Stability of Convective Patterns in Reaction Fronts: A Comparison of Three Models

Chemical fronts propagating in vertical cylinders exhibit flat, nonaxisymmetric, and axisymmetric fronts. The different types of fronts are determined by convection. In the case of flat fronts, there is no convection. For nonaxisymmetric fronts, fluid rises on one side and falls on the opposite side of the tube. For axisymmetric fronts, fluid rises in the middle of the tube and falls on the sides. In this work we model the transition between different fronts using a two-dimensional domain. We compared three different models of front propagation, one based on a reaction-diffusion-advection equation, the other on a front propagation equation, and finally a low-dimensional model. We study the stability of different solutions

Local convergence theorems of Newton’s method for nonlinear equations using outer or generalized inverses

summary:We provide local convergence theorems for Newton’s method in Banach space using outer or generalized inverses. In contrast to earlier results we use hypotheses on the second instead of the first Fréchet-derivative. This way our convergence balls differ from earlier ones. In fact we show that with a simple numerical example that our convergence ball contains earlier ones. This way we have a wider choice of initial guesses than before. Our results can be used to solve undetermined systems, nonlinear least squares problems and ill-posed nonlinear operator equations