# zbMATH — the first resource for mathematics

An efficient Jacobi pseudospectral approximation for nonlinear complex generalized Zakharov system. (English) Zbl 1339.65188
Summary: In this paper, we derive an efficient spectral collocation algorithm to solve numerically the nonlinear complex generalized Zakharov system (GZS) subject to initial-boundary conditions. The Jacobi pseudospectral approximation is investigated for spatial approximation of the GZS. It possesses the spectral accuracy in space. The Jacobi-Gauss-Lobatto quadrature rule is established to treat the boundary conditions, and then the problem with its boundary conditions is reduced to a system of ordinary differential equations in time variable. This scheme has the advantage of allowing us to obtain the spectral solution in terms of the Jacobi parameters $$\alpha$$ and $$\beta$$, which therefore means that the algorithm holds a number of collocation methods as special cases. Finally, two illustrative examples are implemented to assess the efficiency and high accuracy of the Jacobi pseudo-spectral scheme.

##### MSC:
 65M70 Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs 35L70 Second-order nonlinear hyperbolic equations 35Q55 NLS equations (nonlinear Schrödinger equations)
Full Text:
##### References:
 [1] El-Wakil, S. A.; Abdou, M. A.; Elhanbaly, A., New solitons and periodic wave solutions for nonlinear evolution equations, Phys. Lett. A, 353, 40-47, (2006) [2] Mallory, K.; Van Gorder, R. A., Control of error in the homotopy analysis of solutions to the Zakharov system with dissipation, Numer. Algorithms, 64, 633-657, (2013) · Zbl 1283.65090 [3] Ebadi, G.; Biswas, A., The $$G^\prime / G$$ method and 1-soliton solution of the Davey-Stewartson equation, Math. Comput. Model., 53, 694-698, (2011) · Zbl 1217.35171 [4] Zhang, J., Variational approach to solitary wave solution of the generalized Zakharov equation, Comput. Math. Appl., 54, 1043-1046, (2007) · Zbl 1141.65391 [5] Javidi, M.; Golbabai, A., Exact and numerical solitary wave solutions of generalized Zakharov equation by the variational iteration method, Chaos Solitons Fract., 36, 309-313, (2008) · Zbl 1350.35182 [6] Awawdeh, F., New exact solitary wave solutions of the Zakharov-Kuznetsov equation in the electron-positron-ion plasmas, Appl. Math. Comput., 218, 7139-7143, (2012) · Zbl 1441.76144 [7] Song, M.; Yang, C., Exact traveling wave solutions of the Zakharov-Kuznetsov-Benjamin-Bona-Mahony equation, Appl. Math. Comput., 216, 3234-3243, (2010) · Zbl 1193.35199 [8] Chang, Q.; Jiang, H., A conservative difference scheme for the Zakharov equations, J. Comput. Phys., 113, 309, (1994) · Zbl 0807.76050 [9] Chang, Q.; Guo, B.; Jiang, H., Finite difference method for generalized Zakharov equations, Math. Comput., 64, 537, (1995) · Zbl 0827.65138 [10] Abbasbandy, S.; Babolian, E.; Ashtiani, M., Numerical solution of the generalized Zakharov equation by homotopy analysis method, Commun. Nonlinear Sci. Numer. Simul., 14, 4114-4121, (2009) · Zbl 1221.65269 [11] Bao, W.; Sun, F.; Wei, G. W., Numerical methods for the generalized Zakharov system, J. Comput. Phys., 190, 201-228, (2003) · Zbl 1236.76043 [12] Bao, W.; Sun, F., Efficient and stable numerical methods for the generalized and vector Zakharov system, SIAM J. Sci. Comput., 26, 1057-1088, (2005) · Zbl 1076.35114 [13] Wang, J., Multisymplectic numerical method for the Zakharov system, Comput. Phys. Commun., 180, 1063-1071, (2009) · Zbl 1198.82062 [14] Xu, Q.; Song-He, S.; Er, G.; Wei-Bin, L., Explicit multi-symplectic method for the Zakharov Kuznetsov equation, Chin. Phys. B, 21, 070206, (2012) [15] Fayong, Z.; Boling, G., Attractors for fully discrete finite difference scheme of dissipative Zakharov equations, Acta Math. Sci., 32B, 2431-2452, (2012) · Zbl 1289.65198 [16] Canuto, C.; Hussaini, M. Y.; Quarteroni, A.; Zang, T. A., Spectral methods: fundamentals in single domains, (2006), Springer-Verlag New York · Zbl 1093.76002 [17] Guo, B.