**Commenced**in January 2007

**Frequency:**Monthly

**Edition:**International

**Paper Count:**33006

##### Some Solitary Wave Solutions of Generalized Pochhammer-Chree Equation via Exp-function Method

**Authors:**
Kourosh Parand,
Jamal Amani Rad

**Abstract:**

In this paper, Exp-function method is used for some exact solitary solutions of the generalized Pochhammer-Chree equation. It has been shown that the Exp-function method, with the help of symbolic computation, provides a very effective and powerful mathematical tool for solving nonlinear partial differential equations. As a result, some exact solitary solutions are obtained. It is shown that the Exp-function method is direct, effective, succinct and can be used for many other nonlinear partial differential equations.

**Keywords:**
Exp-function method,
generalized Pochhammer- Chree equation,
solitary wave solution,
ODE's.

**Digital Object Identifier (DOI):**
doi.org/10.5281/zenodo.1079408

**References:**

[1] A. Wazwaz, New travelling wave solutions to the boussinesq and the klein-gordon equations, Communications in Nonlinear Science and Numerical Simulation 13 (2008) 889-901.

[2] A. Wazwaz, The variable separated ode and the tanh methods for solving the combined and the double combined sinh-cosh-gordon equations, Applied Mathematics and Computation 177 (2006) 745-754.

[3] A. Wazwaz, Single and multiple-soliton solutions for the (2+1)- dimensional kdv equation, Applied Mathematics and Computation 204 (2008) 20-26.

[4] I. Hashim, M. S. M. Noorani, M. R. S. Hadidi, Solving the generalized burgers-huxley equation using the adomian decomposition method, Mathematical and Computer Modelling 43 (2006) 1404-1411.

[5] M. Tatari, M. Dehghan, M. Razzaghi, Application of the adomian decomposition method for the fokker-planck equation, Mathematical and Computer Modelling 45 (2007) 639-650.

[6] M. Rashidi, D. Ganji, S. Dinarvand, Explicit analytical solutions of the generalized burger and burger-fisher equations by homotopy perturbation method, Numerical Methods for Partial Differential Equations 25 (2009) 409-417.

[7] J. Biazar, F. Badpeimaa, F. Azimi, Application of the homotopy perturbation method to zakharov-kuznetsov equations, Computers and Mathematics with Applications 58 (2009) 2391-2394.

[8] M. Berberler, A. Yildirim, He-s homotopy perturbation method for solving the shock wave equation, Applicable Analysis 88 (2009) 997- 1004.

[9] F. Shakeri, M. Dehghan, Numerical solution of the klein-gordon equation via hes variational iteration method, Nonlinear Dynamics 51 (2008) 89- 97.

[10] A. Soliman, M. Abdou, Numerical solutions of nonlinear evolution equations using variational iteration method, Journal of Computational and Applied Mathematics 207 (2007) 111-120.

[11] E.Yusufoglu, A. Bekir, The variational iteration method for solitary patterns solutions of gbbm equation, Physics Letters A 367 (2007) 461- 464.

[12] A. Taghavi, K. Parand, H. Fani, Lagrangian method for solving unsteady gas equation, International Journal of Computational and Mathematical Sciences 3 (2009) 40-44.

[13] K. Parand, M. Dehghan, A. Pirkhedri, Sinc-collocation method for solving the blasius equation, Physics Letters, Section A: General, Atomic and Solid State Physics 373 (2009) 4060-4065.

[14] K. Parand, M. Dehghan, A. R. Rezaei, S. M. Ghaderi, An approximation algorithm for the solution of the nonlinear lane-emden type equations arising in astrophysics using hermite functions collocation method, Comput Phys Commun DOI =10.1016/j.cpc.2010.02.018.

[15] F. Tascan, A. Bekir, Analytic solutions of the (2 + 1)-dimensional nonlinear evolution equations using the sine-cosine method, Applied Mathematics and Computation 215 (2009) 3134-3139.

[16] A. M. Wazwaz, New solitary wave solutions to the modified kawahara equation, Physics Letters, Section A: General, Atomic and Solid State Physics 360 (2007) 588-592.

[17] M. Tatari, M. M. Dehghan, A method for solving partial differential equations via radial basis functions: Application to the heat equation, Engineering Analysis with Boundary Elements 34 (2010) 206-212.

[18] M. Dehghan, A. Shokri, Numerical solution of the nonlinear kleingordon equation using radial basis functions, Journal of Computational and Applied Mathematics 230 (2009) 400-410.

[19] J. H. He, X. H. Wu, Exp-function method for nonlinear wave equations, Chaos, Solitons and Fractals 30 (2006) 700-708.

[20] S. Zhang, Application of exp-function method to a kdv equation with variable coefficients, Physics Letters, Section A: General, Atomic and Solid State Physics 365 (2007) 448-453.

[21] S. Zhang, Exp-function method for solving maccari-s system, Physics Letters, Section A: General, Atomic and Solid State Physics 371 (2007) 65-71.

