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**Solving heat and wave-like equations using He’s polynomials.**
*(English)*
Zbl 1181.80014

Summary: We use He’s polynomials which are calculated form homotopy perturbation method (HPM) for solving heat and wave-like equations. The proposed iterative scheme finds the solution without any discretization, linearization, or restrictive assumptions. Several examples are given to verify the reliability and efficiency of the method. The fact that suggested technique solves nonlinear problems without using Adomian’s polynomials is a clear advantage of this algorithm over the decomposition method.

### MSC:

80M25 | Other numerical methods (thermodynamics) (MSC2010) |

78M25 | Numerical methods in optics (MSC2010) |

65L15 | Numerical solution of eigenvalue problems involving ordinary differential equations |

35J05 | Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation |

35A25 | Other special methods applied to PDEs |

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\textit{S. T. Mohyud-Din}, Math. Probl. Eng. 2009, Article ID 427516, 12 p. (2009; Zbl 1181.80014)

### References:

[1] | M. A. Noor and S. T. Mohyud-Din, “Modified variational iteration method for heat and wave-like equations,” Acta Applicandae Mathematicae, vol. 104, no. 3, pp. 257-269, 2008. · Zbl 1162.65397 |

[2] | A.-M. Wazwaz and A. Gorguis, “Exact solutions for heat-like and wave-like equations with variable coefficients,” Applied Mathematics and Computation, vol. 149, no. 1, pp. 15-29, 2004. · Zbl 1038.65103 |

[3] | R. Wilcox, “Closed-form solution of the differential equation (\partial 2/\partial x\partial y+ax(\partial /\partial x)+by(\partial /\partial y)+cxy+\partial /\partial t)P(x,y,t)=0 by normal-ordering exponential operators,” Journal of Mathematical Physics, vol. 11, no. 4, pp. 1235-1237, 1970. · Zbl 0191.39603 |

[4] | J.-H. He, “An elementary introduction to recently developed asymptotic methods and nanomechanics in textile engineering,” International Journal of Modern Physics B, vol. 22, no. 21, pp. 3487-3578, 2008. · Zbl 1149.76607 |

[5] | J.-H. He, “Recent development of the homotopy perturbation method,” Topological Methods in Nonlinear Analysis, vol. 31, no. 2, pp. 205-209, 2008. · Zbl 1159.34333 |

[6] | 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 |

[7] | J.-H. He, “Homotopy perturbation method for solving boundary value problems,” Physics Letters A, vol. 350, no. 1-2, pp. 87-88, 2006. · Zbl 1195.65207 |

[8] | J.-H. He, “Comparison of homotopy perturbation method and homotopy analysis method,” Applied Mathematics and Computation, vol. 156, no. 2, pp. 527-539, 2004. · Zbl 1062.65074 |

[9] | J.-H. He, “Homotopy perturbation method for bifurcation of nonlinear problems,” International Journal of Nonlinear Sciences and Numerical Simulation, vol. 6, no. 2, pp. 207-208, 2005. · Zbl 1401.65085 |

[10] | J.-H. He, “The homotopy perturbation method for nonlinear oscillators with discontinuities,” Applied Mathematics and Computation, vol. 151, no. 1, pp. 287-292, 2004. · Zbl 1039.65052 |

[11] | 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 1068.74618 |

[12] | A. Ghorbani and J. Saberi-Nadjafi, “He/s homotopy perturbation method for calculating adomian polynomials,” International Journal of Nonlinear Sciences and Numerical Simulation, vol. 8, no. 2, pp. 229-232, 2007. · Zbl 1401.65056 |

[13] | A. Ghorbani, “Beyond Adomian polynomials: He polynomials,” Chaos, Solitons & Fractals, vol. 39, no. 3, pp. 1486-1492, 2009. · Zbl 1197.65061 |

[14] | S. T. Mohyud-Din, M. A. Noor, and K. I. Noor, “Traveling wave solutions of seventh-order generalized KdV equations using He/s polynomials,” International Journal of Nonlinear Sciences and Numerical Simulation, vol. 10, no. 2, pp. 227-233, 2009. · Zbl 1168.35427 |

[15] | S. T. Mohyud-Din and M. A. Noor, “Homotopy perturbation method for solving fourth-order boundary value problems,” Mathematical Problems in Engineering, vol. 2007, Article ID 98602, 15 pages, 2007. · Zbl 1144.65311 |

[16] | S. T. Mohyud-Din, M. A. Noor, and K. I. Noor, “Homotopy perturbation method for unsteady flow of gas through a porous medium,” International Journal of Modern Physics B. In press. · Zbl 1189.65169 |

[17] | S. T. Mohyud-Din, M. A. Noor, and K. I. Noor, “On the coupling of polynomials with correction functional,” International Journal of Modern Physics B. In press. · Zbl 1206.34024 |

[18] | M. A. Noor and S. T. Mohyud-Din, “Homotopy perturbation method for solving nonlinear higher-order boundary value problems,” International Journal of Nonlinear Sciences and Numerical Simulation, vol. 9, no. 4, pp. 395-408, 2008. · Zbl 1142.65386 |

[19] | L. Xu, “He/s homotopy perturbation method for a boundary layer equation in unbounded domain,” Computers & Mathematics with Applications, vol. 54, no. 7-8, pp. 1067-1070, 2007. · Zbl 1267.76089 |

[20] | M. A. Noor and S. T. Mohyud-Din, “Modified variational iteration method for heat and wave-like equations,” Acta Applicandae Mathematicae, vol. 104, no. 3, pp. 257-269, 2008. · Zbl 1162.65397 |

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