-Y.; Wang, Z.-Q., A collocation method for generalized nonlinear Klein-Gordon equation, Adv. Comput. Math., 40, 377-398, (2014) · Zbl 1300.65072 [18] Yi, Y.-G.; Guo, B.-Y., Generalized Jacobi rational spectral method on the half line, Adv. Comput. Math., 37, 1, 1-37, (2012) · Zbl 1259.65156 [19] Darvishi, M. T.; Kheybari, S.; Khani, F., Spectral collocation method and darvishi’s preconditionings to solve the generalized Burgers-Huxley equation, Commun. Nonlinear Sci. Numer. Simul., 13, 10, 2091-2103, (2008) · Zbl 1221.65261 [20] Javidi, M.; Golbabai, A., Numerical studies on nonlinear Schrödinger equations by spectral collocation method with preconditioning, J. Math. Anal. Appl., 333, 2, 1119-1127, (2007) · Zbl 1117.65141 [21] Tohidi, E.; Bhrawy, A. H.; Erfani, K., A collocation method based on Bernoulli operational matrix for numerical solution of generalized pantograph equation, Appl. Math. Model., 37, 6, 4283-4294, (2013) · Zbl 1273.34082 [22] Kwan, Y.-Y., Efficient spectral-Galerkin methods for polar and cylindrical geometries, Appl. Numer. Math., 59, 1, 170-186, (2009) · Zbl 1180.65160 [23] He, Y.; Liu, Y., Stability and convergence of the spectral Galerkin method for the Cahn-Hilliard equation, Numer. Methods Partial Differ. Equ., 24, 1485-1500, (2008) · Zbl 1157.65053 [24] Karimi Vanani, S.; Soleymani, F., Tau approximate solution of weakly singular Volterra integral equations, Math. Comput. Model., 57, 494-502, (2013) · Zbl 1305.65247 [25] Doha, E. H.; Bhrawy, A. H.; Hafez, R. M., On shifted Jacobi spectral method for high-order multi-point boundary value problems, Commun. Nonlinear Sci. Numer. Simul., 17, 10, 3802-3810, (2012) · Zbl 1251.65112 [26] Parand, K.; Dehghan, M.; Rezaeia, A.; Ghaderi, S., An approximation algorithm for the solution of the nonlinear Lane-Emden type equations arising in astrophysics using Hermite functions collocation method, Comput. Phys. Commun., 181, 1096-1108, (2010) · Zbl 1216.65098 [27] Li, B.-W.; Tian, S.; Sun, Y.-S.; Hu, Z.-M., Schur-decomposition for 3D matrix equations and its application in solving radiative discrete ordinates equations discretized by Chebyshev collocation spectral method, J. Comput. Phys., 229, 4, 1198-1212, (2010) · Zbl 1183.65152 [28] Parand, K.; Shahini, M.; Dehghan, Mehdi, Rational Legendre pseudospectral approach for solving nonlinear differential equations of Lane-Emden type, J. Comput. Phys., 228, 23, 8830-8840, (2009) · Zbl 1177.65100 [29] Dehghan, M.; -Izadi, F. 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., 5, 1865-1877, (2011) · Zbl 1219.65106 [30] Ordokhani, Y.; Razzaghi, M., Solution of nonlinear Volterra-Fredholm-Hammerstein integral equations via a collocation method and rationalized Haar functions, Appl. Math. Lett., 21, 1, 4-9, (2008) · Zbl 1133.65117 [31] Zhang, K.; Li, J.; Song, H., Collocation methods for nonlinear convolution Volterra integral equations with multiple proportional delays, Appl. Math. Comput., 218, 22, 10848-10860, (2012) · Zbl 1280.65148 [32] Jiang, Y.; Ma, J., Spectral collocation methods for Volterra-integro differential equations with noncompact kernels, J. Comput. Appl. Math., 