# zbMATH — the first resource for mathematics

Numerical solution of nonlinear Jaulent-Miodek and Whitham-Broer-Kaup equations. (English) Zbl 1266.65176
Summary: We investigate the numerical solution of Jaulent-Miodek (JM) and Whitham-Broer-Kaup (WBK) equations. The proposed numerical schemes are based on the fourth-order time-stepping schemes in combination with discrete Fourier transform. We discretize the original partial differential equations with discrete Fourier transform in space and obtain a system of ordinary differential equations (ODEs) in Fourier space which are solved with fourth-order time-stepping methods. After transforming the equations to a system of ODEs, the linear operator in the JM equation is diagonal, but in the WBK equation it is not diagonal. However, for the WBK equation we can also implement the methods such as diagonal case which reduces the CPU time. Comparing numerical solutions with analytical solutions demonstrate that those methods are accurate and readily implemented.

##### MSC:
 65M70 Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs 35Q35 PDEs in connection with fluid mechanics 65T50 Numerical methods for discrete and fast Fourier transforms 65M20 Method of lines for initial value and initial-boundary value problems involving PDEs
Matlab
Full Text:
##### References:
 [1] Akhiezer, N.I., Elements of theory of elliptic functions, (1990), American Mathematical Society Providence · Zbl 0694.33001 [2] Boyd, J.P., Chebyshev and Fourier spectral methods, (2001), Dover New York · Zbl 0994.65128 [3] Cox, S.M.; Matthews, P.C., Exponential time differencing for stiff systems, J comput phys, 176, 430-455, (2002) · Zbl 1005.65069 [4] Das, G.C., Explosion of soliton in a multicomponent plasma, Phys plasmas, 4, 2095-2100, (1997) [5] Dehghan, M.; Salehi, R., The solitary wave solution of the two-dimensional regularized long-wave equation in fluids and plasmas, Comput phys commun, 182, 2540-2549, (2011) · Zbl 1263.76047 [6] Dehghan, M.; Mirzaei, D., Meshless local petrov – galerkin (MLPG) method for the unsteady magnetohydrodynamic (MHD) flow through pipe with arbitrary wall conductivity, Appl numer math, 59, 1043-1058, (2009) · Zbl 1159.76034 [7] Dehghan, M.; Manafian Heris, J.; Saadatmandi, A., Application of semi-analytic methods for the fitzhugh – nagumo equation which models the transmission of nerve impulses, Math methods appl sci, 33, 1384-1398, (2010) · Zbl 1196.35025 [8] Dehghan, M.; Shokri, A., A numerical method for solution of the two-dimensional sine – gordon equation using the radial basis functions, Math comput simul, 79, 700-715, (2008) · Zbl 1155.65379 [9] Du, Q.; Zhu, W., Analysis and applications of the exponential time differencing schemes and their contour integration modifications, BIT numer math, 45, 307-328, (2005) · Zbl 1080.65074 [10] El-Sayed, S.M.; Kaya, D., Exact and numerical traveling wave solutions of whitham – broer – kaup equations, Appl math comput, 167, 1339-1349, (2005) · Zbl 1082.65580 [11] Fan, E., Uniformly constructing a series of explicit exact solutions to nonlinear equations in mathematical physics, Chaos solitons fract, 16, 819-839, (2003) · Zbl 1030.35136 [12] Fornberg, B.; Driscoll, T.A., A fast spectral algorithm for nonlinear wave equations with linear dispersion, J comput phys, 155, 456-467, (1999) · Zbl 0937.65109 [13] Ganji, D.D.; Rokni, H.B.; Sfahani, M.G.; Ganji, S.S., Approximate traveling wave solutions for coupled whitham – broer – kaup shallow water, Adv eng soft, 41, 956-961, (2010) · Zbl 1372.76078 [14] Ganji, D.D.; Jannatabadi, M.; Mohseni, E., Application of he’s