AIMS Mathematics, 2020, 5(3): 1642-1662. doi: 10.3934/math.2020111

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Chebyshev pseudospectral approximation of two dimensional fractional Schrodinger equation on a convex and rectangular domain

Discipline of Mathematics, IIITDM Jabalpur, Madhya Pradesh 482005, India

In this article, the authors report the Chebyshev pseudospectral method for solving twodimensional nonlinear Schrodinger equation with fractional order derivative in time and space both. The modified Riemann-Liouville fractional derivatives are used to define the new fractional derivatives matrix at CGL points. Using the Chebyshev fractional derivatives matrices, the given problem is reduced to a diagonally block system of nonlinear algebraic equations, which will be solved using Newton’s Raphson method. The proposed methods have shown error analysis without any dependency on time and space step restrictions. Some model examples of the equations, defined on a convex and rectangular domain, have tested with various values of fractional order α and β. Moreover, numerical solutions are demonstrated to justify the theoretical results.
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1. E. A. B. Abdel-Salam, E. A. Yousif, M. A. El-Aasser, Analytical solution of the space-time fractional nonlinear schrödinger equation, Rep. Math. Phys., 77 (2016), 19-34.    

2. Z. Asgari, S. Hosseini, Efficient numerical schemes for the solution of generalized time fractional burgers type equations, Numer. Algorithms, 77 (2018), 763-792.    

3. A. Bhrawy, M. Abdelkawy, A fully spectral collocation approximation for multi-dimensional fractional schrödinger equations, J. Comput. Phys., 294 (2015), 462-483.    

4. A. Bhrawy, M. A. Zaky, Highly accurate numerical schemes for multi-dimensional space variable-order fractional schrödinger equations, Comput. Math. Appl., 73 (2017), 1100-1117.    

5. J. P. Boyd. Chebyshev and Fourier Spectral Methods, Courier Corporation, 2001.

6. A. Chechkin, R. Gorenflo, I. Sokolov, Retarding subdiffusion and accelerating superdiffusion governed by distributed-order fractional diffusion equations, Phys. Rev. E, 66 (2002), 046129.

7. J. B. Chen, M. Z. Qin, Y. F. Tang, Symplectic and multi-symplectic methods for the nonlinear schrödinger equation, Comput. Math. Appl., 43 (2002), 1095-1106.    

8. X. Cheng, J. Duan, D. Li, A novel compact adi scheme for two-dimensional riesz space fractional nonlinear reaction-diffusion equations, Appl. Math. Comput., 346 (2019), 452-464.

9. M. Dehghan, A. Taleei, A compact split-step finite difference method for solving the nonlinear schrödinger equations with constant and variable coefficients, Comput. Phys. Commun., 181 (2010), 43-51.    

10. K. Diethelm, The Analysis of Fractional Differential Equations: An Application-oriented Exposition Using Differential Operators of Caputo Type, Springer Science and Business Media, 2010.

11. J. Dong, Scattering problems in the fractional quantum mechanics governed by the 2d spacefractional schrödinger equation, J. Math. Phys., 55 (2014), 032102.

12. J. Dong, M. Xu, Some solutions to the space fractional schrödinger equation using momentum representation method, J. Math. Phys., 48 (2007), 072105.

13. W. Fan, F. Liu, A numerical method for solving the two-dimensional distributed order spacefractional diffusion equation on an irregular convex domain, Appl. Math. Lett., 77 (2018), 114-121.    

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

15. B. Jin, B. Li, Z. Zhou, Subdiffusion with a time-dependent coefficient: Analysis and numerical solution, Math. Comput., 88 (2019), 2157-2186.    

16. G. Jumarie, An approach to differential geometry of fractional order via modified riemannliouville derivative, Acta Math. Sin., 28 (2012), 1741-1768.    

17. D. Li, J. Wang, J. Zhang, Unconditionally convergent l1-galerkin fems for nonlinear timefractional schrodinger equations, SIAM J. Sci. Comput., 39 (2017), A3067-A3088.

18. D. Li, C. Wu, Z. Zhang, Linearized galerkin fems for nonlinear time fractional parabolic problems with non-smooth solutions in time direction, J. Sci Comput., 80 (2019), 403-419.    

19. L. Li, D. Li, Exact solutions and numerical study of time fractional burgers' equations, Appl. Math. Lett., 100 (2020), 106011.

20. M. Li, A high-order split-step finite difference method for the system of the space fractional cnls, Eur. Phys. J. Plus, 134 (2019), 244.

