### 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