### Abstract

The study of similarity solutions is necessary for the understanding of the dynamical behaviour of the fluid in the high-frequency excitation of liquid metal embedded in an antisymmetric magnetic field. Such similarity solutions are governed by the third-order non-linear equation: f‴+m+1/2 ff″2 = 0 on (0, + ∞), subject to the boundary conditions f(0) = a ∈ ℝ, f′(0) = -1 and f′(+∞) = 0. The remaining unsolved case is for m ε (- 1, 0). In this paper, we will give an almost complete solution structure of this problem for m ε (- 1, 0), which complements earlier results in literature.

Original language | English |
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Pages (from-to) | 157-195 |

Number of pages | 39 |

Journal | IMA Journal of Applied Mathematics (Institute of Mathematics and Its Applications) |

Volume | 77 |

Issue number | 2 |

DOIs | |

Publication status | Published - 2012 Apr 1 |

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### Keywords

- boundary-value problem
- initial-value problem
- phase plane analysis
- similarity solution
- third-order differential equation

### ASJC Scopus subject areas

- Applied Mathematics

### Cite this

**Similarity solutions for a model arising from high frequency excitation of liquid metal in an antisymmetric magnetic field.** / Brighi, Bernard; Tsai, Je-Chiang.

Research output: Contribution to journal › Article

*IMA Journal of Applied Mathematics (Institute of Mathematics and Its Applications)*, vol. 77, no. 2, pp. 157-195. https://doi.org/10.1093/imamat/hxr012

}

TY - JOUR

T1 - Similarity solutions for a model arising from high frequency excitation of liquid metal in an antisymmetric magnetic field

AU - Brighi, Bernard

AU - Tsai, Je-Chiang

PY - 2012/4/1

Y1 - 2012/4/1

N2 - The study of similarity solutions is necessary for the understanding of the dynamical behaviour of the fluid in the high-frequency excitation of liquid metal embedded in an antisymmetric magnetic field. Such similarity solutions are governed by the third-order non-linear equation: f‴+m+1/2 ff″2 = 0 on (0, + ∞), subject to the boundary conditions f(0) = a ∈ ℝ, f′(0) = -1 and f′(+∞) = 0. The remaining unsolved case is for m ε (- 1, 0). In this paper, we will give an almost complete solution structure of this problem for m ε (- 1, 0), which complements earlier results in literature.

AB - The study of similarity solutions is necessary for the understanding of the dynamical behaviour of the fluid in the high-frequency excitation of liquid metal embedded in an antisymmetric magnetic field. Such similarity solutions are governed by the third-order non-linear equation: f‴+m+1/2 ff″2 = 0 on (0, + ∞), subject to the boundary conditions f(0) = a ∈ ℝ, f′(0) = -1 and f′(+∞) = 0. The remaining unsolved case is for m ε (- 1, 0). In this paper, we will give an almost complete solution structure of this problem for m ε (- 1, 0), which complements earlier results in literature.

KW - boundary-value problem

KW - initial-value problem

KW - phase plane analysis

KW - similarity solution

KW - third-order differential equation

UR - http://www.scopus.com/inward/record.url?scp=84859018283&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84859018283&partnerID=8YFLogxK

U2 - 10.1093/imamat/hxr012

DO - 10.1093/imamat/hxr012

M3 - Article

AN - SCOPUS:84859018283

VL - 77

SP - 157

EP - 195

JO - IMA Journal of Applied Mathematics

JF - IMA Journal of Applied Mathematics

SN - 0272-4960

IS - 2

ER -