Abstract
In this article, the authors proposed Chebyshev pseudospectral method for numerical solutions of two-dimensional nonlinear Schrodinger equation with fractional-order derivative in both time and space. Fractional-order partial differential equations are considered as generalizations of classical integer-order partial differential equations. The proposed method is established in both time and space to approximate the solutions. The Caputo 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 diagonally block system of nonlinear algebraic equations, which will be solved using Newton–Raphson iterative method. Some model examples of the equations, defined on a rectangular domain, have tested with various values of fractional order \(\alpha\) and \(\beta\). Moreover, numerical solutions are demonstrated to justify the theoretical results and confirm the expected convergence rate. For the proposed method, highly accurate numerical results are obtained which are compared with the analytical solution to confirm the accuracy and efficiency of the proposed method.
Similar content being viewed by others
References
Abdel-Salam, E.A., Yousif, E.A., El-Aasser, M.A.: Analytical solution of the space-time fractional nonlinear schrödinger equation. Rep. Math. Phys. 77(1), 19–34 (2016)
Asgari, Z., Hosseini, S.: Efficient numerical schemes for the solution of generalized time fractional Burgers type equations. Numer. Algorithms 77(3), 763–792 (2018)
Atangana, A., Cloot, A.H.: Stability and convergence of the space fractional variable-order Schrödinger equation. Adv. Differ. Equ. 2013(1), 80 (2013)
Boyd, J.P.: Chebyshev and Fourier Spectral Methods. Dover Publications, Mineola (2001)
Chang, Q., Jia, E., Sun, W.: Difference schemes for solving the generalized nonlinear Schrödinger equation. J. Comput. Phys. 148(2), 397–415 (1999)
Chechkin, A., Gorenflo, R., Sokolov, I.: Retarding subdiffusion and accelerating superdiffusion governed by distributed-order fractional diffusion equations. Phys. Rev. E 66(4), 046129 (2002)
Chen, J.-B., Qin, M.-Z., Tang, Y.-F.: Symplectic and multi-symplectic methods for the nonlinear Schrödinger equation. Comput. Math. Appl. 43(8–9), 1095–1106 (2002)
Dehghan, M., Taleei, A.: A compact split-step finite difference method for solving the nonlinear Schrödinger equations with constant and variable coefficients. Comput. Phys. Commun. 181(1), 43–51 (2010)
Diethelm, K.: The Analysis of Fractional Differential Equations: An Application-Oriented Exposition Using Differential Operators of Caputo Type. Springer, Berlin (2010)
Dong, J., Xu, M.: Some solutions to the space fractional Schrödinger equation using momentum representation method. J. Math. Phys. 48(7), 072105 (2007)
Fan, W., Liu, F.: A numerical method for solving the two-dimensional distributed order space-fractional diffusion equation on an irregular convex domain. Appl. Math. Lett. 77, 114–121 (2018)
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)
Fan, W., Liu, F., Jiang, X., Turner, I.: A novel unstructured mesh finite element method for solving the time-space fractional wave equation on a two-dimensional irregular convex domain. Fract. Cal. Appl. Anal. 20(2), 352–383 (2017)
Guo, X., Xu, M.: Some physical applications of fractional Schrödinger equation. J. Math. Phys. 47(8), 082104 (2006)
Jumarie, G.: An approach to differential geometry of fractional order via modified Riemann–Liouville derivative. Acta Math. Sin. Engl. Ser. 28(9), 1741–1768 (2012)
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)
Lin, Y., Xu, C.: Finite difference/spectral approximations for the time-fractional diffusion equation. J. Comput. Phys. 225(2), 1533–1552 (2007)
Luchko, Y.F., Rivero, M., Trujillo, J.J., Velasco, M.P.: Fractional models, non-locality, and complex systems. Comput. Math. Appl. 59(3), 1048–1056 (2010)
Machado, J.T., Kiryakova, V., Mainardi, F.: Recent history of fractional calculus. Commun. Nonlinear Sci. Numer. Simul. 16(3), 1140–1153 (2011)
Mao, Z., Karniadakis, G.E.: Fractional Burgers equation with nonlinear non-locality: spectral vanishing viscosity and local discontinuous Galerkin methods. J. Comput. Phys. 336, 143–163 (2017)
Mohebbi, A.: Analysis of a numerical method for the solution of time fractional Burgers equation. Bull. Iran. Math. Soc. 44(2), 457–480 (2018)
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. 37(2), 475–485 (2013)
Mohebbi, A., Dehghan, M.: The use of compact boundary value method for the solution of two-dimensional Schrödinger equation. J. Comput. Appl. Math. 225(1), 124–134 (2009)
Oldham, K., Spanier, J.: The Fractional Calculus Theory and Applications of Differentiation and Integration to Arbitrary Order, vol. 111. Elsevier, Amsterdam (1974)
Ross, B.: The development of fractional calculus 1695–1900. Hist. Math. 4(1), 75–89 (1977)
Rudolf, H.: Applications of Fractional Calculus in Physics. World Scientific, Singapore (2000)
Samko, S.G., Kilbas, A.A., Marichev, O.I.: Fractional Integrals and Derivatives-Theory and Applications. Gordon and Breach, Linghorne (1993)
Sweilam, N.H., Hasan, M.A.: Numerical solutions for 2-D fractional Schrödinger equation with the Riesz–Feller derivative. Math. Comput. Simul. 140, 53–68 (2017)
Trefethen, L.N.: Finite Difference And Spectral Methods For Ordinary And Partial Differential Equations (1996)
Wei, L., He, Y., Zhang, X., Wang, S.: Analysis of an implicit fully discrete local discontinuous Galerkin method for the time-fractional Schrödinger equation. Finite Elem. Anal. Des. 59, 28–34 (2012)
Yousif, E., Abdel-Salam, E.-B., El-Aasser, M.: On the solution of the space-time fractional cubic nonlinear Schrödinger equation. Results Phys. 8, 702–708 (2018)
Zeng, F., Li, C., Liu, F., Turner, I.: The use of finite difference/element approaches for solving the time-fractional subdiffusion equation. SIAM J. Sci. Comput. 35(6), A2976–A3000 (2013)
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)
Zhao, X., Sun, Z.-Z., Hao, Z.-P.: A fourth-order compact adi scheme for two-dimensional nonlinear space fractional Schrodinger equation. SIAM J. Sci. Comput. 36(6), A2865–A2886 (2014)
Acknowledgements
The first author thankfully acknowledges the Ministry of Human Resource Development, India, for providing financial support for this research.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Mittal, A.K., Balyan, L.K. Numerical solutions of two-dimensional fractional Schrodinger equation. Math Sci 14, 129–136 (2020). https://doi.org/10.1007/s40096-020-00323-y
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s40096-020-00323-y