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

TL;DR

This paper develops a rigorous numerical method for analyzing blow-up solutions in ODEs with exponential nonlinearities, extending techniques used for polynomial cases through phase space compactification and time-scale desingularization.

Contribution

It introduces a novel approach to handle exponential nonlinearities in blow-up problems, enabling validation of blow-up solutions and times similar to polynomial cases.

Findings

Validated blow-up solutions with computed blow-up times.

Extended phase space techniques to exponential nonlinearities.

Provided concrete numerical validation examples.

Abstract

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.