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A class of nonlinear convection-diffusion equation is studied in this paper. The partial differential equation is converted into nonlinear ordinary differential equation by introducing a similarity transformation. The asymptotic analytical solutions are obtained by using double-parameter transformation perturbation expansion method (DPTPEM). The influences of convection functional coefficient

Convection-diffusion equations arose from various fields of applied sciences such as heat transfer problems in a draining film [

In this paper, we consider a class of nonlinear convection-diffusion equation as follows [

Introduce the following similarity variables [

Conversely, if

Let

The method was first proposed by Yan Zhang and Liancun Zheng [

We consider a class of ordinary differential equation initial or boundary value problem without small parameter

where

Introduce an embedded parameter transformation as follows:

where

Combine (

The basic idea of the DPTPEM consists of three steps. Firstly, introducing an artificial small parameter

The initial condition is assumed as follows:

Let

Then, we can obtain

Combine (

Similarly, let

Let

It is obvious that we can promptly obtain the value of

Values of

| | |
---|---|---|

0.2 | | 1.013 |

0.6 | | 0.7815 |

1.0 | | 0.6792 |

2.5 | | 0.5701 |

5.0 | | 0.5342 |

10.0 | | 0.5169 |

Values of

| | |
---|---|---|

0.2 | | 0.9122 |

0.6 | | 0.6532 |

1.0 | | 0.5352 |

2.5 | | 0.4090 |

5.0 | | 0.3691 |

10.0 | | 0.3506 |

Values of

| | |
---|---|---|

0.2 | | 0.8643 |

0.6 | | 0.5927 |

1.0 | | 0.4670 |

2.5 | | 0.3300 |

5.0 | | 0.2870 |

10.0 | | 0.2676 |

Values of

| | |
---|---|---|

2.0 | | 0.5885 |

2.0 | | 0.4300 |

2.0 | | 0.3529 |

Values of

| | |
---|---|---|

8.0 | | 0.5212 |

8.0 | | 0.3551 |

8.0 | | 0.2723 |

Flux distribution for

Flux distribution for

Flux distribution for

Flux distribution for

Flux distribution for

It can be seen from Figures

In order to verify the efficiency and reliability of approximate analytical solution obtained by using DPTPEM, a comparison of approximate analytical solution and numerical one obtained by bvp4c with Matlab is presented in Tables

Comparison of

| | |
---|---|---|

0.0 | 0.5169 | 0.5169 |

0.1 | 0.5119 | 0.5119 |

0.2 | 0.4968 | 0.4968 |

0.3 | 0.4715 | 0.4715 |

0.4 | 0.4359 | 0.4359 |

0.5 | 0.3899 | 0.3899 |

0.6 | 0.3334 | 0.3334 |

0.7 | 0.2663 | 0.2663 |

0.8 | 0.1884 | 0.1884 |

0.9 | 0.0997 | 0.0997 |

1.0 | 0 | 0 |

Comparison of

| | |
---|---|---|

0.0 | 0.3529 | 0.3551 |

0.1 | 0.3528 | 0.3550 |

0.2 | 0.3518 | 0.3540 |

0.3 | 0.3483 | 0.3506 |

0.4 | 0.3405 | 0.3427 |

0.5 | 0.3255 | 0.3278 |

0.6 | 0.3000 | 0.3023 |

0.7 | 0.2601 | 0.2624 |

0.8 | 0.2008 | 0.2031 |

0.9 | 0.1164 | 0.1186 |

1.0 | 0 | 0 |

The comparison of the results of approximate analytical solution and numerical solution for

The comparison of the results of approximate analytical solution and numerical solution for

In the paper, a class of nonlinear convection-diffusion was studied. The partial differential equation and corresponding initial value conditions were transformed into a class of singular nonlinear boundary value problems of ordinary differential equation when similarity variables were introduced. An efficient approximate analytical method named DPTPEM was applied to solve these nonlinear problems. The reliability and effectiveness of the DPTPEM were verified by comparing approximate results with the numerical solutions. The effects of convection functional coefficient

The data used to support the findings of this study are available from the corresponding author upon request.

The author declares that there are no conflicts of interest.

The research was supported by a grant from the National Natural Science Foundation of China (no. 11501496) and the Natural Science Basic Research Plan in Shaanxi Province of China (no. 2014JQ2-1003).

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