On the numerical simulation of time-space fractional coupled nonlinear Schrödinger equations utilizing Wendland’s compactly supported function collocation method
Abstract
This research describes an efficient numerical method based on Wendland’s compactly supported functions to simulate the time-space fractional coupled nonlinear Schrödinger (TSFCNLS) equations. Here, the time and space fractional derivatives are considered in terms of Caputo and Conformable derivatives, respectively. The present numerical discussion is based on the following ways: we first approximate the Caputo fractional derivative of the proposed equation by a scheme order O(∆t2−α), 0 < α < 1 and then the Crank-Nicolson scheme is employed in the mentioned equation to discretize the equations. Second, applying a linear difference scheme to avoid solving nonlinear systems. In this way, we have a linear, suitable calculation scheme. Then, the conformable fractional derivatives of the Wendland’s compactly supported functions are established for the scheme. The stability analysis of the suggested scheme is also examined in a similar way to the classic Von-Neumann technique for the governing equations. The efficiency and accuracy of the present method are verified by solving two examples.
Keyword : fractional coupled nonlinear Schrödinger equations, Crank-Nicolson method, Wendland functions, fractional derivative operator, Von-Neumann stability
How to Cite
Karaman, B. (2021). On the numerical simulation of time-space fractional coupled nonlinear Schrödinger equations utilizing Wendland’s compactly supported function collocation method. Mathematical Modelling and Analysis, 26(1), 94-115. https://doi.org/10.3846/mma.2021.12262
This work is licensed under a Creative Commons Attribution 4.0 International License.
References
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D. Wang, A.G. Xiao and W. Yang. A linearly implicit conservative difference scheme for the space fractional coupled nonlinear Schro¨dinger equations. J. Comput. Phys., 272:644–655, 2014. https://doi.org/10.1016/j.jcp.2014.04.047
S. Wang and M. Xu. Generalized fractional Schrödinger equation with space time fractional derivatives. Journal of Mathematical Physics, 48, 2007. https://doi.org/10.1063/1.2716203
Y. Wang and L. Mei. A conservative spectral Galerkin method for the coupled nonlinear space-fractional Schrödinger equations. Int. J. Comput. Math., 96(12):2387–2410, 2019. https://doi.org/10.1080/00207160.2018.1563687
H. Wendland. Scattered data approximation, volume 17. Cambridge Monographs on Applied and Computational Mathematics, Cambridge University Press, Cambridge, 2005.
O. P. Yadav and R. Jiwari. Some soliton-type analytical solutions and numerical simulation of nonlinear Schrödinger equation. Nonlinear Dynamics, 95(4):2825– 2836, 2019. https://doi.org/10.1007/s11071-018-4724-x
M.A. Akinlar, A. Secer and M. Bayram. Numerical solution of fractional Benney equation. Appl. Math. Inf. Sci., 8:1–5, 2014. https://doi.org/10.12785/amis/080418
A. Bhrawy, E. Doha, S. Ezz-Eldien and R.A. Van Gonder. A new Jacobi spectral collocation method for solving 1+1 dimensional fractional Schrödinger equations and fractional coupled Schrödinger systems. Eur. Phys. J. Plus., 129:1–21, 2014. https://doi.org/10.1140/epjp/i2014-14260-6
M. Caputo. Linear model of dissipation whose Q is almost frequency independent - II. Geophys. J. R. Astron. Soc., 13:529–539, 1967. https://doi.org/10.1111/j.1365-246X.1967.tb02303.x
K.W. Chow. Periodic solutions for a system of four coupled nonlinear Schro¨dinger equations. Physics Lett. A, 285:319–326, 2001. https://doi.org/10.1016/S0375-9601(01)00369-3
K.W. Chow, K.K.Y. Wong and K.Lam. Modulation instabilities in a system of four coupled nonlinear Schrödinger equations. Physics Lett. A, 372:4596–4600, 2008. https://doi.org/10.1016/j.physleta.2008.04.057
Y. Dereli, D. Irk and I. Dağ. Soliton solutions for NLS equation using radial basis functions. Chaos Solitons and Fractals, 42:1227–1233, 2009. https://doi.org/10.1016/j.chaos.2009.03.030
X. Guo and M. Xu. Some physical applications of fractional Schrödinger equation. J. Math. Phys., 47, 2006. https://doi.org/10.1063/1.2235026
B. Hu, Y. Xu and J. Hu. Crank-Nicolson finite difference scheme for the Rosenau-Burgers equation. Appl. Math. Comput., 204(1):311–316, 2008. https://doi.org/10.1016/j.amc.2008.06.051
X. Hu and L. Zhang. Conservative compact difference scheme for the coupled nonlinear Schro¨dinger equation system. Numer. Methods Partial Differ. Eq., 30:749–772, 2014. https://doi.org/10.1002/num.21826
M. Ismail and T. Taha. A linearly implicit conservative scheme for the coupled nonlinear Schrödinger equation. Math. Comput. Sim., 74:302–311, 2007. https://doi.org/10.1016/j.matcom.2006.10.020
R. Jiwari, S. Kumar, R. C. Mittal and J. Awrejcewicz. A meshfree approach for analysis and computational modeling of non-linear Schrödinger equation. Nonlinear Dynamics, 39(95), 2020. https://doi.org/10.1007/s40314-020-1113-0
B. Karaman and Y. Dereli. A Meshless method for the coupled nonlinear Schrödinger equations. Sleyman Demirel University Faculty of Arts and Sciences Journal of Science, 14(2):418–435, 2019. https://doi.org/10.29233/sdufeffd.592437
B. Peterson M. Lisak and H. Wilhelmsson. Coupled nonlinear Schrödinger equations including growth and damping. Physics Lett. A, 66:83–85, 1978. https://doi.org/10.1016/0375-9601(78)90002-6
S.V. Manakov. On the theory of two-dimensional stationary self focusing of electromagnetic waves. Soviet Physics JETP, 38(2):248–253, 1974.
