http://hdl.handle.net/2440/8 Applied Mathematics 2013-05-21T18:40:18Z http://hdl.handle.net/2440/76595 Title: Carbon nanotori as traps for atoms and ions Author: Chan, Yue; Cox, Barry James; Hill, James Murray 2011-12-31T13:30:00Z http://hdl.handle.net/2440/71981 Title: Patch dynamics for macroscale modelling in one dimension Author: Bunder, Judith; Roberts, Anthony J. Abstract: We discuss efficient macroscale modelling of microscale systems using patch dynamics. This pilot study effectively homogenises microscale varying diffusion in one dimension. The `equation free' approach requires that the microscale model be solved only on small spatial patches. Suitable boundary conditions ensure that these patches are well coupled. By centre manifold theory, an emergent closed model exists on the macroscale. Patch dynamics systematically approximates this macroscale model. The modelling is readily adaptable to higher dimensions and to reaction-diffusion equations. Description: Proceedings of the 10th Biennial Engineering Mathematics and Applications Conference (EMAC2011) held at University Technology Sydney in December 2011 2011-12-31T13:30:00Z http://hdl.handle.net/2440/71466 Title: Averaging approximation to singularly perturbed nonlinear stochastic wave equations Author: Lv, Yan; Roberts, Anthony John Abstract: An averaging method is applied to derive effective approximation to a singularly perturbed nonlinear stochastic damped wave equation. Small parameter ν > 0 characterizes the singular perturbation, and νᵅ, 0 ≤ α ≤ 1/2, parametrizes the strength of the noise. Some scaling transformations and the martingale representation theorem yield the effective approximation, a stochastic nonlinear heat equation, for small ν in the sense of distribution. 2011-12-31T13:30:00Z http://hdl.handle.net/2440/71453 Title: Average and deviation for slow-fast stochastic partial differential equations Author: Wang, Wei; Roberts, Anthony John Abstract: Averaging is an important method to extract effective macroscopic dynamics from complex systems with slow modes and fast modes. This article derives an averaged equation for a class of stochastic partial differential equations without any Lipschitz assumption on the slow modes. The rate of convergence in probability is obtained as a byproduct. Importantly, the stochastic deviation between the original equation and the averaged equation is also studied. A martingale approach proves that the deviation is described by a Gaussian process. This gives an approximation to errors of order O(ε) instead of order O(√ε) attained in previous averaging. 2011-12-31T13:30:00Z