zbMATH — the first resource for mathematics

Jacobi-Gauss-Lobatto collocation method for the numerical solution of \(1+1\) nonlinear Schrödinger equations. (English) Zbl 1349.65511
Summary: A Jacobi-Gauss-Lobatto collocation (J-GL-C) method, used in combination with the implicit Runge-Kutta method of fourth order, is proposed as a numerical algorithm for the approximation of solutions to nonlinear Schrödinger equations (NLSE) with initial-boundary data in \(1+1\) dimensions. Our procedure is implemented in two successive steps. In the first one, the J-GL-C is employed for approximating the functional dependence on the spatial variable, using \((N-1)\) nodes of the Jacobi-Gauss-Lobatto interpolation which depends upon two general Jacobi parameters. The resulting equations together with the two-point boundary conditions induce a system of \(2(N-1)\) first-order ordinary differential equations (ODEs) in time. In the second step, the implicit Runge-Kutta method of fourth order is applied to solve this temporal system. The proposed J-GL-C method, used in combination with the implicit Runge-Kutta method of fourth order, is employed to obtain highly accurate numerical approximations to four types of NLSE, including the attractive and repulsive NLSE and a Gross-Pitaevskii equation with space-periodic potential. The numerical results obtained by this algorithm have been compared with various exact solutions in order to demonstrate the accuracy and efficiency of the proposed method. Indeed, for relatively few nodes used, the absolute error in our numerical solutions is sufficiently small.

65M70 Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs
35Q55 NLS equations (nonlinear Schrödinger equations)
Full Text: DOI
[1] Islam, S.; Haq, S.; Uddin, M., A mesh free interpolation method for the numerical solution of the coupled nonlinear partial differential equations, Eng. Anal. Bound. Elem., 33, 399-409, (2009) · Zbl 1244.65193
[2] Mohebbi, A.; Asgari, Z.; Dehghan, M., Numerical solution of nonlinear Jaulent-Miodek and Whitham-Broer-Kaup equations, Commun. Nonlinear Sci. Numer. Simul., 17, 4602-4610, (2012) · Zbl 1266.65176
[3] Mittal, R. C.; Jain, R. K., Numerical solutions of nonlinear burgersʼ equation with modified cubic B-splines collocation method, Appl. Math. Comput., 218, 7839-7855, (2012) · Zbl 1242.65209
[4] 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 schrodinger equation arising in quantum mechanics, Eng. Anal. Bound. Elem., 37, 475-485, (2013) · Zbl 1352.65397
[5] Gurefe, Y.; Misirli, E.; Sonmezoglu, A.; Ekici, M., Extended trial equation method to generalized nonlinear partial differential equations, Appl. Math. Comput., 219, 5253-5260, (2013) · Zbl 1284.35371
[6] Wang, H., Numerical studies on the split step finite difference method for the nonlinear Schrödinger equations, Appl. Math. Comput., 170, 17-35, (2005) · Zbl 1082.65570
[7] Dehghan, M.; Taleei, A., Numerical solution of nonlinear Schrödinger equation by using time-space pseudo-spectral method, Numer. Methods Partial Differ. Equ., 26, 979-992, (2010) · Zbl 1195.65137
[8] Dehghan, M.; Taleei, A., A Chebyshev pseudo-spectral multi-domain method for the soliton solution of coupled nonlinear schrodinger equations, Comput. Phys. Commun., 182, 2519-2529, (2011) · Zbl 1261.65103
[9] Ismail, M. S., A fourth order explicit scheme for the coupled nonlinear Schrödinger equation, Appl. Math. Comput., 196, 273-284, (2008) · Zbl 1133.65063
[10] Ismail, M. S., Numerical solution of coupled nonlinear Schrödinger equation by Galerkin method, Math. Comput. Simul., 78, 532-547, (2008) · Zbl 1145.65075
[11] Xu, Y.; Zhang, L., Alternating direction implicit method for solving two-dimensional cubic nonlinear Schrödinger equation, Comput. Phys. Commun., 183, 1082-1093, (2012) · Zbl 1277.65073
[12] Smadi, M.; Bahloul, D., A compact split step pade scheme for higher-order nonlinear Schrödinger equation (HNLS) with power law nonlinearity and fourth order dispersion, Comput. Phys. Commun., 182, 366-371, (2011) · Zbl 1217.65177
[13] Duan, Y.; Rong, F., A numerical scheme for nonlinear Schrödinger equation by MQ quasi-interpolation, Eng. Anal. Bound. Elem., 37, 89-94, (2013) · Zbl 1352.65391
[14] Sakaguchi, H.; Higashiuchi, T., Two-dimensional dark soliton in the nonlinear Schrödinger equation, Phys. Lett. A, 359, 647-651, (2006) · Zbl 1236.35140
[15] Kalogiratou, Z.; Monovasilis, Th.; Simos, T. E., Numerical solution of the two-dimensional time independent Schrödinger equation with numerov-type methods, J. Math. Chem., 37, 3, 271-279, (2005) · Zbl 1077.65110
[16] Javidi, M.; Golbabai, A., Numerical studies on nonlinear Schrödinger equations by spectral collocation method with preconditioning, J. Math. Anal. Appl., 333, 1119-1127, (2007) · Zbl 1117.65141
