### Abstract

We present constructive a posteriori estimates of inverse operators for initial value problems in linear ordinary differential equations (ODEs) on a bounded interval. Here, "constructive" indicates that we can obtain bounds of the operator norm in which all constants are explicitly given or are represented in a numerically computable form. In general, it is difficult to estimate these inverse operators a priori. We, therefore, propose a technique for obtaining a posteriori estimates by using Galerkin approximation of inverse operators. This type of estimation will play an important role in the numerical verification of solutions for initial value problems in nonlinear ODEs as well as for parabolic initial boundary value problems.

Original language | English |
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Pages (from-to) | 1622-1636 |

Number of pages | 15 |

Journal | Journal of Computational and Applied Mathematics |

Volume | 236 |

Issue number | 6 |

DOIs | |

Publication status | Published - 2011 Oct 15 |

Externally published | Yes |

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### Keywords

- Constructive a posteriori estimates
- Finite element method
- Linear ODEs

### ASJC Scopus subject areas

- Computational Mathematics
- Applied Mathematics

### Cite this

*Journal of Computational and Applied Mathematics*,

*236*(6), 1622-1636. https://doi.org/10.1016/j.cam.2011.09.026

**A posteriori estimates of inverse operators for initial value problems in linear ordinary differential equations.** / Kinoshita, T.; Kimura, T.; Nakao, M. T.

Research output: Contribution to journal › Article

*Journal of Computational and Applied Mathematics*, vol. 236, no. 6, pp. 1622-1636. https://doi.org/10.1016/j.cam.2011.09.026

}

TY - JOUR

T1 - A posteriori estimates of inverse operators for initial value problems in linear ordinary differential equations

AU - Kinoshita, T.

AU - Kimura, T.

AU - Nakao, M. T.

PY - 2011/10/15

Y1 - 2011/10/15

N2 - We present constructive a posteriori estimates of inverse operators for initial value problems in linear ordinary differential equations (ODEs) on a bounded interval. Here, "constructive" indicates that we can obtain bounds of the operator norm in which all constants are explicitly given or are represented in a numerically computable form. In general, it is difficult to estimate these inverse operators a priori. We, therefore, propose a technique for obtaining a posteriori estimates by using Galerkin approximation of inverse operators. This type of estimation will play an important role in the numerical verification of solutions for initial value problems in nonlinear ODEs as well as for parabolic initial boundary value problems.

AB - We present constructive a posteriori estimates of inverse operators for initial value problems in linear ordinary differential equations (ODEs) on a bounded interval. Here, "constructive" indicates that we can obtain bounds of the operator norm in which all constants are explicitly given or are represented in a numerically computable form. In general, it is difficult to estimate these inverse operators a priori. We, therefore, propose a technique for obtaining a posteriori estimates by using Galerkin approximation of inverse operators. This type of estimation will play an important role in the numerical verification of solutions for initial value problems in nonlinear ODEs as well as for parabolic initial boundary value problems.

KW - Constructive a posteriori estimates

KW - Finite element method

KW - Linear ODEs

UR - http://www.scopus.com/inward/record.url?scp=80955144822&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=80955144822&partnerID=8YFLogxK

U2 - 10.1016/j.cam.2011.09.026

DO - 10.1016/j.cam.2011.09.026

M3 - Article

AN - SCOPUS:80955144822

VL - 236

SP - 1622

EP - 1636

JO - Journal of Computational and Applied Mathematics

JF - Journal of Computational and Applied Mathematics

SN - 0377-0427

IS - 6

ER -