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Prabhakar, Nidhi, Sharma, Seema. (1405). Numerical solution of linear and nonlinear Schrödinger equations via the shifted Chebyshev collocation method. سامانه مدیریت نشریات علمی, 14(1), 316-336. doi: 10.22034/cmde.2024.57475.2405
Nidhi Prabhakar; Seema Sharma. "Numerical solution of linear and nonlinear Schrödinger equations via the shifted Chebyshev collocation method". سامانه مدیریت نشریات علمی, 14, 1, 1405, 316-336. doi: 10.22034/cmde.2024.57475.2405
Prabhakar, Nidhi, Sharma, Seema. (1405). 'Numerical solution of linear and nonlinear Schrödinger equations via the shifted Chebyshev collocation method', سامانه مدیریت نشریات علمی, 14(1), pp. 316-336. doi: 10.22034/cmde.2024.57475.2405
Prabhakar, Nidhi, Sharma, Seema. Numerical solution of linear and nonlinear Schrödinger equations via the shifted Chebyshev collocation method. سامانه مدیریت نشریات علمی, 1405; 14(1): 316-336. doi: 10.22034/cmde.2024.57475.2405

Numerical solution of linear and nonlinear Schrödinger equations via the shifted Chebyshev collocation method

Computational Methods for Differential Equations
مقاله 22، دوره 14، شماره 1، فروردین 2026، صفحه 316-336    اصل مقاله (549.18 K)
نوع مقاله: Research Paper
شناسه دیجیتال (DOI): 10.22034/cmde.2024.57475.2405
نویسندگان
Nidhi Prabhakar؛ Seema Sharma*
Department of Mathematics and Statistics, Gurukul Kangri (Deemed to be University), Haridwar, Uttarakhand.
چکیده
The present study focuses on numerical solutions of linear and nonlinear Schrödinger equations subject to initial and boundary conditions employing the shifted Chebyshev spectral collocation method (SCSCM). In the solution procedure, unknown function and its space derivatives have been approximated employing shifted Chebyshev polynomials and their derivatives, respectively, together with Chebyshev-Gauss-Lobatto points. The present collocation method transforms the Schrödinger equation into a system of ordinary differential equations (ODEs). Thereafter, the obtained system has been solved employing the fourth-order Runge-Kutta scheme. To demonstrate the accuracy and efficiency of the present method, a comparison of the present numerical solutions of different examples of the Schrödinger equation with exact and approximate solutions available in the literature has been discussed. The SCSCM can be implemented to solve second and higher-order linear and nonlinear partial differential equations (PDEs) arising in physics, mechanics, and biophysics.
کلیدواژه‌ها
Shifted Chebyshev polynomials؛ Spectral collocation method؛ Schrödinger equation؛ Runge-Kutta method
مراجع
  • [1] Y. E. Aghdam, H. Safdari, Y. Azari, H. Jafari, and D. Baleanu, Numerical investigation of space fractional order diffusion equation by the Chebyshev collocation method of the fourth kind and compact finite difference scheme, Discrete Contin. Dyn. Syst. S, 14(7) (2021), 2025–2039.
  • [2] Y. E. Aghdam, H. Safdari, and M. Javidi, Numerical approach of the space fractional order diffusion equation based on the third kind of Chebyshev polynomials, In 4th International Conference on Combinatorics, Cryptography, Computer Science and Computing, (2019), 1231–1237.
  • [3] M. Ahsan, I. Ahmad, M. Ahmad, and I. Hussain, A numerical Haar wavelet finite difference hybrid method for linear and non-linear Schrödinger equation, Math. Comput. Simul., 169 (2019), 13–25.
  • [4] S. Ashrafi, M. Alineia, H. Kheiri, and G. Hojjati, Spectral collocation method for the numerical solution of the Gardner and Huxley equations, Int. J. Nonlinear Sci., 18 (2014), 71–77.
  • [5] Z. Avazzadeh, O. Nikan, and J. A. T. Machado, Solitary wave solutions of the generalized Rosenau-KdV-RLW equation, Mathematics., 8(9) (2020), 1601.
  • [6] A. H. Bhrawy, A Jacobi spectral collocation method for solving multi-dimensional nonlinear fractional sub-diffusion equations, Numer. Algor., 73 (2016), 91–113.
  • [7] J. P. Boyd, Chebyshev and Fourier spectral methods, Dover Publications Inc, New York, 2000.
  • [8] C. Canuto, M. Y. Hussaini, A. Quarteroni, and T. A. Zang, Spectral methods in fluid dynamics, Springer-Verlag, Berlin Heidelberg, 1988.
  • [9] Q. Chang, E. Jia, and W. Sun, Difference schemes for solving the generalized nonlinear Schrödinger equation, J. Comput. Phys., 148(2) (1999), 397–415.
  • [10] I. Dag, A quadratic B-spline finite element method for solving nonlinear Schrödinger equation, Comput. Methods Appl. Mech. Eng., 174(1-2) (1999), 247-258.
  • [11] M. Dehghan and A. Taleei, A Chebyshev Pseudospectral multidomain method for the soliton solution of the coupled nonlinear Schrödinger equations, Comput. Phys. Commun., 182(12) (2011), 2519–2529.
  • [12] M. Dehghan and A. Taleei, Numerical solution of nonlinear Schrödinger equation by using time-space pseudospectral method, Numer. Methods Partial Differ. Equ., 26(4) (2010), 979–992.
  • [13] E. H. Doha, A. H. Bhrawy, M. A. Abdelkawy, and R. A. Van Gorder, Jacobi-Gauss-Lobatto collocation method for the numerical solution of 1+1 nonlinear Schrödinger equations, J. Comput. Phys., 261 (2014), 244–255.
