Crank-Nicolson difference scheme for the coupled nonlinear Schrödinger equations with the Riesz space fractional derivative

Journal of Computational Physics

Volumn 242, Issue , 2013, Pages 670-681

  Wang, Dongling ^a   Xiao, Aiguo ^a   Yang, Wei ^a  

^a XIANGTAN UNIVERSITY   (China)

Author keywords

Crank Nicolson scheme; Fractional centered difference; Fractional Schr dinger equation

Indexed keywords

FIXED POINT ARITHMETIC;

BROUWER FIXED POINT THEOREM; CRANK-NICOLSON; CRANK-NICOLSON SCHEME; DIFFERENCE SCHEMES; DIFFERENCE SOLUTIONS; FRACTIONAL CENTERED DIFFERENCE; FRACTIONAL DERIVATIVES; FRACTIONAL SCHRÖDINGE EQUATION; RIESZ SPACES; STABILITY AND CONVERGENCE;

NONLINEAR EQUATIONS;

EID: 84875797647     PISSN: 00219991     EISSN: 10902716     Source Type: Journal    
DOI: 10.1016/j.jcp.2013.02.037     Document Type: Article

References (32)

1. Laskin N. Fractional quantum mechanics. Phys. Rev. E 2000, 62:3135-3145.
2. Laskin N. Fractional quantum mechanics and Lévy path integrals. Phys. Lett. A 2000, 268(4-6):298-305.
3. Guo X., Xu M. Some physical applications of fractional Schrödinger equation. J. Math. Phys. 2006, 47:82-104.
4. Fujioka J., Espinosa A., Rodríguez R.F. Fractional optical solitons. Phys. Lett. A 2010, 374:1126-1134.
5. Amore P., Fernández F.M., Hofmann C.P., Sáenz R.A. Collocation method for fractional quantum mechanics. J. Math. Phys. 2010, 51:122101.
6. Leble S., Reichel B. Coupled nonlinear Schrödinger equations in optic fibers theory: from general to solitonic aspects. Eur. Phys. J. Spec. Top. 2009, 173:5-55.
7. Bao W., Jaksch D. An explicit unconditionally stable numerical method for solving damped nonlinear Schrödinger equations with a focusing nonlinearity. SIAM J. Numer. Anal. 2003, 41:1406-1426.
8. Tarasov V.E. Fractional Dynamics: Application of Fractional Calculus to Dynamics of Particles, Fields and Media 2010, Higher Education Press and Springer.
9. Hilfer R. Applications of Fractional Calculus in Physics 1999, World Scientific, Singapore.
10. Kilbas A.A., Srivastava H.M., Trujillo J.J. Theory and Applications of Fractional Dfferential Equations 2006, Elsevier, Amsterdam.
11. 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.
12. 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.
13. Wang T., Nie T., Zhang L. Analysis of a symplectic difference scheme for a coupled nonlinear Schrödinger system. J. Comput. Appl. Math. 2009, 231(2):745-759.
14. Wang T., Guo B., Zhang L. New conservative difference schemes for a coupled nonlinear Schrödinger system. Appl. Math. Comput. 2010, 217:1604-1619.
15. L∞ convergence of a difference scheme for coupled nonlinear Schrödinger equations. Comput. Math. Appl. 2010, 59:3286-3300.
16. Ismail M.S. A fourth-order explicit schemes for the coupled nonlinear Schrödinger equation. Appl. Math. Comput. 2008, 196:273-284.
17. F. Ivanauskas, M. Radžiunas, On convergence and stability of the explicit difference method for solution of nonlinear Schrödinger equations, SIAM J. Numer. Anal. 36 (1999) 1466-1481.
18. Gauckler L., Lubich C. Nonlinear Schrödinger equations and their spectral discretizations over long times. Found. Comput. Math. 2010, 10:141-169.
19. Sun J., Qin M. Multi-symplectic methods for the coupled 1D nonlinear Schrödinger system. Comput. Phys. Commun. 2003, 155:221-235.
20. Aydin A., Karasözen B. Multi-symplectic integration of coupled nonlinear Schrödinger system with soliton solutions. Int. J. Comput. Math. 2009, 86:864-882.
21. Zhuang P., Liu F., Anh V., Turner I. Numerical methods for the variable-order fractional advection-diffusion equation with a nonlinear source term. SIAM J. Numer. Anal. 2009, 47:1760-1781.
22. Yang Q., Liu F., Turner I. Numerical methods for fractional partial differential equations with Riesz space fractional derivatives. Appl. Math. Model. 2010, 34(1):200-218.
23. Ilić M., Liu F., Turner I., Anh V. Numerical approximation of a fractional-in-space diffusion equation. I. Fract. Calc. Appl. Anal. 2005, 8(3):323-341.
24. S. Shen, F. Liu, V. Anh, I. Turner, J. Chen, A novel numerical approximation for the Riesz space fractional advection-dispersion equation, IMA J. Appl. Math., in press, doi:10.1093/imamat/hxs073.
25. Yang Q., Turner I., Liu F., Ilis Milos Novel numerical methods for solving the time-space fractional diffusion equation in 2D. SIAM J. Sci. Comput. 2011, 33:1159-1180.
26. Zhang H., Liu F., Anh V. Galerkin finite element approximations of symmetric space-fractional partial differential equations. Appl. Math. Comput. 2010, 217(6):2534-2545.
27. Ortigueira M.D. Riesz potential operators and inverses via fractional centred derivatives. Int. J. Math. Math. Sci. 2006, 1-12.
28. Çelik C., Duman M. Crank-Nicolson method for the fractional diffusion equation with the Riesz fractional derivative. J. Comput. Phys. 2012, 231:1743-1750.
29. Yu Q., Liu F., Turner I., Burrage K. A computationally effective alternating direction method for the space and time fractional Bloch-Torrey equation in 3-D. Appl. Math. Comput. 2012, 219:4082-4095.
30. Shen S., Liu F., Anh V. Numerical approximations and solution techniques for the space-time Riesz-Caputo fractional advection-diffusion equation. Numer. Algorithm 2011, 56(3):383-404.
31. Zhang H., Liu F. Numerical simulation of the Riesz fractional diffusion equation with a nonlinear source term. J. Appl. Math. Inf. 2008, 26(1-2):1-14.
32. Shen S., Liu F., Anh V., Turner I. The fundamental solution and numerical solution of the Riesz fractional advection-dispersion equation. IMA J. Appl. Math. 2008, 73:850-872.