TY - JOUR
T1 - Numerical validation of blow-up solutions with quasi-homogeneous compactifications
AU - Matsue, Kaname
AU - Takayasu, Akitoshi
N1 - Funding Information:
KM was partially supported by Program for Promoting the reform of national universities (Kyushu University), Ministry of Education, Culture, Sports, Science and Technology (MEXT), Japan, World Premier International Research Center Initiative (WPI), MEXT, Japan, and JSPS Grant-in-Aid for Young Scientists (B) (No. 17K14235). AT was partially supported by JSPS Grant-in-Aid for Young Scientists (B) (No. 15K17596).
Publisher Copyright:
© 2020, Springer-Verlag GmbH Germany, part of Springer Nature.
PY - 2020/7/1
Y1 - 2020/7/1
N2 - 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.
AB - 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.
UR - http://www.scopus.com/inward/record.url?scp=85086116080&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=85086116080&partnerID=8YFLogxK
U2 - 10.1007/s00211-020-01125-z
DO - 10.1007/s00211-020-01125-z
M3 - Article
AN - SCOPUS:85086116080
SN - 0029-599X
VL - 145
SP - 605
EP - 654
JO - Numerische Mathematik
JF - Numerische Mathematik
IS - 3
ER -