Partial differential equations (PDEs) model numerous phenomena from physics, biology, and our daily life; we may even model the distribution of milk in our morning coffee by a convection-diffusion PDE. Partial differential equations are also used in image processing and training of neural nets.
In studying partial differential equations, several questions are asked, which may pose interesting mathematical challenges. First of all: do solutions to this PDE exist? In which sense? Is the solution unique? After that, we may study of qualitative and quantitative properties of solutions, e.g. their regularity or long-time behavior, or focus on certain special solutions and their practical applications.
In this project we want to focus on the study of wave-type solutions in reaction-diffusion equations. These model, among others, signal propagation in nerve fibres. An interesting point of study here is the difference in phenomena between the space-discrete equation (also called „lattice differential equation“) and the continuous equation (PDE). In the discrete setting, traveling waves may become stationary, a phenomena which is called „pinning“ or „propagation failure“. Another issue which may occurr under discretization is a change in the shape of the wave. It may even happen that the wave exists in the PDE setting, but not in the discrete setting.
Several questions appear: On which other factors does the pinning depend? We know that, as the mesh size becomes smaller, the wave may start to travel (de-pinning) - at what speed does this happen? When doing simulations, how small does the mesh size have to be in order for the solutions to reflect the continuous phenomenology?
All these questions can be addressed in a deterministic setting, using PDE and dynamical systems techniques, or we may study them in the presence of noise, which is challenging but gives rise to many more interesting questions.
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