Numerical validation of blow-up solutions with quasi-homogeneous compactifications

Kaname Matsue, Akitoshi Takayasu

Research output: Contribution to journalArticlepeer-review

Abstract

We provide a numerical validation method of blow-up solutions for finite dimensional vector fields admitting asymptotic quasi-homogeneity at infinity. Our methodology is based on quasi-homogeneous compactifications containing quasi-parabolic-type and directional-type compactifications. Divergent solutions including blow-up solutions then correspond to global trajectories of associated vector fields with appropriate time-variable transformation tending to equilibria on invariant manifolds representing infinity. We combine standard methodology of rigorous numerical integration of differential equations with Lyapunov function validations around equilibria corresponding to divergent directions, which yields rigorous upper and lower bounds of blow-up time as well as rigorous profile enclosures of blow-up solutions.

Original languageEnglish
Pages (from-to)605-654
Number of pages50
JournalNumerische Mathematik
Volume145
Issue number3
DOIs
Publication statusPublished - Jul 1 2020

All Science Journal Classification (ASJC) codes

  • Computational Mathematics
  • Applied Mathematics

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