Numerical validation of blow-up solutions of ordinary differential equations

Akitoshi Takayasu; Kaname Matsue; Takiko Sasaki; Kazuaki Tanaka; Makoto Mizuguchi; Shin'ichi Oishi

doi:10.1016/j.cam.2016.10.013

Numerical validation of blow-up solutions of ordinary differential equations

Akitoshi Takayasu^*, Kaname Matsue, Takiko Sasaki, Kazuaki Tanaka, Makoto Mizuguchi, Shin'ichi Oishi

^*Corresponding author for this work

Research output: Contribution to journal › Article › peer-review

20 Citations (Scopus)

Abstract

This paper focuses on blow-up solutions of ordinary differential equations (ODEs). We present a method for validating blow-up solutions and their blow-up times, which is based on compactifications and the Lyapunov function validation method. The necessary criteria for this construction can be verified using interval arithmetic techniques. Some numerical examples are presented to demonstrate the applicability of our method.

Original language	English
Pages (from-to)	10-29
Number of pages	20
Journal	Journal of Computational and Applied Mathematics
Volume	314
DOIs	https://doi.org/10.1016/j.cam.2016.10.013
Publication status	Published - 2017 Apr 1

Keywords

Blow-up solutions
Compactifications
Lyapunov functions
Ordinary differential equations
Validated computations

ASJC Scopus subject areas

Computational Mathematics
Applied Mathematics

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10.1016/j.cam.2016.10.013

Cite this

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title = "Numerical validation of blow-up solutions of ordinary differential equations",

abstract = "This paper focuses on blow-up solutions of ordinary differential equations (ODEs). We present a method for validating blow-up solutions and their blow-up times, which is based on compactifications and the Lyapunov function validation method. The necessary criteria for this construction can be verified using interval arithmetic techniques. Some numerical examples are presented to demonstrate the applicability of our method.",

keywords = "Blow-up solutions, Compactifications, Lyapunov functions, Ordinary differential equations, Validated computations",

author = "Akitoshi Takayasu and Kaname Matsue and Takiko Sasaki and Kazuaki Tanaka and Makoto Mizuguchi and Shin'ichi Oishi",

note = "Funding Information: AT was partially supported by JSPS Grant-in-Aid for Young Scientists (B), No. 15K17596 . KM was partially supported by The Coop with Math Program, a commissioned project by Ministry of Education, Culture, Sports, Science and Technology (MEXT) . SO was partially supported by CREST, JST . The authors also wish to thank Prof. Nobito Yamamoto, who provided them with the essential suggestion of re-parameterization in the integrals of ODEs via Lyapunov functions. Publisher Copyright: {\textcopyright} 2016 Elsevier B.V.",

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AU - Takayasu, Akitoshi

AU - Matsue, Kaname

AU - Sasaki, Takiko

AU - Tanaka, Kazuaki

AU - Mizuguchi, Makoto

AU - Oishi, Shin'ichi

N1 - Funding Information: AT was partially supported by JSPS Grant-in-Aid for Young Scientists (B), No. 15K17596 . KM was partially supported by The Coop with Math Program, a commissioned project by Ministry of Education, Culture, Sports, Science and Technology (MEXT) . SO was partially supported by CREST, JST . The authors also wish to thank Prof. Nobito Yamamoto, who provided them with the essential suggestion of re-parameterization in the integrals of ODEs via Lyapunov functions. Publisher Copyright: © 2016 Elsevier B.V.

PY - 2017/4/1

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N2 - This paper focuses on blow-up solutions of ordinary differential equations (ODEs). We present a method for validating blow-up solutions and their blow-up times, which is based on compactifications and the Lyapunov function validation method. The necessary criteria for this construction can be verified using interval arithmetic techniques. Some numerical examples are presented to demonstrate the applicability of our method.

AB - This paper focuses on blow-up solutions of ordinary differential equations (ODEs). We present a method for validating blow-up solutions and their blow-up times, which is based on compactifications and the Lyapunov function validation method. The necessary criteria for this construction can be verified using interval arithmetic techniques. Some numerical examples are presented to demonstrate the applicability of our method.

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KW - Lyapunov functions

KW - Ordinary differential equations

KW - Validated computations

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JF - Journal of Computational and Applied Mathematics

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Numerical validation of blow-up solutions of ordinary differential equations

Abstract

Keywords

ASJC Scopus subject areas

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