### Abstract

In this paper, we present a numerical method with guaranteed accuracy to solve initial value problems (IVPs) of normal form simultaneous first order ordinary differential equations (ODEs) which have wide domain. Our method is based on the algorithm proposed by Kashiwagi, by which we can obtain inclusions of exact values at several discrete points of the solution curve of ODEs. The method can be regarded as an extension of the Lohner's method. But the algorithm is not efficient for equations which have wide domain, because the error bounds become too wide from a practical point of view. Our purpose is to produce tight bounds even for such equations. We realize it by combining Kashiwagi's algorithm with the mean value form. We also consider the wrapping effects to obtain tighter bounds.

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

Pages (from-to) | 2230-2237 |

Number of pages | 8 |

Journal | IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences |

Volume | E84-A |

Issue number | 9 |

Publication status | Published - 2001 Sep |

### Fingerprint

### Keywords

- Mean value form
- Numerical method with guaranteed accuracy
- Ordinary differential equation
- Power series arithmetic
- Wrapping effect

### ASJC Scopus subject areas

- Electrical and Electronic Engineering
- Hardware and Architecture
- Information Systems

### Cite this

*IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences*,

*E84-A*(9), 2230-2237.

**Long time integration for initial value problems of ordinary differential equations using power series arithmetic.** / Miyata, T.; Nagatomo, Y.; Kashiwagi, Masahide.

Research output: Contribution to journal › Article

*IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences*, vol. E84-A, no. 9, pp. 2230-2237.

}

TY - JOUR

T1 - Long time integration for initial value problems of ordinary differential equations using power series arithmetic

AU - Miyata, T.

AU - Nagatomo, Y.

AU - Kashiwagi, Masahide

PY - 2001/9

Y1 - 2001/9

N2 - In this paper, we present a numerical method with guaranteed accuracy to solve initial value problems (IVPs) of normal form simultaneous first order ordinary differential equations (ODEs) which have wide domain. Our method is based on the algorithm proposed by Kashiwagi, by which we can obtain inclusions of exact values at several discrete points of the solution curve of ODEs. The method can be regarded as an extension of the Lohner's method. But the algorithm is not efficient for equations which have wide domain, because the error bounds become too wide from a practical point of view. Our purpose is to produce tight bounds even for such equations. We realize it by combining Kashiwagi's algorithm with the mean value form. We also consider the wrapping effects to obtain tighter bounds.

AB - In this paper, we present a numerical method with guaranteed accuracy to solve initial value problems (IVPs) of normal form simultaneous first order ordinary differential equations (ODEs) which have wide domain. Our method is based on the algorithm proposed by Kashiwagi, by which we can obtain inclusions of exact values at several discrete points of the solution curve of ODEs. The method can be regarded as an extension of the Lohner's method. But the algorithm is not efficient for equations which have wide domain, because the error bounds become too wide from a practical point of view. Our purpose is to produce tight bounds even for such equations. We realize it by combining Kashiwagi's algorithm with the mean value form. We also consider the wrapping effects to obtain tighter bounds.

KW - Mean value form

KW - Numerical method with guaranteed accuracy

KW - Ordinary differential equation

KW - Power series arithmetic

KW - Wrapping effect

UR - http://www.scopus.com/inward/record.url?scp=0035446001&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0035446001&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:0035446001

VL - E84-A

SP - 2230

EP - 2237

JO - IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences

JF - IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences

SN - 0916-8508

IS - 9

ER -