Rigorous numerics of blow-up solutions for ODEs with exponential nonlinearity

JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS

卷:374

Article

Matsue, Kaname^1,2  Takayasu, Akitoshi³ 

[1] Kyushu Univ, Inst Math Ind, Fukuoka 8190395, Japan

[2] Kyushu Univ, Int Inst Carbon Neutral Energy Res WPI I2CNER, Fukuoka 8190395, Japan

[3] Univ Tsukuba, Fac Engn Informat & Syst, 1-1-1 Tennodai, Tsukuba, Ibaraki 3058573, Japan

关键词: Blow-up solutions for ODEs;    Numerical validation;    Compactification;    Exponential nonlinearity;   

DOI  :  10.1016/j.cam.2019.112607

来源: Elsevier

【 摘 要 】

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 u(t) = u(xx) + e(um) 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. (C) 2019 Elsevier B.V. All rights reserved.

【 授权许可】

Free