  • [14] Y. Duan and F. Rong, A numerical scheme for nonlinear Schrödinger equation by MQ quasi-interpolation, Eng. Anal. Bound. Elem., 37(1) (2013), 89–94.
  • [15] S. Jaiswal, M. Chopra, and S. Das, Numerical solution of non-linear partial differential equation for porous media using operational matrices, Math. Comput. Simul., 160 (2019), 138–154.
  • [16] A. Khan, M. Ahsan, E. Bonyah, R. Jan, M. Nisar, A. H. Abdel-Aty, and I.S. Yahia, Numerical solution of Schrödinger equation by Crank-Nicolson method, Math. Probl. Eng., 2022 (2022), 991067.
  • [17] A. H. Khater, R. S. Temsah, and M. M. Hassan, A Chebyshev spectral collocation method for solving Burgers’-type equations, J. Comput. Appl. Math., 222(2) (2008), 333-350.
  • [18] H. Li and Y. Wang, An averaged vector field Legendre spectral element method for the nonlinear Schrödinger equation, Int. J. Comput. Math., 94(6) (2017), 1196–1218.
  • [19] X. Liu, M. Ahsan, M. Ahmad, M. Nisar, X. Liu, I. Ahmad, and H. Ahmad, Applications of Haar wavelet-finite difference hybrid method and its convergence for hyperbolic nonlinear Schrödinger equation with energy and mass conversion, Energies., 14(23) (2021), 7831.
  • [20] H. Mesgarani, A. Beiranvand, and Y. Esmaeelzade Aghdam, The impact of the Chebyshev collocation method on solutions of the time-fractional Black-Scholes, Math. Sci., 15 (2021), 137–143.
  • [21] O., Nikan and Z. Avazzadeh, A locally stabilized radial basis function partition of unity technique for the sine–Gordon system in nonlinear optics, Math. Comput. Simul., 199 (2022), 394–413.
  • [22] O. Nikan, Z. Avazzadeh, and M. N. Rasoulizadeh, Soliton wave solutions of nonlinear mathematical models in elastic rods and bistable surfaces, Eng. Anal. Bound. Elem., 143 (2022), 14–27.
  • [23] O. Oruc, A. Esen, and F. Bulut, A Haar wavelet collocation method for coupled nonlinear Schr¨odinger–KdV equations, Int. J. Mod. Phys. C., 27(9) (2016), 1650103.
  • [24] N. Pervaiz and I. Aziz, Haar wavelet approximation for the solution of cubic nonlinear Schrödinger equations, Physica A: Stat. Mech. Appl., 545 (2020), 123738.
  • [25] M. N. Rasoulizadeh, Z. Avazzadeh, and O. Nikan, Solitary wave propagation of the generalized Kuramoto-Sivashinsky equation in fragmented porous media, Int. J. Appl. Comput. Math., 8 (2022), 252.
  • [26] M. P. Robinson, The solution of nonlinear Schr¨odinger equations using orthogonal spline collocation, Comput. Math. Appl., 33(7) (1997), 39–57.
  • [27] H. Safdari, Y. E. Aghdam, and J. F. G´omez-Aguilar, Shifted Chebyshev collocation of the fourth kind with convergence analysis for the space–time fractional advection-diffusion equation, Eng. Comput., 38 (2022), 1409–1420.
  • [28] B. Saka, A quintic B-spline finite element method for solving the nonlinear Schrödinger equation, Phys. Wave Phenom., 20 (2012), 107–117.
  • [29] S. Sharma, U. S. Gupta, and R. Lal, Effect of Pasternak foundation on axisymmetric vibration of polar orthotropic annular plates of varying thickness, J. Vib. Acoust., 132(4) (2010), 041001.
  • [30] Q. Sheng, A. Q. M. Khaliq, and E. A. Al-Said, Solving the generalized nonlinear Schrödinger equation via quartic spline approximation, J. Comput. Phys., 166(2) (2001), 400–417.
  • [31] M. Smadi and D. Bahloul, A compact split step Pade scheme for higher order nonlinear Schrödinger equation with power law nonlinearity and fourth order dispersion, Comput. Phys. Commun., 182(2) (2011), 366–371.
  • [32] M. A. Snyder, Chebyshev Methods in Numerical Approximation, Prentice-Hall, Inc., 1996.
  • [33] A. Taleei and M. Dehghan, Time-splitting pseudo-spectral domain decomposition method for the soliton solutions of the one and multi-dimensional nonlinear Schrödinger equations, Comput. Phys. Commun., 185(6) (2014), 1515–1528.
  • [34] L. N. Trefethen, Spectral Methods in Matlab, SIAM, 2000.
  • [35] H. Wang, Numerical studies on the split-step finite difference method for nonlinear Schr¨odinger equations, Appl. Math. Comput., 170(1) (2005), 17–35.
  • [36] Y. Wang, T. Wang, and G. H. Gao, Series solution and Chebyshev collocation method for the initial value problem of Emden-Fowler equation, Int. J. Comput. Math., 100(2) (2023), 233–252.
  • [37] M. Zarebnia and S. Jalili, Application of spectral collocation method to a class of nonlinear PDEs, Commun. Numer. Anal., 2013 (2013), 1–14.
  • [38] H. Zhang, X. Jiang, C. Wang, and S. Chen, Crank–Nicolson Fourier spectral methods for the space fractional nonlinear Schrödinger equation and its parameter estimation. Int. J. Comput. Math., 96(2) (2019), 238–263.
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