variational iteration method to nonlinear jaulent – miodek equations and comparing it with ADM, J comput appl math, 207, 35-45, (2007) · Zbl 1120.65107 [15] He, J-H.; Zhang, L-N., Generalized solitary solution and compacton-like solution of the jaulent – miodek equations using the exp-function method, Phys lett A, 372, 7, 1044-1047, (2008) · Zbl 1217.35152 [16] Hong, T.; Wang, Y.Z.; Huo, Y.S., Bogoliubov quasiparticles carried by dark solitonic excitations in nonuniform bose – einstein condensates, Chin phys lett, 15, 550-552, (1998) [17] Kassam, A.K.; Trefethen, L.N., Fourth-order time-stepping for stiff pdes, SIAM J sci comput, 26, 1214-1233, (2005) · Zbl 1077.65105 [18] Kaya, D.; El-Sayed, S.M., A numerical method for solving jaulent – miodek equation, Phys lett A, 318, 345-353, (2003) · Zbl 1045.35065 [19] Kavitha, L.; Satishkumar, P.; Nathiyaa, T.; Gopi, D., Cusp-like singular soliton solutions of jaulent – miodek equation using symbolic computation, Phys scripta, 79, 035403, (2009) · Zbl 1170.81361 [20] Khatack, A.J.; Tirmizi, S.I.A.; Siraj-ul-Islam, Application of meshfree collocation method to a class of nonlinear partial differential equations, Eng anal bound elem, 33, 661-667, (2009) · Zbl 1244.65149 [21] Krogstad, S., Generalized integrating factor methods for stiff pdes, J comput phys, 203, 72-88, (2005) · Zbl 1063.65097 [22] Kupershmidt, B.A., Mathematics of dispersive water waves, Commun math phys, 99, 51-73, (1985) · Zbl 1093.37511 [23] Lou, S.Y., A direct perturbation method: nonlinear schrodinger equation with loss, Chin phys lett, 16, 659-661, (1999) [24] Ma, W.X.; Li, C.X.; He, J.S., A second Wronskian formulation of the Boussinesq equation, Nonlinear anal, 70, 4245-4258, (2009) · Zbl 1159.37425 [25] Ma, W.X.; Zhang, Y.; Tang, Y.; Tu, J., Hirota bilinear equations with linear subspaces of solutions, Appl math comput, 218, 174-7183, (2012) · Zbl 1245.35109 [26] Mohebbi, A.; Asgari, Z., Efficient numerical algorithms for the solution of “good” Boussinesq equation in water wave propagation, Comput phys commun, 182, 2464-2470, (2011) · Zbl 1457.65147 [27] Shokri, A.; Dehghan, M., A not-a-knot meshless method using radial basis functions and predictor – corrector scheme to the numerical solution of improved Boussinesq equation, Comput phys commun, 181, 1990-2000, (2010) · Zbl 1426.76569 [28] Tatari, M.; Dehghan, M., On the convergence of he’s variational iteration method, J comput appl math, 207, 121-128, (2007) · Zbl 1120.65112 [29] Ozer, H.T.; Salihoglu, S., Nonlinear schrodinger equations and N=1 superconformal algebra, Chaos solitons fract, 33, 1417-1423, (2007) · Zbl 1138.37329 [30] Trefethen, L.N., Spectral methods in Matlab, (2000), SIAM Philadelphia · Zbl 0953.68643 [31] Wazwaz, A.M., The tanh – coth and the sech methods for exact solutions of the jaulent – miodek equation, Phys lett A, 366, 85-90, (2007) · Zbl 1203.81069 [32] Xie, F.; Yan, Z.; Zhang, H., Explicit and exact traveling wave solutions of whitham – broer – kaup shallow water equations, Phys lett A, 28, 76-80, (2001) · Zbl 0969.76517 [33] Xu, G.; Li, Z., Exact traveling wave solutions of the whitham – broer – kaup and broer – kaup – kupershmidt equations, Chaos solitons fract, 24, 549-556, (2005) · Zbl 1069.35067 [34] Zhang, J.F., Multiple soliton solutions of the dispersive long-wave equations, Chin phys lett, 16, 4-6, (1999) [35] Zheng, Z.; Shan, W.R., Application of exp-function method to the whitham – broer – kaup shallow water model using symbolic computation, Appl math comput, 215, 2390-2396, (2009) · Zbl 1422.76146
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.