TY - JOUR
T1 - Rigorous numerics of blow-up solutions for ODEs with exponential nonlinearity
AU - Matsue, Kaname
AU - Takayasu, Akitoshi
N1 - Funding Information:
KM was partially supported by World Premier International Research Center Initiative (WPI) , Ministry of Education, Culture, Sports, Science and Technology (MEXT), Japan , and JSPS Grant-in-Aid for Young Scientists (B) (No. 17K14235 ). AT was partially supported by JSPS Grant-in-Aid for Early-Career Scientists (No. 18K13453 ).
Publisher Copyright:
© 2019 Elsevier B.V.
PY - 2020/8/15
Y1 - 2020/8/15
N2 - Our concerns here are blow-up solutions for ODEs with exponential nonlinearity from the viewpoint of dynamical systems and their numerical validations. As an example, the finite difference discretization of ut=uxx+eum with the homogeneous Dirichlet boundary condition is considered. Our idea is based on compactification of phase spaces and time-scale desingularization as in previous works. In the present case, treatment of exponential nonlinearity is the main issue. Fortunately, under a kind of exponential homogeneity of vector field, we can treat the problem in the same way as polynomial vector fields. In particular, we can characterize and validate blow-up solutions with their blow-up times for differential equations with such exponential nonlinearity in the similar way to previous works. A series of technical treatments of exponential nonlinearity in blow-up problems is also shown with concrete validation examples.
AB - Our concerns here are blow-up solutions for ODEs with exponential nonlinearity from the viewpoint of dynamical systems and their numerical validations. As an example, the finite difference discretization of ut=uxx+eum with the homogeneous Dirichlet boundary condition is considered. Our idea is based on compactification of phase spaces and time-scale desingularization as in previous works. In the present case, treatment of exponential nonlinearity is the main issue. Fortunately, under a kind of exponential homogeneity of vector field, we can treat the problem in the same way as polynomial vector fields. In particular, we can characterize and validate blow-up solutions with their blow-up times for differential equations with such exponential nonlinearity in the similar way to previous works. A series of technical treatments of exponential nonlinearity in blow-up problems is also shown with concrete validation examples.
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U2 - 10.1016/j.cam.2019.112607
DO - 10.1016/j.cam.2019.112607
M3 - Article
AN - SCOPUS:85079164894
SN - 0377-0427
VL - 374
JO - Journal of Computational and Applied Mathematics
JF - Journal of Computational and Applied Mathematics
M1 - 112607
ER -