Some applications of computational mathematics: Tumor angiogenesis &amp; Bose-Einstein condensates
This diploma thesis studies the application of computational mathematics to two different fields of research: tumor angiogenesis and Bose-Einstein condensates (BECs).
In Chapter 2, a minimal model for describing the effect of an angiogenic inhibitor upon tumor-induced angiogenesis is examined. Simulations of the model will make use of the Deterministic Cellular Automata technique developed by Dr. Sandy Anderson.
Chapter 3 examines steady states and their dynamics for four different scenarios related to Bose-Einstein condensates. The first two scenarios address the possibility of manipulating a condensate via the use of a laser in the form of either a localized inhomogeneity or optical lattice. A system of linearly coupled, discrete nonlinear Schrödinger equations is the subject of the third scenario, which can be used to describe the behavior of two different BECs contained within a strong optical lattice and subject to an external microwave or radio-frequency field. The final scenario expands upon previous research into radial BEC solutions by including numerical simulations achieved through the use of spectral methods.
Both of these problems provide interesting systems for study and draw upon different aspects of computational mathematics, along with other areas of analytical mathematics, including bifurcation and perturbation theory.