Canards in advection-reaction-diffusion systems in one spatial dimension

This thesis contains a mathematical investigation of the existence of travelling wave solutions to singularly perturbed advection-reaction-diffusion models of biological processes. An enhanced mathematical understanding of these solutions and models is gained via the identification of canards (special solutions of fast/slow dynamical systems) and their role in the existence of the most biologically relevant, shock-like solutions. The analysis focuses on two existing models. A new proof of existence of a whole family of travelling waves is provided for a model describing malignant tumour invasion, while new solutions are identified for a model describing wound healing angiogenesis.

Diffusion-driven formation of MoS2 nanotube bundles containing MoS2 nanopods

MoS2 nanotube bundles along with embedded nested fullerenes were formed in a gas phase reaction of molybdenum carbonyl and H2S gas with the assistance of I2. The amorphous Mo-S-I intermediates obtained through quenching a modified MOCVD reaction in a large temperature gradient were annealed at elevated temperature in an inert atmosphere. Under the influence of the iodine the amorphous precursor formed a surface film with an enhanced mobility of the molybdenum and sulfur components. Point defects within the MoS2 layers combined with the enhanced surface diffusion lead to a scrolling of the inherently instable MoS2 lamellae.

Delivering crushed tablets using thickened fluids : is drug diffusion into gastric fluids restricted?

Solid medications are often crushed and mixed with food or thickened water to aid drug delivery for those who cannot or prefer not to swallow whole tablets or capsules. Dysphagic patients have the added problem of being unable to safely swallow thin fluids so water thickened with polysaccharides is used to deliver crushed medications and ensure safe swallowing. It is postulated that these polysaccharide systems may restrict drug release by reducing the diffusion of the drug into gastric fluids. METHODS By using a vertical diffusion cell separated with a synthetic membrane, the diffusion of a model drug (atenolol) was studied from a donor system containing the drug dispersed into thickened water with xanthan gum (concentration range from 0.005%-2.2%) into a receptor system containing simulated gastric fluid (SGF) at 37°C. The amount of drug transferred was measured over 8 hours and diffusion coefficients estimated using the Higuchi model approach. RESULTS Atenolol diffusion decreased with increasing xanthan gum concentration up to 1.0%, above which diffusion remained around 300 μ2s-1. The rheological measurements captured the influence of the structure and conformation of the polysaccharide in water on the movement and availability of the drug in SGF. DISCUSSION Dose form administration for dysphagic patients’ needs special attention from general practitioners, pharmacist and patients. Improving drug release of crushed tablets from thickening agents requires a reduction in the diffusion pathway (e.g. by decreasing drop size radius). This approach could make the drug available in SGF in a short time without compromising the mechanical aspects of thickening agents that guarantee safe swallowing.

A numerical method for the fractional Fitzhugh–Nagumo monodomain model

A fractional FitzHugh–Nagumo monodomain model with zero Dirichlet boundary conditions is presented, generalising the standard monodomain model that describes the propagation of the electrical potential in heterogeneous cardiac tissue. The model consists of a coupled fractional Riesz space nonlinear reaction-diffusion model and a system of ordinary differential equations, describing the ionic fluxes as a function of the membrane potential. We solve this model by decoupling the space-fractional partial differential equation and the system of ordinary differential equations at each time step. Thus, this means treating the fractional Riesz space nonlinear reaction-diffusion model as if the nonlinear source term is only locally Lipschitz. The fractional Riesz space nonlinear reaction-diffusion model is solved using an implicit numerical method with the shifted Grunwald–Letnikov approximation, and the stability and convergence are discussed in detail in the context of the local Lipschitz property. Some numerical examples are given to show the consistency of our computational approach.

A reduction of diffusion in PVA Fricke hydrogels

A modification to the PVA-FX hydrogel whereby the chelating agent, xylenol orange, was partially bonded to the gelling agent, poly-vinyl alcohol, resulted in an 8% reduction in the post irradiation Fe3+ diffusion, adding approximately 1 hour to the useful timespan between irradiation and readout. This xylenol orange functionalised poly-vinyl alcohol hydrogel had an OD dose sensitivity of 0.014 Gy−1 and a diffusion rate of 0.133 mm2 h−1. As this partial bond yields only incremental improvement, it is proposed that more efficient methods of bonding xylenol orange to poly-vinyl alcohol be investigated to further reduce the diffusion in Fricke gels.

Green urbanism and diffusion issues

This paper addresses the research question, ‘What are the diffusion determinants for green urbanism innovations in Australia?’ This is a significant topic given the global movement towards green urbanism. The study reported here is based on desktop research that provides new insights through (1) synthesis of the latest research findings on green urbanism innovations and (2) interpretation of diffusion issues through our innovation system model. Although innovation determinants have been studied extensively overseas and in Australia, there is presently a gap in the literature when it comes to these determinants for green urbanism in Australia. The current paper fills this gap. Using a conceptual framework drawn from the innovation systems literature, this paper synthesises and interprets the literature to map the current state of green urbanism innovations in Australia and to analyse the drivers for, and obstacles to, their optimal diffusion. The results point to the importance of collaboration between project-based actors in the implementation of green urbanism. Education, training and regulation across the product system is also required to improve the cultural and technical context for implementation. The results are limited by their exploratory nature and future research is planned to quantify barriers to green urbanism.

Further development of discrete computational techniques for calculation of restricted diffusion propagators in porous media

Magnetic resonance is a well-established tool for structural characterisation of porous media. Features of pore-space morphology can be inferred from NMR diffusion-diffraction plots or the time-dependence of the apparent diffusion coefficient. Diffusion NMR signal attenuation can be computed from the restricted diffusion propagator, which describes the distribution of diffusing particles for a given starting position and diffusion time. We present two techniques for efficient evaluation of restricted diffusion propagators for use in NMR porous-media characterisation. The first is the Lattice Path Count (LPC). Its physical essence is that the restricted diffusion propagator connecting points A and B in time t is proportional to the number of distinct length-t paths from A to B. By using a discrete lattice, the number of such paths can be counted exactly. The second technique is the Markov transition matrix (MTM). The matrix represents the probabilities of jumps between every pair of lattice nodes within a single timestep. The propagator for an arbitrary diffusion time can be calculated as the appropriate matrix power. For periodic geometries, the transition matrix needs to be defined only for a single unit cell. This makes MTM ideally suited for periodic systems. Both LPC and MTM are closely related to existing computational techniques: LPC, to combinatorial techniques; and MTM, to the Fokker-Planck master equation. The relationship between LPC, MTM and other computational techniques is briefly discussed in the paper. Both LPC and MTM perform favourably compared to Monte Carlo sampling, yielding highly accurate and almost noiseless restricted diffusion propagators. Initial tests indicate that their computational performance is comparable to that of finite element methods. Both LPC and MTM can be applied to complicated pore-space geometries with no analytic solution. We discuss the new methods in the context of diffusion propagator calculation in porous materials and model biological tissues.

A meshless method based on Point Interpolation Method (PIM) for the space fractional diffusion equation

This paper aims to develop a meshless approach based on the Point Interpolation Method (PIM) for numerical simulation of a space fractional diffusion equation. Two fully-discrete schemes for the one-dimensional space fractional diffusion equation are obtained by using the PIM and the strong-forms of the space diffusion equation. Numerical examples with different nodal distributions are studied to validate and investigate the accuracy and efficiency of the newly developed meshless approach.