##
**Study of convergence of homotopy perturbation method for systems of partial differential equations.**
*(English)*
Zbl 1189.65246

Summary: The aim of this paper is convergence study of homotopy perturbation method for systems of nonlinear partial differential equations. The sufficient condition for convergence of the method is addressed. Since mathematical modeling of numerous scientific and engineering experiments lead to Brusselator and Burgers’ system of equations, it is worth trying new methods to solve these systems. We construct a new efficient recurrent relation to solve nonlinear Burgers’ and Brusselator systems of equations. Comparison of the results obtained by homotopy perturbation method with those of Adomian’s decomposition method and dual-reciprocity boundary element method leads to significant consequences. Two standard problems are used to validate the method.

### MSC:

65M99 | Numerical methods for partial differential equations, initial value and time-dependent initial-boundary value problems |

35Q53 | KdV equations (Korteweg-de Vries equations) |

### Keywords:

Brusselator equations; Burgers’ equations; homotopy perturbation method; convergence sequence
PDF
BibTeX
XML
Cite

\textit{J. Biazar} and \textit{H. Aminikhah}, Comput. Math. Appl. 58, No. 11--12, 2221--2230 (2009; Zbl 1189.65246)

Full Text:
DOI

### References:

[1] | Wazwaz, Abdul-Majid, The decomposition method applied to systems of partial differential equations and to the reaction-diffusion Brusselator model, Applied Mathematics and Computation, 110, 251-264 (2000) · Zbl 1023.65109 |

[2] | Ang, Whye-Teong, The two-dimensional reaction-diffusion Brusselator system: A dual-reciprocity boundary element solution, Engineering Analysis with Boundary Elements, 27, 897-903 (2003) · Zbl 1060.76599 |

[3] | Burger, J. M., A Mathematical Model Illustrating the Theory of Turbulence (1948), Academic Press: Academic Press New York |

[4] | Refik Bahadir, A., A fully implicit finite-difference scheme for two dimensional Burgers’ equations, Applied Mathematics and Computation, 137, 131-137 (2003) · Zbl 1027.65111 |

[5] | He, J. H., Homotopy perturbation technique, Computer Methods in Applied Mechanics and Engineering, 178, 257-262 (1999) · Zbl 0956.70017 |

[6] | He, J. H., A coupling method of homotopy technique and perturbation technique for nonlinear problems, International Journal of Non-Linear Mechanics, 35, 1, 37-43 (2000) · Zbl 1068.74618 |

[7] | He, J. H., Comparison of homotopy perturbation method and homotopy analysis method, Applied Mathematics and Computation, 156, 527-539 (2004) · Zbl 1062.65074 |

[8] | He, J. H., Homotopy perturbation method: A new nonlinear analytical technique, Applied Mathematics and Computation, 135, 73-79 (2003) · Zbl 1030.34013 |

[9] | He, J. H., The homotopy perturbation method for nonlinear oscillators with discontinuities, Applied Mathematics and Computation, 151, 287-292 (2004) · Zbl 1039.65052 |

[10] | He, J. H., Application of homotopy perturbation method to nonlinear wave equations, Chaos, Solitons and Fractals, 26, 695-700 (2005) · Zbl 1072.35502 |

[11] | He, J. H., Homotopy perturbation method for solving boundary value problems, Physics Letters A, 350, 87-88 (2006) · Zbl 1195.65207 |

[12] | Wang, Q., Homotopy perturbation method for fractional KdV-Burgers equation, Chaos, Solitons and Fractals, 190, 1795-1802 (2007) · Zbl 1122.65397 |

[13] | Ganji, D. D.; Rafei, M., Solitary wave solutions for a generalized Hirota-Satsuma coupled KdV equation by homotopy perturbation method, Physics Letters A, 356, 131-137 (2006) · Zbl 1160.35517 |

