Giovanni Samaey
Travelling waves appear as solutions of mathematical models in a wide range of applications, including chemical reactions, epidemiology and semiconductor lasers. They are characterized by a spatial pattern that does not change shape and that moves at a constant speed. This thesis describes the numerical study of such travelling wave solutions in a class of reaction-diffusion equations in which the reaction term also depends on the state of the system in the past.Filling in the (unknown) travelling wave solution in the reaction-diffusion equation reduces the problem to the computation of homoclinic and heteroclinic orbits in delay differential equations (DDEs). The computation of these so-called connecting orbits is also important in itself, as it might explain possible chaotic behaviour of systems described by this type of equations.
We adapted the strategy developed for ordinary differential equations, using projection boundary conditions with respect to a special bilinear form. This was necessary in order to circumvent the difficulties that arise from the infinite-dimensional nature of DDEs. The method was tested on a number of example systems, and we made an analysis of convergence properties. Our experiments indicate convergence behaviour that corresponds to known theoretical results for ordinary differential equations and periodic boundary value problems in DDEs.
The algorithm is also incorporated in DDE-BIFTOOL, a publicly available Matlab package for the numerical bifurcation analysis of DDEs.
In a second part, we used this method to compute travelling waves in two model problems which arise frequently in mathematical biology: the Fisher equation and the Huxley equation. We then investigated linear stability of these waves. Stability is important, as a travelling wave must be (linearly) stable in order to observe it in practice. The numerically obtained stability properties on finite intervals can however differ drastically from the theoretically proved stability properties of the wave on the infinite domain. Indeed, perturbations can grow very large, but may disappear through the boundaries of the interval, giving an approximation of the absolute, rather than the essential spectrum.