The homotopy analysis method for solving higher dimensional initial boundary value problems of variable coefficients.

*(English)*Zbl 1197.65151Summary: Higher dimensional initial boundary value problems with variable coefficients are solved by means of an analytic technique, namely the homotopy analysis method. Comparisons are made between the Adomian decomposition method, the exact solution and the homotopy analysis method. The results reveal that the proposed method is very effective and simple.

##### MSC:

65M70 | Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs |

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

##### Keywords:

Adomian decomposition method; higher dimensional initial boundary value problems of variable coefficients; homotopy analysis method; homotopy perturbation method; comparison of methods
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\textit{H. Jafari} et al., Numer. Methods Partial Differ. Equations 26, No. 5, 1021--1032 (2010; Zbl 1197.65151)

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##### References:

[1] | Miller, Symmetries of differential equations: the hypergeometric and Euler-Darboux equations, It S I N J Math Anal 4 pp 314– (1973) · Zbl 0254.35080 |

[2] | Wilcox, Closed form solution of the differential equation uxy + axux + byuy + cxyu + ut = 0 by normal-ordering exponential operators, J Math Phys 11 pp 1235– (1970) |

[3] | Manwell 35 (1979) |

[4] | Nimala, A variable coefficient Kortweg-de Vries equation: similarity analysis and exact solution, J Math Phys 27 pp 2644– (1986) |

[5] | Iyanaga, Encyclopedi dictionary of mathematics (1962) |

[6] | Zwillinger, Handbook of differential equations (1992) · Zbl 0678.34001 |

[7] | S. J. Liao, The proposed homotopy analysis technique for the solution of nonlinear problems, Ph.D. Thesis, Shanghai Jiao Tong University, 1992. |

[8] | Liao, Beyond perturbation: introduction to the homotopy analysis method (2003) · Zbl 1051.76001 |

[9] | Liao, A general approach to obtain series solutions of nonlinear differential equations, Studies Appl Mathe 119 pp 297– (2007) |

[10] | Liao, Notes on the homotopy analysis method: some definitions and theorems, Commun Nonlinear Sci Numerical Simul · Zbl 1221.65126 |

[11] | Ayub, Exact flow of a third grade fluid past a porous plate using homotopy analysis method, Int J Eng Sci 41 pp 2091– (2003) · Zbl 1211.76076 |

[12] | Hayat, Homotopy solutions for a generalized second-grade fluid past a porous plate, Nonlinear Dyn 42 pp 395– (2005) · Zbl 1094.76005 |

[13] | Hayat, On non-linear flows with slip boundary condition, ZAMP 56 pp 1012– (2005) · Zbl 1097.76007 |

[14] | Liao, A new branch of solutions of boundary-layer flows over an impermeable stretched plate, Int J Heat Mass Transfer 48 pp 2529– (2005) · Zbl 1189.76142 |

[15] | Liao, Pop I. Explicit analytic solution for similarity boundary layer equations, Int J Heat Mass Transfer 47 pp 75– (2004) · Zbl 1045.76008 |

[16] | Liao, Series solution of nonlinear eigenvalue problems by means of the homotopy analysis method, Nonlinear Anal B: Real World Appl |

[17] | He, A coupling method of homotopy technique and perturbation technique for nonlinear problems, Int J Nonlinear Mech 35 pp 37– (2000) · Zbl 1068.74618 |

[18] | He, Homotopy perturbation method for solving boundary value problems, Phys Lett A 350 pp 87– (2006) · Zbl 1195.65207 |

[19] | He, Some asymptotic methods for strongly nonlinear equations, Int J Mod Phys B 20 pp 1141– (2006) · Zbl 1102.34039 |

[20] | Sajid, Comparison of HAM and HPM methods in nonlinear heat conduction and convection equations, Nonlinear Analysis: Real World Appl 9 pp 2296– (2008) · Zbl 1156.76436 |

[21] | Abdul-Majid, The decomposition method for solving higher dimensional initial boundary value problems of variable coefficients, Int J Comput Math 76 pp 159– (2000) · Zbl 0970.65106 |

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