K.S. Miller and B. Ross. An introduction to the fractional calculus and fractional differential equations. John Wiley and Sons Inc., New York, 1993.
K.B. Oldham and J. Spanier. The fractional calculus. Academic Press, New York - London, 1974.
I. Podlubny. Fractional differential equations. Academic Press, San Diego, 1999.
F. Zheng F Q. Liu and C. Li. Finite difference method for time-space fractional Schro¨dinger equations. Int. J. Comput. Math., 242:670–681, 2013.
A. Yousef A R. Khalil, M.Al. Horani and M. Sababheh. A new definition of fractional derivative. J. Comput. Appl. Math., 264:65–70, 2014. https://doi.org/10.1016/j.cam.2014.01.002
M. Ran and C. Zhang. A conservative difference scheme for solving the strongly coupled nonlinear fractional Schrödinger equations. Commun. Numer. Sci. Numer. Simul., 41:64–83, 2016. https://doi.org//10.1016/j.cnsns.2016.04.026
M. Ran and C. Zhang. Linearized Crank-Nicolson scheme for the nonlinear timespace fractional Schrödinger equations. J. Comput. Appl. Math., 355:218–231, 2019. https://doi.org/10.1016/j.cam.2019.01.045
R. Schaback. Creating surfaces from scattered data using radial basis functions. Mathematical Methods for Curves and Surfaces, pp. 477–496, 1995.
R. Schaback. Error estimates and condition numbers for radial basis function interpolation. Advances in Comput. Math., 32(2):251–264, 1995. https://doi.org/10.1007/BF02432002
Z. Sun and X. Wu. A fully discrete difference scheme for a diffusion-wave system. Appl. Num. Math., 56(2):193–209, 2006. https://doi.org/10.1016/j.apnum.2005.03.003
N.H. Sweilam and M.M. Abou Hasan. Numerical solutions of a general coupled nonlinear system of parabolic and hyperbolic equations of thermoelasticity. Eur. Phys. J. Plus, 132:212–228, 2017. https://doi.org/10.1140/epjp/i2017-11484-x
M. Wadati, T. Izuka and M. Hisakado. A coupled nonlinear Schrödinger equation and optical solitons. J. Phys. Soc., Japan, 61:2241–2245, 1992. https://doi.org/10.1143/JPSJ.61.2241
D. Wang, A.G. Xiao and W. Yang. Crank-Nicolson difference scheme for the coupled nonlinear Schrödinger equations with the Riesz space fractional derivative. J. Comput. Phys., 242:670–681, 2013. https://doi.org/10.1016/j.jcp.2013.02.037
D. Wang, A.G. Xiao and W. Yang. A linearly implicit conservative difference scheme for the space fractional coupled nonlinear Schro¨dinger equations. J. Comput. Phys., 272:644–655, 2014. https://doi.org/10.1016/j.jcp.2014.04.047
S. Wang and M. Xu. Generalized fractional Schrödinger equation with space time fractional derivatives. Journal of Mathematical Physics, 48, 2007. https://doi.org/10.1063/1.2716203
Y. Wang and L. Mei. A conservative spectral Galerkin method for the coupled nonlinear space-fractional Schrödinger equations. Int. J. Comput. Math., 96(12):2387–2410, 2019. https://doi.org/10.1080/00207160.2018.1563687
H. Wendland. Scattered data approximation, volume 17. Cambridge Monographs on Applied and Computational Mathematics, Cambridge University Press, Cambridge, 2005.
O. P. Yadav and R. Jiwari. Some soliton-type analytical solutions and numerical simulation of nonlinear Schrödinger equation. Nonlinear Dynamics, 95(4):2825– 2836, 2019. https://doi.org/10.1007/s11071-018-4724-x