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**New exact solutions of the Brusselator reaction diffusion model using the exp-function method.**
*(English)*
Zbl 1184.35099

Summary: Firstly, using a series of transformations, the Brusselator reaction diffusion model is reduced into a nonlinear reaction diffusion equation, and then through using Exp-function method, more new exact solutions are found which contain soliton solutions. 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.

### MSC:

35C05 | Solutions to PDEs in closed form |

35K57 | Reaction-diffusion equations |

35Q51 | Soliton equations |

35A25 | Other special methods applied to PDEs |

### Keywords:

series of transformation
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XMLCite

\textit{F. Khani} et al., Math. Probl. Eng. 2009, Article ID 346461, 9 p. (2009; Zbl 1184.35099)

### References:

[1] | R. Lefever, M. Herschkowitz-Kaufman, and J. W. Turner, “Dissipative structures in a soluble non-linear reaction-diffusion system,” Physics Letters A, vol. 60, no. 5, pp. 389-391, 1977. |

[2] | P. K. Vani, G. A. Ramanujam, and P. Kaliappan, “Painleve analysis and particular solutions of a coupled nonlinear reaction diffusion system,” Journal of Physics A, vol. 26, no. 3, pp. L97-L99, 1993. · Zbl 0797.35157 |

[3] | A. Pickering, “A new truncation in Painlevé analysis,” Journal of Physics A, vol. 26, no. 17, pp. 4395-4405, 1993. · Zbl 0803.35131 |

[4] | I. Ndayirinde and W. Malfliet, “New special solutions of the “Brusselator” reaction model,” Journal of Physics A, vol. 30, no. 14, pp. 5151-5157, 1997. · Zbl 0924.35184 |

[5] | A. L. Larsen, “Weiss approach to a pair of coupled nonlinear reaction-diffusion equations,” Physics Letters A, vol. 179, no. 4-5, pp. 284-290, 1993. |

[6] | J.-H. He, “Variational iteration method: a kind of non-linear analytical technique: some examples,” International Journal of Non-Linear Mechanics, vol. 34, no. 4, pp. 699-708, 1999. · Zbl 1342.34005 |

[7] | M. T. Darvishi, F. Khani, and A. A. Soliman, “The numerical simulation for stiff systems of ordinary differential equations,” Computers & Mathematics with Applications, vol. 54, no. 7-8, pp. 1055-1063, 2007. · Zbl 1141.65371 |

[8] | M. T. Darvishi and F. Khani, “Numerical and explicit solutions of the fifth-order Korteweg-de Vries equations,” Chaos, Solitons & Fractals, vol. 39, no. 5, pp. 2484-2490, 2009. · Zbl 1197.65165 |

[9] | A. Molabahrami, F. Khani, and S. Hamedi-Nezhad, “Soliton solutions of the two-dimensional KdV-Burgers equation by homotopy perturbation method,” Physics Letters A, vol. 370, no. 5-6, pp. 433-436, 2007. · Zbl 1209.65113 |

[10] | J.-H. He, “New interpretation of homotopy perturbation method,” International Journal of Modern Physics B, vol. 20, no. 18, pp. 2561-2568, 2006. |

[11] | M. T. Darvishi and F. Khani, “Application of He’s homotopy perturbation method to stiff systems of ordinary differential equations,” Zeitschrift fur Naturforschung. Section A, vol. 63, no. 1-2, pp. 19-23, 2008. |

[12] | A.-M. Wazwaz, “The tanh method: solitons and periodic solutions for the Dodd-Bullough-Mikhailov and the Tzitzeica-Dodd-Bullough equations,” Chaos, Solitons & Fractals, vol. 25, no. 1, pp. 55-63, 2005. · Zbl 1070.35076 |

[13] | C.-L. Bai and H. Zhao, “Generalized extended tanh-function method and its application,” Chaos, Solitons & Fractals, vol. 27, no. 4, pp. 1026-1035, 2006. · Zbl 1088.35534 |

[14] | F. Pedit and H. Wu, “Discretizing constant curvature surfaces via loop group factorizations: the discrete sine- and sinh-Gordon equations,” Journal of Geometry and Physics, vol. 17, no. 3, pp. 245-260, 1995. · Zbl 0856.58020 |

[15] | X. Zhao, L. Wang, and W. Sun, “The repeated homogeneous balance method and its application to nonlinear partial differential equations,” Chaos, Solitons & Fractals, vol. 28, no. 2, pp. 448-453, 2006. · Zbl 1082.35014 |

[16] | J.-F. Zhang, “Homogeneous balance method and chaotic and fractal solutions for the Nizhnik-Novikov-Veselov equation,” Physics Letters A, vol. 313, no. 5-6, pp. 401-407, 2003. · Zbl 1040.35105 |

