Some Solitary Wave Solutions of Generalized Pochhammer-Chree Equation via Exp-function Method
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.
[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.
[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.
@article{"International Journal of Engineering, Mathematical and Physical Sciences:61575", author = "Kourosh Parand and Jamal Amani Rad", title = "Some Solitary Wave Solutions of Generalized Pochhammer-Chree Equation via Exp-function Method", 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.", volume = "4", number = "7", pages = "995-6", }