[17] Canuto, C.; Hussaini, M. Y.; Quarteroni, A.; Zang, T. A., Spectral methods: fundamentals in single domains, (2006), Springer-Verlag New York · Zbl 1093.76002
[18] Doha, E. H.; Bhrawy, A. H., An efficient direct solver for multidimensional elliptic Robin boundary value problems using a Legendre spectral-Galerkin method, Comput. Math. Appl., 64, 558-571, (2012) · Zbl 1252.65194
[19] Karimi Vanani, S.; Soleymani, F., Tau approximate solution of weakly singular Volterra integral equations, Math. Comput. Model., 57, 3-4, 494-502, (2013) · Zbl 1305.65247
[20] Shao, W.; Wu, X., Chebyshev tau meshless method based on the highest derivative for fourth order equations, Appl. Math. Model., 37, 3, 1413-1430, (1 February 2013)
[21] Doha, E. H.; Abd-Elhameed, W. M.; Bassuony, M. A., New algorithms for solving high even-order differential equations using third and fourth Chebyshev-Galerkin methods, J. Comput. Phys., 236, 563-579, (2013) · Zbl 1286.65093
[22] Gheorghiu, C. I., Spectral methods for differential problems, (2007), T. Popoviciu Institute of Numerical Analysis Cluj-Napoca, Romaina · Zbl 1122.65118
[23] Gheorghiu, C. I.; Hochstenbach, M. E.; Plestenjak, B.; Rommes, J., Spectral collocation solutions to multiparameter mathieuʼs system, Appl. Math. Comput., 218, 24, 11990-12000, (2012) · Zbl 1280.65078
[24] Bhrawy, A. H.; Alghamdi, M. A., A shifted Jacobi-Gauss-lobatto collocation method for solving nonlinear factional Langevin equation involving two fractional orders in different intervals, Bound. Value Probl., 2012, 62, (2012) · Zbl 1280.65079
[25] Zhang, K.; Li, J.; Song, H., Collocation methods for nonlinear convolution Volterra integral equations with multiple proportional delays, Appl. Math. Comput., 218, 10848-10860, (2012) · Zbl 1280.65148
[26] Saadatmandi, A.; Dehghan, M., The use of sinc-collocation method for solving multi-point boundary value problems, Commun. Nonlinear Sci. Numer. Simul., 17, 593-601, (2012) · Zbl 1244.65114
[27] Bhrawy, A. H.; Alofi, A. S., A Jacobi-Gauss collocation method for solving nonlinear Lane-Emden type equations, Commun. Nonlinear Sci. Numer. Simul., 17, 62-70, (2012) · Zbl 1244.65099
[28] Shamsi, M.; Dehghan, M., Determination of a control function in three-dimensional parabolic equations by Legendre pseudospectral method, Numer. Methods Partial Differ. Equ., 28, 74-93, (2012) · Zbl 1252.65161
[29] Eslahchi, M. R.; Dehghan, M.; Ahmadi-Asl, S., The general Jacobi matrix method for solving some nonlinear ordinary differential equations, Appl. Math. Model., 36, 3387-3398, (2012) · Zbl 1252.65121
[30] Eslahchi, M. R.; Dehghan, M.; Amani, S., The third and fourth kinds Chebyshev polynomials and best uniform approximation, Math. Comput. Model., 55, 1746-1762, (2012) · Zbl 1255.41015
[31] Dehghan, M.; Fakhar-Izadi, F., The spectral collocation method with three different bases for solving a nonlinear partial differential equation arising in modeling of nonlinear waves, Math. Comput. Model., 53, 1865-1877, (2011) · Zbl 1219.65106
[32] Dehghan, M.; Jazlanian, R., A fourth-order central Runge-Kutta scheme for hyperbolic conservation laws, Numer. Methods Partial Differ. Equ., 26, 1675-1692, (2010) · Zbl 1200.65074
[33] Mohebbi, A.; Dehghan, M., The use of compact boundary value method for the solution of two-dimensional Schrödinger equation, J. Comput. Appl. Math., 225, 124-134, (2009) · Zbl 1159.65081
[34] Dehghan, M.; Mirzaei, D., Numerical solution to the unsteady two-dimensional Schrödinger equation using meshless local boundary integral equation method, Int. J. Numer. Methods Eng., 76, 501-520, (2008) · Zbl 1195.81007
[35] Szegő, G., Orthogonal polynomials, Colloquium Publications, vol. XXIII, (1939), American Mathematical Society, MR 0372517 · JFM 65.0286.02
[36] Doha, E. H.; Bhrawy, A. H., Efficient spectral-Galerkin algorithms for direct solution of fourth-order differential equations using Jacobi polynomials, Appl. Numer. Math., 58, 1224-1244, (2008) · Zbl 1152.65112
[37] El-Kady, M., Jacobi discrete approximation for solving optimal control problems, J. Korean Math. Soc., 49, 99-112, (2012) · Zbl 1236.65072
[38] Ablowitz, M. J.; Segur, H., Solitons and the inverse scattering transform, (1981), Society for Industrial and Applied Mathematics (SIAM) Philadelphia · Zbl 0472.35002
[39] Mallory, K.; Van Gorder, R. A., Stationary solutions for the \(1 + 1\) nonlinear Schrödinger equation modeling repulsive Bose-Einstein condensates in small potentials, Phys. Rev. E, 88, 013205, (2013)
[40] Gross, E. P., Structure of a quantized vortex in boson systems, Nuovo Cimento, 20, 454-457, (1961) · Zbl 0100.42403
[41] Pitaevsk, L. P., Vortex lines in an imperfect Bose gas, Sov. Phys. JETP, 13, 451-454, (1961)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.