We will consider variational iteration method (VIM) and Padé approximant, for finding analytical solutions of the Glauert-jet (self-similar wall jet over an impermeable, resting plane surface) problem. The solutions are compared with the exact solution. The results illustrate that VIM is an attractive method in solving the systems of nonlinear equations. It is predicted that VIM can have a found wide application in engineering problems.

Nonlinear phenomena play a crucial role in applied mathematics and physics. We know that most of engineering problems are nonlinear, and it is difficult to solve them analytically. Various powerful mathematical methods have been proposed for obtaining exact and approximate analytic solutions.

The VIM was first proposed by He [

The motivation of this letter is to extend variational iteration method and Padé approximant to solve Glauert-jet problem [

We consider the self-similar plane wall jet formed over an impermeable resting wall governed by the equation of a steady boundary layer over a flat plate (see [

To illustrate the basic concepts of VIM, we consider the following differential equation:

In order to obtain VIM solution of (

Its stationary conditions can be obtained as follows:

After obtaining the result of 7th iteration, we will apply the padé approximation using symbolic software such as Mathematica; we have the following:

In this paper, VIM and Padé approximants are used to find approximate solutions of the famous Glauert-jet problem. The problem of fluid jet along an impermeable, resting wall surrounded by fluid of the same type at rest has been considered. The closed-form solution of the corresponding boundary layer equations (

The VIM-Padé solution of

Comparison of the exact solution (

VIM-Padé | Exact results | Absolute error | |
---|---|---|---|

0.5 | 0.0277 | 0.0277 | 0 |

1 | 0.1091 | 0.1091 | 0 |

1.5 | 0.2354 | 0.2354 | 0 |

2 | 0.3879 | 0.3879 | 0 |

2.5 | 0.5421 | 0.5421 | 0 |

3 | 0.6774 | 0.6774 | 0 |

3.5 | 0.7833 | 0.7833 | 0 |

4 | 0.8595 | 0.8595 | 0 |

4.5 | 0.9111 | 0.9111 | 0 |

5 | 0.9446 | 0.9446 | 0 |

5.5 | 0.9657 | 0.9658 | 0.0001 |

6 | 0.9786 | 0.9791 | 0.0005 |

6.5 | 0.9863 | 0.9872 | 0.0009 |

7 | 0.9901 | 0.9922 | 0.0021 |

Comparison of the exact derivative solution (

VIM-Padé | Exact results | Absolute error | |
---|---|---|---|

0.5 | 0.1105 | 0.1105 | 0 |

1 | 0.2123 | 0.2123 | 0 |

1.5 | 0.2865 | 0.2865 | 0 |

2 | 0.3149 | 0.3149 | 0 |

2.5 | 0.2949 | 0.2949 | 0 |

3 | 0.2428 | 0.2428 | 0 |

3.5 | 0.1810 | 0.1810 | 0 |

4 | 0.1256 | 0.1256 | 0 |

4.5 | 0.0829 | 0.0830 | 0.0001 |

5 | 0.0530 | 0.0531 | 0.0001 |

5.5 | 0.0329 | 0.0333 | 0.0004 |

6 | 0.0198 | 0.0206 | 0.0008 |

6.5 | 0.0111 | 0.0126 | 0.0015 |

7 | 0.0049 | 0.0077 | 0.0028 |

Comparison of the exact solution (

Comparison of the exact derivative solution (