Rigorous numerics of blow-up solutions for ODEs with exponential nonlinearity
For researchers in dynamical systems and numerical analysis, it incrementally extends existing compactification and desingularization techniques to handle exponential nonlinearities.
This paper extends a rigorous numerical validation method for blow-up solutions from polynomial to exponential nonlinearities in ODEs, demonstrating it on a discretized PDE with exponential term and providing concrete validation examples.
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 $u_t = u_{xx} + e^{u^m}$ 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.