Exp-function method for solving Kuramoto-Sivashinsky and Boussinesq equations. (English) Zbl 1176.65115

Summary: We use the exp-function method to construct the generalized solitary and periodic solution of the Kuramoto-Sivashinsky and Boussinesq equations. These equations play very important role in mathematical physics and engineering sciences. The suggested algorithm is quite efficient and is practically well suited for use in these problems. The results show the reliability and efficiency of the proposed method.


65M70 Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs
35Q53 KdV equations (Korteweg-de Vries equations)
Full Text: DOI


[1] Abdou, M.A., Soliman, A.A., Basyony, S.T.: New application of Exp-function method for improved Boussinesq equation. Phys. Lett. A 369, 469–475 (2007) · Zbl 1209.81091
[2] Daripa, P., Hua, W.: A numerical method for solving an ill posed Boussinesq equation arising in water waves and nonlinear lattices. Appl. Math. Comput. 101, 159–207 (1999) · Zbl 0937.76050
[3] Dash, R.K., Daripa, P.: Analytical and numerical studies of a singularly perturbed Boussinesq equation. Appl. Math. Comput. 126(1), 1–30 (2002) · Zbl 1126.76308
[4] El-Sayad, S.M., Kaya, D.: The decomposition method for solving (2+1)-dimensional Boussinesq equation and (3+1)-dimensional KP equation. Appl. Math. Comput. 157(2), 523–534 (2004) · Zbl 1058.65111
[5] El-Wakil, S.A., Madkour, M.A., Abdou, M.A.: Application of Exp-function method for nonlinear evolution equations with variable co-efficient. Phys. Lett. A 369, 62–69 (2007) · Zbl 1209.81097
[6] Feng, Z.: Traveling solitary wave equations to the generalized Boussinesq equation. Wave Motion 37, 17–23 (2003) · Zbl 1163.74348
[7] Hajji, M.A., Al-Khalid, K.: Analytic studies and numerical simulation of the generalized Boussinesq equation. Appl. Math. Comput. 191, 332–340 (2007) · Zbl 1193.35188
[8] He, J.H., Abdou, M.A.: New periodic solutions for nonlinear evolution equation using Exp-method. Chaos Solitons Fractals 34, 1421–1429 (2007) · Zbl 1152.35441
[9] He, J.H., Wu, X.H.: Exp-function method for nonlinear wave equations. Chaos Solitons Fractals 30(3), 700–708 (2006) · Zbl 1141.35448
[10] Hirota, R.: Exact N-Soliton solution of the wave equation of long waves in shallow-water and in nonlinear lattices. J. Math. Phys. 14, 810–814 (1973) · Zbl 0261.76008
[11] Hyman, J.M., Nicolaenko, B.: The Kuramoto–Sivashinsky equation: A bridge between PDEs and dynamical systems. Phys. D 18, 113–126 (1986) · Zbl 0602.58033
[12] Kaya, D., El-Sayed, S.M.: An application of the decomposition method for the generalized KdV and Rlw equations. Chaos Solitons Fractals 17(5), 869–877 (2003) · Zbl 1030.35139
[13] Kuramoto, Y., Tsuzuki, T.: Persistent propagation of concentration waves in dissipative media far from thermal equilibrium. Prog. Theor. Phys. 55, 356–369 (1976)
[14] Manoranjan, V.S., Mitchell, A.S., Morris, J.L.: Numerical solution of the good Boussinesq equation. SIAM J. Sci. Stat. Comput. 5, 946–957 (1984) · Zbl 0555.65080
[15] Nicolaenko, B., Scheurer, B., Temam, T.: Some global properties of the Kuramoto–Sivashinsky equation: Nonlinear stability and attractors. Phys. D 16, 155–183 (1985) · Zbl 0592.35013
[16] Noor, M.A., Mohyud-Din, S.T.: Variational iteration technique for solving Boussinesq equations. Int. J. Appl. Math. Eng. Sci. 2 (2008, in press) · Zbl 1163.76045
[17] Noor, M.A., Mohyud-Din, S.T.: Variational iteration method for solving higher-order boundary value problems using He’s polynomial. Int. J. Nonlinear Sci. Numer. Simul. 9 (2008, in press) · Zbl 1151.65334
[18] Smith, L.M., Waleffe, F.: Transfer of energy to two-dimensional large scales in forced, rotating three-dimensional turbulence. Phys. Fluids 11, 1608–1622 (1999) · Zbl 1147.76500
[19] Smith, L.M., Waleffe, F.: Generation of slow large scales in forced rotating stratified turbulence. J. Fluid Mech. 451, 145–169 (2002) · Zbl 1009.76040
[20] Wazwaz, A.M.: The decomposition method applied to systems of partial differential equations and to the reaction-diffusion Brusselator model. Appl. Math. Comput. 110, 251–264 (2000) · Zbl 1023.65109
[21] Wu, X.H., He, J.H.: Solitary solutions, periodic solutions and compacton like solutions using the Exp-function method. Comput. Math. Appl. 54, 966–986 (2007) · Zbl 1143.35360
[22] Wu, X.H., He, J.H.: Exp-function method and its applications to nonlinear equations. Chaos Solitons Fractals (2007, in press)
[23] Yusufoglu, E.: New solitonary solutions for the MBBN equations using Exp-function method. Phys. Lett. A 372, 442–446 (2008) · Zbl 1217.35156
[24] Zhang, S.: Application of Exp-function method to high-dimensional nonlinear evolution equation. Chaos Solitons Fractals 365, 448–455 (2007)
[25] Zhu, S.D.: Exp-function method for the hybrid-lattice system. Int. J. Nonlinear Sci. Numer. Simul. 8, 461–464 (2007) · Zbl 06942293
[26] Zhu, S.D.: Exp-function method for the discrete mKdV lattice. Int. J. Nonlinear Sci. Numer. Simul. 8, 465–468 (2007) · Zbl 06942294
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.