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This paper is devoted to the study of a free-convective boundary-layer flow modeled by a system of nonlinear ordinary differential equations. We apply a modified variational iteration method (MVIM) coupled with He's polynomials and Padé approximation to solve free-convective boundary-layer equation. It is observed that the combination of MVIM and the Padé approximation improves the accuracy and enlarges the convergence domain.

The boundary-layer flows of viscous fluids are of utmost importance for industry and applied sciences. These flows can be modeled by systems of nonlinear ordinary differential equations on an unbounded domain, see [

To illustrate the basic concept of the modified variational iteration method (MVIM), we consider the following general differential equation:

Let us consider the problem of cooling of a low-heat-resistance sheet that moves downwards in a viscous fluid [

We denote

From (

Consider problems (

The correction functional is given by

It is observed in Figures

Variation of

Variation of

Figure

Variation of

In this study, we employed modified variational iteration method (MVIM) coupled with Padé approximation to solve a system of two nonlinear ordinary differential equations that describes a free-convective boundary-layer in glass-fiber production process. The results show strong effects of the Prandtl number on the velocity and temperature profiles since the two model equations are coupled.

Numerical values of

[ | [ | |||
---|---|---|---|---|

0.001 | 1.1135529418 | 1.1272760416 | 1.1252849854 | 1.1231381347 |

0.01 | 1.0631737963 | 1.0741895683 | 1.0638385351 | 1.0633808585 |

0.1 | 0.9128082210 | 0.9238226280 | 0.9242158493 | 0.9240830397 |

1 | 0.6941230861 | 0.6929598014 | 0.6932195158 | 0.6932116298 |

10 | 0.4511240728 | 0.4502429544 | 0.4476712316 | 0.4471165250 |

100 | 0.2679197151 | 0.2681474363 | 0.2641295627 | 0.2645235434 |

1000 | 0.2204061432 | 0.1524783266 | 0.1500456755 | 0.1512901971 |

10000 | 0.0858587180 | 0.0858519249 | 0.0844775473 | 0.0855408524 |

Numerical values of

[ | [ | [ | ||
---|---|---|---|---|

0.001 | −0.0371141028 | −0.0415417739 | −0.0436188230 | −0.0468074648 |

0.01 | −0.1274922800 | −0.1221616907 | −0.1351353865 | −0.1357607439 |

0.1 | −0.3621215470 | −0.3505589981 | −0.3499273453 | −0.3500596733 |

1 | −0.7694165843 | −0.7695971295 | −0.7698955992 | −0.7698611967 |

10 | −1.5028543431 | −1.5007437650 | −1.4985484075 | −1.4970992078 |

100 | −2.7627624234 | −2.7637067330 | −2.7445541894 | −2.7468855016 |

1000 | −5.7787858408 | −4.9468469883 | −4.9104728566 | −4.9349476252 |

10000 | −8.8057265644 | −8.8032691004 | −8.7384279086 | −8.8044492660 |

The authors are highly grateful to the referee for his/her very constructive comments.