21. M. Li, X. M. Gu, C. Huang, et al. A fast linearized conservative finite element method for the strongly coupled nonlinear fractional schrödinger equations, J. Comput. Phys., 358 (2018), 256-282.    

22. M. Li, C. Huang, W. Ming, A relaxation-type galerkin fem for nonlinear fractional schrödinger equations. Numer. Algorithms, 83 (2019), 99-124.

23. M. Li, C. Huang, P. Wang, Galerkin finite element method for nonlinear fractional schrödinger equations, Numer. Algorithms, 74 (2017), 499-525.    

24. M. Li, C. Huang, Z. Zhang, Unconditional error analysis of galerkin fems for nonlinear fractional schrödinger equation, Appl. Anal., 97 (2018), 295-315.    

25. M. Li, C. Huang, Y. Zhao, Fast conservative numerical algorithm for the coupled fractional kleingordon-schrödinger equation, Numer. Algorithms, 82 (2019), 1-39.    

26. M. Li, D. Shi, J. Wang, et al. Unconditional superconvergence analysis of the conservative linearized galerkin fems for nonlinear klein-gordon-schrödinger equation, Appl. Numer. Math., 142 (2019), 47-63.    

27. Y. Lin, C. Xu, Finite difference/spectral approximations for the time-fractional diffusion equation, J. Comput. Phys., 225 (2007), 1533-1552.    

28. Y. F. Luchko, M. Rivero, J. J. Trujillo, et al. Fractional models, non-locality, and complex systems, Comput. Math. Appl., 59 (2010), 1048-1056.    

29. J. T. Machado, V. Kiryakova, F. Mainardi, Recent history of fractional calculus, Commun. Nonlinear Sci., 16 (2011), 1140-1153.    

30. Z. Mao, G. E. Karniadakis, Fractional burgers equation with nonlinear non-locality: Spectral vanishing viscosity and local discontinuous galerkin methods, J. Comput. Phys., 336 (2017), 143-163.    

31. A. K. Mittal, A stable time-space Jacobi pseudospectral method for two-dimensional sine-Gordon equation, J. Appl. Math. Comput., 63 (2020), 1-26.

32. A. K. Mittal, L. K. Balyan, A highly accurate time-space pseudospectral approximation and stability analysis of two dimensional brusselator model for chemical systems, Int. J. Appl. Comput. Math., 5 (2019), 140.

33. A. Mohebbi, Analysis of a numerical method for the solution of time fractional burgers equation, B. Iran. Math. Soc., 44 (2018), 457-480.    

34. A. Mohebbi, M. Dehghan, The use of compact boundary value method for the solution of twodimensional schrödinger equation, J. Comput. Appl. Math., 225 (2009), 124-134.    

35. K. Oldham, J. Spanier, The Fractional Calculus Theory and Applications of Differentiation and Integration to Arbitrary Order, Elsevier, 1974.

36. B. Ross, The development of fractional calculus, Hist. Math., 4 (1977), 75-89.    

37. H. Rudolf, Applications of Fractional Calculus in Physics, World Scientific, 2000.

38. L. N. Trefethen, Finite Difference and Spectral Methods for Ordinary and Partial Differential Equations, 1996.

39. L. Wei, Y. He, X. Zhang, et al. Analysis of an implicit fully discrete local discontinuous galerkin method for the time-fractional schrödinger equation, Finite Elem. Anal. Des., 59 (2012), 28-34.    

40. E. Yousif, E. B. Abdel-Salam, M. El-Aasser, On the solution of the space-time fractional cubic nonlinear schrödinger equation, Results Phys., 8 (2018),702-708.    

41. F. Zeng, C. Li, F. Liu, et al. The use of finite difference/element approaches for solving the timefractional subdiffusion equation, SIAM J. Sci. Comput., 35 (2013), A2976-A3000.

42. G. Zhang, C. Huang, M. Li, A mass-energy preserving galerkin fem for the coupled nonlinear fractional schrödinger equations, Eur. Phys. J. Plus, 133 (2018), 155.

43. H. Zhang, X. Jiang, C. Wang, et al. Galerkin-legendre spectral schemes for nonlinear space fractional schrödinger equation, Numer. Algorithms, 79 (2018), 337-356.    

44. X. Zhao, Z. Z. Sun, Z. P. Hao, A fourth-order compact adi scheme for two-dimensional nonlinear space fractional schrodinger equation, SIAM J. Sci. Comput., 36 (2014), A2865-A2886.

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