[14] | Cveticanin, L., Homotopy perturbation method for pure nonlinear differential equation, Chaos, Solitons and Fractals, 30, 1221-1230 (2006) · Zbl 1142.65418 |

[15] | Cveticanin, L., The homotopy-perturbation method applied for solving complex-valued differential equations with strong cubic nonlinearity, Journal of Sound and Vibration, 285, 4-5, 1171-1179 (2005) · Zbl 1238.65085 |

[16] | Biazar, J.; Ghazvini, H.; Eslami, M., He’s homotopy perturbation method for systems of integro-differential equations, Chaos, Solitons and Fractals, 39, 3, 1253-1258 (2009) · Zbl 1197.65106 |

[17] | He, J. H., Application of homotopy perturbation method to nonlinear wave equations, Chaos, Solitons and Fractals, 26, 695-700 (2005) · Zbl 1072.35502 |

[18] | Ganji, D. D.; Sadighi, A., Application of homotopy-perturbation and variational iteration methods to nonlinear heat transfer and porous media equations, Journal of Computational and Applied Mathematics, 34, 1003-1016 (2007) · Zbl 1120.65108 |

[19] | Wang, Q., Homotopy perturbation method for fractional KdV-Burgers equation, Chaos, Solitons and Fractals, 190, 1795-1802 (2007) · Zbl 1122.65397 |

[20] | Biazar, J.; Ghazvini, H., He’s homotopy perturbation method for solving systems of Volterra integral equations of the second kind, Chaos, Solitons and Fractals, 39, 2, 770-777 (2009) · Zbl 1197.65219 |

[21] | Biazar, J.; Ghazvini, H., Exact solutions for non-linear Schrödinger equations by He’s homotopy perturbation method, Physics Letters A, 366, 79-84 (2007) · Zbl 1203.65207 |

[22] | Ganji, D. D., The application of He’s homotopy perturbation method to nonlinear equations arising in heat transfer, Physics Letters A, 355, 337-341 (2006) · Zbl 1255.80026 |

[23] | Siddiqui, A. M.; Mahmood, R.; Ghori, Q. K., Homotopy perturbation method for thin film flow of a fourth grade fluid down a vertical cylinder, Physics Letters A, 352, 404-410 (2006) · Zbl 1187.76622 |

[24] | Odibat, Z. M.; Momani, S., Application of variational iteration method to nonlinear differential equations of fractional order, International Journal of Nonlinear Sciences and Numerical Simulation, 7, 1, 27-34 (2006) · Zbl 1401.65087 |

[25] | Bildik, N.; Konuralp, A., The use of variational iteration method differential transform method and Adomian decomposition method for solving different types of nonlinear partial differential equations, International Journal of Nonlinear Sciences and Numerical Simulation, 7, 1, 65-70 (2006) · Zbl 1401.35010 |

[26] | Ariel, P. D.; Hayat, T.; Asghar, S., Homotopy perturbation method and axisymmetric flow over a stretching sheet, International Journal of Nonlinear Sciences and Numerical Simulation, 7, 4, 399-406 (2006) |

[27] | Biazar, J.; Ghazvini, H., Numerical solution for special non-linear Fredholm integral equation by HPM, Applied Mathematics and Computation, 195, 681-687 (2008) · Zbl 1132.65115 |

[28] | He, J. H., Homotopy perturbation method for bifurcation of nonlinear problems, International Journal of Nonlinear Sciences and Numerical Simulation, 6, 2, 207-208 (2005) · Zbl 1401.65085 |

[29] | Cai, X. C.; Wu, W. Y.; Li, M. S., Approximate period solution for a kind of nonlinear oscillator by He’s perturbation method, International Journal of Nonlinear Sciences and Numerical Simulation, 7, 1, 109-112 (2006) |

[30] | Biazar, J.; Ghazvini, H., Homotopy perturbation method for solving hyperbolic partial differential equations, Computers and Mathematics with Applications, 54, 7-8, 1047-1054 (2007) · Zbl 1267.65147 |

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.