Study of Singular Capillary Surfaces and Development of the Cluster Newton Method

Abstract

In this thesis, we explore two important aspects of study of differential equations: analytical and computational aspects. We first consider a partial differential equation model for a static liquid surface (capillary surface). We prove through mathematical analyses that the solution of this mathematical model (the Laplace-Young equation) in a cusp domain can be bounded or unbounded depending on the boundary conditions. By utilizing the knowledge we have obtained about the singular behaviour of the solution through mathematical analysis, we then construct a numerical methodology to accurately approximate unbounded solutions of the Laplace-Young equation. Using this accurate numerical methodology, we explore some remaining open problems on singular solutions of the Laplace-Young equation. Lastly, we consider ordinary differential equation models used in the pharmaceutical industry and develop a numerical method for estimating model parameters from incomplete experimental data. With our numerical method, the parameter estimation can be done significantly faster and more robustly than with conventional methods.