Homotopy perturbation method for solving reaction-diffusion equations.

*(English)*Zbl 1173.35551Summary: The homotopy perturbation method is applied to solve reaction-diffusion equations. In this method, the trial function (initial solution) is chosen with some unknown parameters, which are identified using the method of weighted residuals. Some examples are given. The obtained results are compared with the exact solutions, revealing that this method is very efficient and the obtained solutions are of high accuracy.

##### MSC:

35K57 | Reaction-diffusion equations |

35A25 | Other special methods applied to PDEs |

35K15 | Initial value problems for second-order parabolic equations |

##### References:

[1] | D. Lesnic, “A nonlinear reaction-diffusion process using the Adomian decomposition method,” International Communications in Heat and Mass Transfer, vol. 34, no. 2, pp. 129-135, 2007. · Zbl 1102.34039 |

[2] | J.-H. He, “New interpretation of homotopy perturbation method,” International Journal of Modern Physics B, vol. 20, no. 18, pp. 2561-2568, 2006. · Zbl 1102.34039 |

[3] | J.-H. He, “A coupling method of a homotopy technique and a perturbation technique for non-linear problems,” International Journal of Non-Linear Mechanics, vol. 35, no. 1, pp. 37-43, 2000. · Zbl 1102.34039 |

[4] | T. Ozis and A. Yildirim, “A comparative study of He/s homotopy perturbation method for determining frequency-amplitude relation of a nonlinear oscillator with discontinuities,” International Journal of Nonlinear Sciences and Numerical Simulation, vol. 8, no. 2, pp. 243-248, 2007. · Zbl 1102.34039 |

[5] | A. Belendez, A. Hernandez, T. Belendez, et al., “Application of He/s homotopy perturbation method to the Duffing-harmonic oscillator,” International Journal of Nonlinear Sciences and Numerical Simulation, vol. 8, no. 1, pp. 79-88, 2007. · Zbl 1102.34039 |

[6] | X.-C. Cai, W.-Y. Wu, and M.-S. Li, “Approximate period solution for a kind of nonlinear oscillator by He/s perturbation method,” International Journal of Nonlinear Sciences and Numerical Simulation, vol. 7, no. 1, pp. 109-112, 2006. · Zbl 1102.34039 |

[7] | M. Dehghan and F. Shakeri, “Solution of an integro-differential equation arising in oscillating magnetic fields using He/s homotopy perturbation method,” Progress in Electromagnetics Research, vol. 78, pp. 361-376, 2008. |

[8] | F. Shakeri and M. Dehghan, “Inverse problem of diffusion equation by He/s homotopy perturbation method,” Physica Scripta, vol. 75, no. 4, pp. 551-556, 2007. · Zbl 1110.35354 |

[9] | M. Dehghan and F. Shakeri, “Solution of a partial differential equation subject to temperature overspecification by He/s homotopy perturbation method,” Physica Scripta, vol. 75, no. 6, pp. 778-787, 2007. · Zbl 1117.35326 |

[10] | D. D. Ganji and A. Sadighi, “Application of He/s homotopy-perturbation method to nonlinear coupled systems of reaction-diffusion equations,” International Journal of Nonlinear Sciences and Numerical Simulation, vol. 7, no. 4, pp. 411-418, 2006. · Zbl 1102.34039 |

[11] | 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 1102.34039 |

[12] | 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 1102.34039 |

[13] | J.-H. He, “Some asymptotic methods for strongly nonlinear equations,” International Journal of Modern Physics B, vol. 20, no. 10, pp. 1141-1199, 2006. · Zbl 1102.34039 |

[14] | L.-F. Mo, “Variational approach to reaction-diffusion process,” Physics Letters A, vol. 368, no. 3-4, pp. 263-265, 2007. · Zbl 1209.65112 |

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