### Abstract

In this paper, we study a boundary value problem for a third order differential equation which arises in the study of self-similar solutions of the steady free convection problem for a vertical heated impermeable flat plate embedded in a porous medium. We consider the structure of solutions of the initial value problem for this third order differential equation. First, we classify the solutions into 6 different types. Then, by transforming the third order equation into a second order equation, with the help of some comparison principle we are able to derive the structure of solutions. This answers some of the open questions proposed by Belhachmi, Brighi, and Taous in 2001. To obtain a further distinctions of the solution structure, we introduce a new change of variables to transform the third order equation into a system of two first order equations. Then by the phase plane analysis we can obtain more information on the structure of solutions.

Original language | English |
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Pages (from-to) | 311-351 |

Number of pages | 41 |

Journal | Japan Journal of Industrial and Applied Mathematics |

Volume | 22 |

Issue number | 3 |

DOIs | |

Publication status | Published - 2005 Oct |

### Fingerprint

### Keywords

- Comparison principle
- Phase plane analysis
- Porous medium
- Self-similar solution
- The third order ODE

### ASJC Scopus subject areas

- Engineering(all)
- Applied Mathematics

### Cite this

**The structure of solutions for a third order differential equation in boundary layer theory.** / Guo, Jong Shenq; Tsai, Je Chiang.

Research output: Contribution to journal › Article

*Japan Journal of Industrial and Applied Mathematics*, vol. 22, no. 3, pp. 311-351. https://doi.org/10.1007/BF03167488

}

TY - JOUR

T1 - The structure of solutions for a third order differential equation in boundary layer theory

AU - Guo, Jong Shenq

AU - Tsai, Je Chiang

PY - 2005/10

Y1 - 2005/10

N2 - In this paper, we study a boundary value problem for a third order differential equation which arises in the study of self-similar solutions of the steady free convection problem for a vertical heated impermeable flat plate embedded in a porous medium. We consider the structure of solutions of the initial value problem for this third order differential equation. First, we classify the solutions into 6 different types. Then, by transforming the third order equation into a second order equation, with the help of some comparison principle we are able to derive the structure of solutions. This answers some of the open questions proposed by Belhachmi, Brighi, and Taous in 2001. To obtain a further distinctions of the solution structure, we introduce a new change of variables to transform the third order equation into a system of two first order equations. Then by the phase plane analysis we can obtain more information on the structure of solutions.

AB - In this paper, we study a boundary value problem for a third order differential equation which arises in the study of self-similar solutions of the steady free convection problem for a vertical heated impermeable flat plate embedded in a porous medium. We consider the structure of solutions of the initial value problem for this third order differential equation. First, we classify the solutions into 6 different types. Then, by transforming the third order equation into a second order equation, with the help of some comparison principle we are able to derive the structure of solutions. This answers some of the open questions proposed by Belhachmi, Brighi, and Taous in 2001. To obtain a further distinctions of the solution structure, we introduce a new change of variables to transform the third order equation into a system of two first order equations. Then by the phase plane analysis we can obtain more information on the structure of solutions.

KW - Comparison principle

KW - Phase plane analysis

KW - Porous medium

KW - Self-similar solution

KW - The third order ODE

UR - http://www.scopus.com/inward/record.url?scp=27744526037&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=27744526037&partnerID=8YFLogxK

U2 - 10.1007/BF03167488

DO - 10.1007/BF03167488

M3 - Article

AN - SCOPUS:27744526037

VL - 22

SP - 311

EP - 351

JO - Japan Journal of Industrial and Applied Mathematics

JF - Japan Journal of Industrial and Applied Mathematics

SN - 0916-7005

IS - 3

ER -