A new iteration method for solving space fractional coupled nonlinear Schrödinger equations

Document Type : Research Article

Authors

• H. Aslani ¹
• D. Khojasteh Salkuyeh ²
• M. Taghipour ¹

¹ Faculty of Mathematical Sciences, University of Guilan, Rasht, Iran.

² Faculty of Mathematical Sciences, and Center of Excellence for Mathematical Modelling Optimization and Combinational Computing (MMOCC), University of Guilan, Rasht, Iran.

10.22067/ijnao.2022.77745.1163

Abstract

A linearly implicit difference scheme for the space fractional coupled nonlinear Schrödinger equation is proposed. The resulting coefficient matrix of the discretized linear system consists of the sum of a complex scaled identity and a symmetric positive definite, diagonal-plus-Toeplitz, matrix. An efficient block Gauss–Seidel over-relaxation (BGSOR) method has been established to solve the discretized linear system. It is worth noting that the proposed method solves the linear equations without the need for any system solution, which is beneficial for reducing computational cost and memory requirements. Theoretical analysis implies that the BGSOR method is convergent under a suitable condition. Moreover, an appropriate approach to compute the optimal parameter in the BGSOR method is exploited. Finally, the theoretical analysis is validated by some numerical experiments.

Keywords

• The space fractional Schrödinger equations
• Toeplitz matrix
• Block Gauss-Seidel over-relaxation method
• Convergence analysis

Main Subjects

• Numerical methods for differential equations and its applications

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References

1. Atangana, A. and Cloot, A.H. Stability and convergence of the space fractional variable-order Schrödinger equation, Adv. Diff. Equ. 2013 (2013) 80.

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

3. Chang, Y. and Chen, H. Fourth-order finite difference scheme and efficient algorithm for nonlinear fractional Schrödinger equations, Adv. 

4. Dai, P. and Wu, Q. An efficient block Gauss-Seidel iteration method forthe space fractional coupled nonlinear Schrödinger equations, Appl. Math. Lett. 117 (2021) 107–116.

5. Demengel, F. and Demengel, G. Fractional sobolev spaces, in: Functional spaces for the theory of elliptic partial differential equations, Springer, London, (2012) 179–228.

6. Laskin, N. Fractional quantum mechanics and Lévy path integrals, Phys. Lett. A 268 (2000) 298–305.
7. Ortigueira, M.D. Riesz potential operators and inverses via fractional centred derivatives, Int. J. Math. Math. Sci. (2006) 1–12.

8. Saad, Y. Iterative methods for sparse linear systems, Second edition PWS, New York, 1995.
9. Saad, Y. and Schultz, M.H. GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems, SIAM J. Sci. and Stat. Comput. 7 (1986) 856–869.

10. Sun, Z.Z. and Gao, G.h. Fractional differential equations, De Gruyter, 2020.
11. Wang, D., Xiao, A. and Yang, W. Crank-Nicolson difference scheme for the coupled nonlinear Schrödinger equations with the Riesz space fractional derivative, J. Comput. Phys. 242 (2013) 670–681.

12. Wang, D., Xiao, A. and Yang, W. A linearly implicit conservative difference scheme for the space fractional coupled nonlinear Schrödinger equations, J. Comput. Phys. 272 (2014) 644–655.
13. Wang, D., Xiao, A. and Yang, W. Maximum-norm error analysis of a difference scheme for the space fractional CNLS, Appl. Math. Comput. 257 (2015) 241–251.

14. Young, D.M. Iterative solution or large linear systems, Academic Press, New York, 1971.