[17] | S. Zhang, “New exact solutions of the KdV-Burgers-Kuramoto equation,” Physics Letters A, vol. 358, no. 5-6, pp. 414-420, 2006. · Zbl 1142.35592 |

[18] | E. Yomba, “The modified extended Fan’s sub-equation method and its application to (2+1)-dimensional dispersive long wave equation,” Chaos, Solitons & Fractals, vol. 26, no. 3, pp. 785-794, 2005. · Zbl 1080.35096 |

[19] | J.-H. He and X.-H. Wu, “Exp-function method for nonlinear wave equations,” Chaos, Solitons & Fractals, vol. 30, no. 3, pp. 700-708, 2006. · Zbl 1141.35448 |

[20] | S.-D. Zhu, “Exp-function method for the Hybrid-Lattice system,” International Journal of Nonlinear Sciences and Numerical Simulation, vol. 8, no. 3, pp. 461-464, 2007. · Zbl 06942293 |

[21] | S.-D. Zhu, “Exp-function method for the discrete mKdV lattice,” International Journal of Nonlinear Sciences and Numerical Simulation, vol. 8, no. 3, pp. 465-468, 2007. · Zbl 06942294 |

[22] | A. Bekir and A. Boz, “Exact solutions for a class of nonlinear partial differential equations using exp-function method,” International Journal of Nonlinear Sciences and Numerical Simulation, vol. 8, no. 4, pp. 505-512, 2007. · Zbl 06942300 |

[23] | J.-H. He and L.-N. Zhang, “Generalized solitary solution and compacton-like solution of the Jaulent-Miodek equations using the Exp-function method,” Physics Letters A, vol. 372, no. 7, pp. 1044-1047, 2008. · Zbl 1217.35152 |

[24] | F. Khani, S. Hamedi-Nezhad, and A. Molabahrami, “A reliable treatment for nonlinear Schrödinger equations,” Physics Letters A, vol. 371, no. 3, pp. 234-240, 2007. · Zbl 1209.81104 |

[25] | F. Khani, S. Hamedi-Nezhad, M. T. Darvishi, and S.-W. Ryu, “New solitary wave and periodic solutions of the foam drainage equation using the Exp-function method,” Nonlinear Analysis: Real World Applications, vol. 10, no. 3, pp. 1904-1911, 2009. · Zbl 1168.35302 |

[26] | F. Khani, “Analytic study on the higher order Ito equations: new solitary wave solutions using the Exp-function method,” Chaos, Solitons & Fractals, vol. 41, no. 4, pp. 2128-2134, 2009. · Zbl 1198.35224 |

[27] | F. Khani and S. Hamedi-Nezhad, “Some new exact solutions of the (2+1)-dimensional variable coefficient Broer-Kaup system using the Exp-function method,” Computers & Mathematics with Applications. In press. · Zbl 1189.35278 |

[28] | J.-H. He and M. A. Abdou, “New periodic solutions for nonlinear evolution equations using Exp-function method,” Chaos, Solitons & Fractals, vol. 34, no. 5, pp. 1421-1429, 2007. · Zbl 1152.35441 |

[29] | X.-H. Wu and J.-H. He, “Solitary solutions, periodic solutions and compacton-like solutions using the Exp-function method,” Computers & Mathematics with Applications, vol. 54, no. 7-8, pp. 966-986, 2007. · Zbl 1143.35360 |

[30] | S. A. El-Wakil, M. A. Madkour, and M. A. Abdou, “New traveling wave solutions for nonlinear evolution equations,” Physics Letters A, vol. 365, no. 5-6, pp. 429-438, 2007. · Zbl 1203.35214 |

[31] | M. A. Abdou, A. A. Soliman, and S. T. El-Basyony, “New application of Exp-function method for improved Boussinesq equation,” Physics Letters A, vol. 369, no. 5-6, pp. 469-475, 2007. · Zbl 1209.81091 |

[32] | S. Zhang, “Application of Exp-function method to high-dimensional nonlinear evolution equation,” Chaos, Solitons & Fractals, vol. 38, no. 1, pp. 270-276, 2008. · Zbl 1142.35593 |

[33] | A. Ebaid, “Exact solitary wave solutions for some nonlinear evolution equations via Exp-function method,” Physics Letters A, vol. 365, no. 3, pp. 213-219, 2007. · Zbl 1203.35213 |

[34] | Z.-Y. Yan and H.-Q. Zhang, “Auto-Darboux transformation and exact solutions of the Brusselator reaction diffusion model,” Applied Mathematics and Mechanics, vol. 22, no. 5, pp. 541-546, 2001. · Zbl 0989.92028 |

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