Riesz space fractional derivative. J. Comput. Phys. 242, 670–681 (2013)

    MathSciNet  MATH  Google Scholar 

  44. Wang, P., Huang, C.: A conservative linearized difference scheme for the nonlinear fractional Schrödinger equation. Numer. Algorithms 69(3), 625–641 (2015)

    MathSciNet  MATH  Google Scholar 

  45. Wang, S., Xu, M.: Generalized fractional Schrödinger equation with space–time fractional derivatives. J. Math. Phys. 48(4), 043502 (2007)

    MathSciNet  MATH  Google Scholar 

  46. Yan, S., Zhao, F., Li, C., Zhao, L.: High order WSGL difference operators combined with Sinc–Galerkin method for time fractional Schrödinger equation. Int. J. Comput. Math. 97(11), 2259–2286 (2020)

    MathSciNet  Google Scholar 

  47. Zaky, M.A., Hendy, A.S.: Convergence analysis of an L1-continuous Galerkin method for nonlinear time–space fractional Schrödinger equations. Int. J. Comput. Math. (2020). https://doi.org/10.1080/00207160.2020.1822994

    Article  MATH  Google Scholar 

  48. Zaky, M.A., Hendy, A.S.: An efficient dissipation-preserving Legendre–Galerkin spectral method for the Higgs boson equation in the de Sitter spacetime universe. Appl. Numer. Math. 160, 281–295 (2021)

    MathSciNet  MATH  Google Scholar 

  49. Zaky, M.A., Hendy, A.S.: Graded mesh discretization for coupled system of nonlinear multi-term time–space fractional diffusion equations. Eng. Comput. (2020). https://doi.org/10.1007/s00366-020-01095-8

    Article  Google Scholar 

  50. Zaky, M.A., Hendy, A.S., Macías-Diaz, J.E.: High-order finite difference/spectral-Galerkin approximations for the nonlinear time–space fractional Ginzburg–Landau equation. Numer. Methods Part. Differ. Equ. (2020). https://doi.org/10.1002/NUM.22630

    Article  MATH  Google Scholar 

  51. Zeng, F., Liu, F., Li, C., Burrage, K., Turner, I., Anh, V.: A Crank–Nicolson ADI spectral method for a two-dimensional Riesz space fractional nonlinear reaction–diffusion equation. SIAM J. Numer. Anal. 52(6), 2599–2622 (2014)

    MathSciNet  MATH  Google Scholar 

  52. Zhang, H., Jiang, X.: Spectral method and bayesian parameter estimation for the space fractional coupled nonlinear Schrödinger equations. Nonlinear Dyn. 95(2), 1599–1614 (2019)

    MATH  Google Scholar 

  53. Zhang, H., Jiang, X., Wang, C., Chen, S.: Crank–Nicolson Fourier spectral methods for the space fractional nonlinear Schrödinger equation and its parameter estimation. Int. J. Comput. Math. 96(2), 238–263 (2019)

    MathSciNet  Google Scholar 

  54. Zhang, H., Jiang, X., Wang, C., Fan, W.: Galerkin–Legendre spectral schemes for nonlinear space fractional Schrödinger equation. Numer. Algorithms 79(1), 337–356 (2018)

    MathSciNet  MATH  Google Scholar 

  55. Zhang, H., Jiang, X., Yang, X.: A time–space spectral method for the time–space fractional Fokker–Planck equation and its inverse problem. Appl. Math. Comput. 320, 302–318 (2018)

    MathSciNet  MATH  Google Scholar 

  56. Zheng, M., Liu, F., Jin, Z.: The global analysis on the spectral collocation method for time fractional Schrödinger equation. Appl. Math. Comput. 365, 124689 (2020)

    MathSciNet  MATH  Google Scholar 

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Acknowledgements

The authors wish to thank the editor-in-chief Professor YangQuan Chen and the anonymous referees for their comments and criticism. All of their comments were taken into account in the revised version of the paper, resulting in a substantial improvement with respect to the original submission. Ahmed S. Hendy wishes to acknowledge the support of RFBR Grant 19-01-00019. Mahmoud A. Zaky wishes to acknowledge the financial support of the National Research Centre of Egypt (NRC).

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Authors and Affiliations

  1. Department of Computational Mathematics and Computer Science, Institute of Natural Sciences and Mathematics, Ural Federal University, 19 Mira St., Yekaterinburg, Russia, 620002

    Ahmed S. Hendy

  2. Department of Mathematics, Faculty of Science Benha University, Benha, 13511, Egypt

    Ahmed S. Hendy

  3. Department of Applied Mathematics, National Research Centre, Dokki, Cairo, 12622, Egypt

    Mahmoud A. Zaky

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  1. Ahmed S. Hendy
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ASH and MAZ contributed to writing—review and editing.

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Hendy, A.S., Zaky, M.A. Combined Galerkin spectral/finite difference method over graded meshes for the generalized nonlinear fractional Schrödinger equation. Nonlinear Dyn 103, 2493–2507 (2021). https://doi.org/10.1007/s11071-021-06249-x

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