244, 115-124, (2013) · Zbl 1263.65134 [33] Yüzbasi, S.; Sezer, M.; Kemanci, B., Numerical solutions of integro-differential equations and application of a population model with an improved Legendre method, Appl. Math. Model., 37, 4, 2086-2101, (2013) · Zbl 1349.65728 [34] Doha, E. H.; Bhrawy, A. H.; Ezz-Eldien, S. S., Efficient Chebyshev spectral methods for solving multi-term fractional orders differential equations, Appl. Math. Model., 35, 12, 5662-5672, (2011) · Zbl 1228.65126 [35] Bhrawy, A. H.; Tharwat, M. M.; Yildirim, A., A new formula for fractional integrals of Chebyshev polynomials: application for solving multi-term fractional differential equations, Appl. Math. Model., 37, 6, 4245-4252, (2013) · Zbl 1278.65096 [36] Wang, Z.; Guo, B.-Y., Jacobi rational approximation and spectral method for differential equations of degenerate type, Math. Comput., 77, 883-907, (2008) · Zbl 1132.41315 [37] Doha, E. H.; Bhrawy, A. H.; Hafez, R. M.; Abdelkawy, M. A., A Chebyshev-Gauss-radau scheme for nonlinear hyperbolic system of first order, Appl. Math. Inform. Sci., 8, 535-544, (2014) [38] Doha, E. H.; Bhrawy, A. H.; Baleanu, D.; Hafez, R. M., A new Jacobi rational-Gauss collocation method for numerical solution of generalized pantograph equations, Appl. Numer. Math., 77, 43-54, (2014) · Zbl 1302.65175 [39] Doha, E. H.; Bhrawy, A. H.; Abdelkawy, M. A.; Van Gorder, R. A., Jacobi-Gauss-lobatto collocation method for the numerical solution of $$1 + 1$$ nonlinear Schrödinger equations, J. Comput. Phys., 261, 244-255, (2014) · Zbl 1349.65511 [40] Christara, C. C.; Chen, T.; Dang, D. M., Quadratic spline collocation for one-dimensional linear parabolic partial differential equations, Numer. Algorithms, 53, 4, 511-553, (2010) · Zbl 1189.65235 [41] Bhrawy, A. H., A Jacobi-Gauss-lobatto collocation method for solving generalized Fitzhugh-Nagumo equation with time-dependent coefficients, Appl. Math. Comput., 222, 255-264, (2013) · Zbl 1329.65234 [42] Szego, G., Orthogonal polynomials, (1959), Amer. Math. Soc. · JFM 61.0386.03 [43] Guo, B.-Y.; Wang, L., Jacobi interpolation approximations and their applications to singular differential equations, Adv. Comput. Math., 14, 227-276, (2001) · Zbl 0984.41004 [44] Butcher, J. C., The numerical analysis of ordinary differential equations, Runge-Kutta and general linear methods, (1987), Wiley New York · Zbl 0616.65072 [45] Doha, E. H., On the Chebyshev method to linear parabolic P.D.E.s with inhomogeneous mixed boundary conditions, Arab. J. Math., 4, 31-57, (1983) · Zbl 0602.65088 [46] Attili, B. S.; Furati, K.; Syam, M. I., An efficient implicit Runge-Kutta method for second order systems, Appl. Math. Comput., 178, 229-238, (2006) · Zbl 1100.65062 [47] Ehle, B.; Picel, Z., Two parameter arbitrary order exponential approximations for stiff equations, Math. Comput., 29, 501-511, (1975) · Zbl 0302.65059 [48] Van Gorder, R. A.; Vajravelu, K., A general class of coupled nonlinear differential equations arising in self-similar solutions of convective heat transfer problems, Appl. Math. Comput., 217, 460-465, (2010) · Zbl 1211.34037 [49] Van Gorder, R. A.; Sweet, E.; Vajravelu, K., Analytical solutions of a coupled nonlinear system arising in a flow between stretching disks, Appl. Math. Comput., 216, 1513-1523, (2010) · Zbl 1277.76024 [50] Mallory, K.; Van Gorder, R. A., A transformed time-dependent Michaelis-Menten enzymatic reaction model and its asymptotic stability, J. Math. Chem., 52, 222-230, (2014) · Zbl 1311.92072
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.