### Abstract

Some numerical schemes, based upon Newton’s and chord methods, for the computations of the perturbed bifurcation points as well as the solution curves through them, are presented. The “initial” guesses for Newton’s and chord methods are obtained using the local analysis techniques and proved to fall into the neighborhoods of contraction for these methods. In applications the “perturbation” parameter represents a physical quantity and it is desirable to use it to parameterize the solution curves near the perturbed bifurcation point. In this regard, it is shown that, for certain classes of the perturbed bifurcation problems, Newton’s and chord methods can be used to follow the solution curves in a neighborhood of the perturbed bifurcation point while the perturbation parameter is kept fixed.

Original language | English |
---|---|

Pages (from-to) | 561-570 |

Number of pages | 10 |

Journal | International Journal of Mathematics and Mathematical Sciences |

Volume | 18 |

Issue number | 3 |

DOIs | |

Publication status | Published - 1995 |

Externally published | Yes |

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### Keywords

- bifurcation from the trivial solution
- Newton's and chord methods
- Perturbed bifurcation
- singular perturbation

### ASJC Scopus subject areas

- Mathematics (miscellaneous)

### Cite this

*International Journal of Mathematics and Mathematical Sciences*,

*18*(3), 561-570. https://doi.org/10.1155/S0161171295000718

**On the Numerical Solution of Perturbed Bifurcation Problems.** / Elgindi, Mohamed; Langer, R. W.

Research output: Contribution to journal › Article

*International Journal of Mathematics and Mathematical Sciences*, vol. 18, no. 3, pp. 561-570. https://doi.org/10.1155/S0161171295000718

}

TY - JOUR

T1 - On the Numerical Solution of Perturbed Bifurcation Problems

AU - Elgindi, Mohamed

AU - Langer, R. W.

PY - 1995

Y1 - 1995

N2 - Some numerical schemes, based upon Newton’s and chord methods, for the computations of the perturbed bifurcation points as well as the solution curves through them, are presented. The “initial” guesses for Newton’s and chord methods are obtained using the local analysis techniques and proved to fall into the neighborhoods of contraction for these methods. In applications the “perturbation” parameter represents a physical quantity and it is desirable to use it to parameterize the solution curves near the perturbed bifurcation point. In this regard, it is shown that, for certain classes of the perturbed bifurcation problems, Newton’s and chord methods can be used to follow the solution curves in a neighborhood of the perturbed bifurcation point while the perturbation parameter is kept fixed.

AB - Some numerical schemes, based upon Newton’s and chord methods, for the computations of the perturbed bifurcation points as well as the solution curves through them, are presented. The “initial” guesses for Newton’s and chord methods are obtained using the local analysis techniques and proved to fall into the neighborhoods of contraction for these methods. In applications the “perturbation” parameter represents a physical quantity and it is desirable to use it to parameterize the solution curves near the perturbed bifurcation point. In this regard, it is shown that, for certain classes of the perturbed bifurcation problems, Newton’s and chord methods can be used to follow the solution curves in a neighborhood of the perturbed bifurcation point while the perturbation parameter is kept fixed.

KW - bifurcation from the trivial solution

KW - Newton's and chord methods

KW - Perturbed bifurcation

KW - singular perturbation

UR - http://www.scopus.com/inward/record.url?scp=77953023521&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=77953023521&partnerID=8YFLogxK

U2 - 10.1155/S0161171295000718

DO - 10.1155/S0161171295000718

M3 - Article

AN - SCOPUS:77953023521

VL - 18

SP - 561

EP - 570

JO - International Journal of Mathematics and Mathematical Sciences

JF - International Journal of Mathematics and Mathematical Sciences

SN - 0161-1712

IS - 3

ER -