[22] L. Assas, New exact solutions for the kawahara equation using expfunction method, Journal of Computational and Applied Mathematics 233 (2009) 97-102.

[23] M. A. Abdou, A. A. Soliman, S. T. El-Basyony, New application of exp-function method for improved boussinesq equation, Physics Letters, Section As : General, Atomic and Solid State Physics 369 (2007) 469- 475.

[24] A. Ebaid, Exact solitary wave solutions for some nonlinear evolution equations via exp-function method, Physics Letters, Section As : General, Atomic and Solid State Physics 365 (2007) 213-219.

[25] J. Biazar, Z. Ayati, Extension of the exp-function method for systems of two-dimensional burger-s equations, Computers and Mathematics with Applications 58 (2009) 2103-2106.

[26] A. Ebaid, Generalization of he-s exp-function method and new exact solutions for burgers equation, Zeitschrift fur Naturforschung - Section A Journal of Physical Sciences 64 (2009) 604-608.

[27] G. Domairry, A. G. Davodi, A. G. Davodi, Solutions for the double sine-gordon equation by exp-function, tanh, and extended tanh methods, Numerical Methods for Partial Differential Equations 26 (2010) 384- 398.

[28] J. H. He, M. A. Abdou, New periodic solutions for nonlinear evolution equations using exp-function method, Chaos, Solitons and Fractals 34 (2007) 1421-1429.

[29] T. Ozis, C. Koroglu, A novel approach for solving the fisher equation using exp-function method, Physics Letters, Section As : General, Atomic and Solid State Physics 372 (2008) 3836-3840.

[30] J. H. He, L. Zhang, Generalized solitary solution and compacton-like solution of the jaulent-miodek equations using the exp-function method, Physics Letters, Section As : General, Atomic and Solid State Physics 372 (2008) 1044-1047.

[31] C. Koroglu, T. Ozis, A novel traveling wave solution for ostrovsky equation using exp-function method, Computers and Mathematics with Applications 58 (2009) 2142-2146.

[32] B. Shin, M. Darvishi, A. Barati, Some exact and new solutions of the nizhnik-novikov-vesselov equation using the exp-function method, Computers and Mathematics with Applications 58 (2009) 2147-2151.

[33] S. Zhang, Application of exp-function method to high-dimensional nonlinear evolution equation, Chaos, Solitons and Fractals 38 (2008) 270-276.

[34] C. Chree, Longitudinal vibrations of a circular bar, The Quarterly Journal of Mathematics 21 (1886) 287-298.

[35] L. Pochhammer, Biegung des kreiscylinders-fortpflanzungsgeschwindigkeit kleiner schwingungen in einem kreiscylinder, Journal fr die reine und angewandte Mathematik 81 (1876) 326-336.

[36] W. L. Zhang, Solitary wave solutions and kink wave solutions for a generalized pc equation, Acta Mathematicae Applicatae Sinica 21 (2005) 125-134.

[37] N. Shawagfeh, D. Kaya, Series solution to the pochhammer-chree equation and comparison with exact solutions, Computers and Mathematics with Applications 47 (2004) 1915-1920.

[38] B. Li, Y. Chen, H. Zhang, Travelling wave solutions for generalized pochhammer-chree equations, Zeitschrift fur Naturforschung - Section A Journal of Physical Sciences 57 (2002) 874-882.

[39] Y. Liu, Existence and blow up of solutions of a nonlinear pochhammerchree equation, Indiana University Mathematics Journal 45 (1996) 797- 816.

[40] I. Bagolubasky, Some examples of inelastic soliton interaction, Computer Physics Communications 13 (1977) 149-155.

[41] P. A. Clarkson, R. J. LeVaque, R. Saxton, Solitary wave interactions in elastic rods, Studies in Applied Mathematics 75 (1986) 95-122.

[42] A. Parker, On exact solutions of the regularized long wave equation: a direct approach to partially integrable equations, Journal of Mathematical Physics 36 (1995) 3498-3505.

[43] W. Zhang, M. Wenxiu, Explicit solitary wave solutions to generalized pochhammer-chree equation, Journal of Applied Mathematics and Mechanics 20 (1999) 666-674.

[44] L. Jibin, Z. Lijun, Bifurcations of travelling wave solutions in generalized pochhammer-chree equation, Chaos, Solitons and Fractals 14 (2002) 581-593.

[45] Z. Feng, On explicit exact solutions for the lienard equation and its applications, Physics Letters A 293 (2002) 50-56.

[46] A. M. Wazwaz, The tanh-coth and the sine-cosine methods for kinks, solitons, and periodic solutions for the pochhammer-chree equations, Applied Mathematics and Computation 195 (2008) 24-33.

[47] J. H. He, X. H. Wu, Exp-function method for nonlinear wave equations, Chaos, Solitons and Fractals 30 (2006) 700-708.

[48] J. H. He, X. H. Wu, Exp-function method and its application to nonlinear equations, Chaos, Solitons and Fractals 38 (2008) 903-910.