2019/02/05 by Matsue, Kaname, Takayasu, Akitoshi
#34A26 #34C08 #35B44 #37B25 #37C99 #37M99 #58K55 #65D30 #65G30 #65L99 #65P99 #Dynamical Systems (math.DS) #FOS: Mathematics #Numerical Analysis (math.NA)
paper · doi:10.48550/arxiv.1902.01842
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.