### 抄録

We propose a computer-assisted method for excluding eigenvalues of an elliptic operator linearized at a solution of a nonlinear problem. The method works in both the one-dimensional and the two-dimensional case. We begin by finding an approximate solution to a nonlinear problem, and we then enclose the solution by using Nakao’s numerical verification method. Instead of considering directly the eigenvalues for the elliptic operator linearized at the verified solution, we linearize the operator at the approximate solution. We present a theorem that allows us to determine under which conditions and in which disks there will be no eigenvalues. Thus, if any of those disks are contained in the enclosed area, we can exclude those eigenvalues. Next, we construct various computable criteria that allow us to use a computer program to find these disks. Finally, we use our results to determine which eigenvalues to exclude for the operator linearized at the verified solution. We present some verified results.

元の言語 | 英語 |
---|---|

ページ（範囲） | 263-294 |

ページ数 | 32 |

ジャーナル | Japan Journal of Industrial and Applied Mathematics |

巻 | 32 |

発行部数 | 1 |

DOI | |

出版物ステータス | 出版済み - 1 1 2015 |

### Fingerprint

### All Science Journal Classification (ASJC) codes

- Engineering(all)
- Applied Mathematics

### これを引用

**A computer-assisted method for excluding eigenvalues of an elliptic operator linearized at a solution of a nonlinear problem.** / Cai, Shuting; Watanabe, Yoshitaka.

研究成果: ジャーナルへの寄稿 › 記事

}

TY - JOUR

T1 - A computer-assisted method for excluding eigenvalues of an elliptic operator linearized at a solution of a nonlinear problem

AU - Cai, Shuting

AU - Watanabe, Yoshitaka

PY - 2015/1/1

Y1 - 2015/1/1

N2 - We propose a computer-assisted method for excluding eigenvalues of an elliptic operator linearized at a solution of a nonlinear problem. The method works in both the one-dimensional and the two-dimensional case. We begin by finding an approximate solution to a nonlinear problem, and we then enclose the solution by using Nakao’s numerical verification method. Instead of considering directly the eigenvalues for the elliptic operator linearized at the verified solution, we linearize the operator at the approximate solution. We present a theorem that allows us to determine under which conditions and in which disks there will be no eigenvalues. Thus, if any of those disks are contained in the enclosed area, we can exclude those eigenvalues. Next, we construct various computable criteria that allow us to use a computer program to find these disks. Finally, we use our results to determine which eigenvalues to exclude for the operator linearized at the verified solution. We present some verified results.

AB - We propose a computer-assisted method for excluding eigenvalues of an elliptic operator linearized at a solution of a nonlinear problem. The method works in both the one-dimensional and the two-dimensional case. We begin by finding an approximate solution to a nonlinear problem, and we then enclose the solution by using Nakao’s numerical verification method. Instead of considering directly the eigenvalues for the elliptic operator linearized at the verified solution, we linearize the operator at the approximate solution. We present a theorem that allows us to determine under which conditions and in which disks there will be no eigenvalues. Thus, if any of those disks are contained in the enclosed area, we can exclude those eigenvalues. Next, we construct various computable criteria that allow us to use a computer program to find these disks. Finally, we use our results to determine which eigenvalues to exclude for the operator linearized at the verified solution. We present some verified results.

UR - http://www.scopus.com/inward/record.url?scp=84925520075&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84925520075&partnerID=8YFLogxK

U2 - 10.1007/s13160-015-0167-7

DO - 10.1007/s13160-015-0167-7

M3 - Article

AN - SCOPUS:84925520075

VL - 32

SP - 263

EP - 294

JO - Japan Journal of Industrial and Applied Mathematics

JF - Japan Journal of Industrial and Applied Mathematics

SN - 0916-7005

IS - 1

ER -