Non-local differential operators are a common mathematical tool to spatially model the risk of spread of an invasive species, an epidemic, or any other system where the outliers dominate the dynamics of spread. It is the one dandelion seed or the one sick traveller that establishes the new colony, not the growth of the colony, that is responsible for the speed of the invasion. While non-local operators are flexible enough to take into account some local topography the models are usually run on an infinite domain as the connection between boundary conditions for non-local differential operators on a finite domain and the underlying stochastic process remains largely unknown. This not only leads to numerical inefficiencies but also hinders the applicability and accuracy of the model. Over the last few years the connection between boundary conditions and the underlying stochastic processes were established for a very special but important case in one dimension. We generalised this approach to a much larger class of non-local differential operators to greatly enhance their applicability. This PhD project is either about numerical investigations into efficient algorithms for solving these non-local differential equations with boundary conditions and their application.
Interested applicants are encouraged to apply for a University of Otago PhD scholarship. Extra travel funding to attend conferences, etc. available.