Numerical methods for the variable-order fractional advection-diffusion equation with a nonlinear source term

In this paper, we consider a variable-order fractional advection-diffusion equation with a nonlinear source term on a finite domain. Explicit and implicit Euler approximations for the equation are proposed. Stability and convergence of the methods are discussed. Moreover, we also present a fractional method of lines, a matrix transfer technique, and an extrapolation method for the equation. Some numerical examples are given, and the results demonstrate the effectiveness of theoretical analysis.

Stability and convergence of a new explicit finite-difference approximation for the variable-order nonlinear fractional diffusion equation

In this paper, we consider the variable-order nonlinear fractional diffusion equation View the MathML source where xRα(x,t) is a generalized Riesz fractional derivative of variable order View the MathML source and the nonlinear reaction term f(u,x,t) satisfies the Lipschitz condition |f(u1,x,t)-f(u2,x,t)|less-than-or-equals, slantL|u1-u2|. A new explicit finite-difference approximation is introduced. The convergence and stability of this approximation are proved. Finally, some numerical examples are provided to show that this method is computationally efficient. The proposed method and techniques are applicable to other variable-order nonlinear fractional differential equations.

Latent variable models in statistical genetics

Understanding the complexities that are involved in the genetics of multifactorial diseases is still a monumental task. In addition to environmental factors that can influence the risk of disease, there is also a number of other complicating factors. Genetic variants associated with age of disease onset may be different from those variants associated with overall risk of disease, and variants may be located in positions that are not consistent with the traditional protein coding genetic paradigm. Latent Variable Models are well suited for the analysis of genetic data. A latent variable is one that we do not directly observe, but which is believed to exist or is included for computational or analytic convenience in a model. This thesis presents a mixture of methodological developments utilising latent variables, and results from case studies in genetic epidemiology and comparative genomics. Epidemiological studies have identified a number of environmental risk factors for appendicitis, but the disease aetiology of this oft thought useless vestige remains largely a mystery. The effects of smoking on other gastrointestinal disorders are well documented, and in light of this, the thesis investigates the association between smoking and appendicitis through the use of latent variables. By utilising data from a large Australian twin study questionnaire as both cohort and case-control, evidence is found for the association between tobacco smoking and appendicitis. Twin and family studies have also found evidence for the role of heredity in the risk of appendicitis. Results from previous studies are extended here to estimate the heritability of age-at-onset and account for the eect of smoking. This thesis presents a novel approach for performing a genome-wide variance components linkage analysis on transformed residuals from a Cox regression. This method finds evidence for a dierent subset of genes responsible for variation in age at onset than those associated with overall risk of appendicitis. Motivated by increasing evidence of functional activity in regions of the genome once thought of as evolutionary graveyards, this thesis develops a generalisation to the Bayesian multiple changepoint model on aligned DNA sequences for more than two species. This sensitive technique is applied to evaluating the distributions of evolutionary rates, with the finding that they are much more complex than previously apparent. We show strong evidence for at least 9 well-resolved evolutionary rate classes in an alignment of four Drosophila species and at least 7 classes in an alignment of four mammals, including human. A pattern of enrichment and depletion of genic regions in the profiled segments suggests they are functionally significant, and most likely consist of various functional classes. Furthermore, a method of incorporating alignment characteristics representative of function such as GC content and type of mutation into the segmentation model is developed within this thesis. Evidence of fine-structured segmental variation is presented.

A two-Dimensional finite volume method for variable density flow and solute transport through saturated-unsaturated media

Extensive groundwater withdrawal has resulted in a severe seawater intrusion problem in the Gooburrum aquifers at Bundaberg, Queensland, Australia. Better management strategies can be implemented by understanding the seawater intrusion processes in those aquifers. To study the seawater intrusion process in the region, a two-dimensional density-dependent, saturated and unsaturated flow and transport computational model is used. The model consists of a coupled system of two non-linear partial differential equations. The first equation describes the flow of a variable-density fluid, and the second equation describes the transport of dissolved salt. A two-dimensional control volume finite element model is developed for simulating the seawater intrusion into the heterogeneous aquifer system at Gooburrum. The simulation results provide a realistic mechanism by which to study the convoluted transport phenomena evolving in this complex heterogeneous coastal aquifer.

Models of collective cell spreading with variable cell aspect ratio : a motivation for degenerate diffusion models

Continuum diffusion models are often used to represent the collective motion of cell populations. Most previous studies have simply used linear diffusion to represent collective cell spreading, while others found that degenerate nonlinear diffusion provides a better match to experimental cell density profiles. In the cell modeling literature there is no guidance available with regard to which approach is more appropriate for representing the spreading of cell populations. Furthermore, there is no knowledge of particular experimental measurements that can be made to distinguish between situations where these two models are appropriate. Here we provide a link between individual-based and continuum models using a multi-scale approach in which we analyze the collective motion of a population of interacting agents in a generalized lattice-based exclusion process. For round agents that occupy a single lattice site, we find that the relevant continuum description of the system is a linear diffusion equation, whereas for elongated rod-shaped agents that occupy L adjacent lattice sites we find that the relevant continuum description is connected to the porous media equation (pme). The exponent in the nonlinear diffusivity function is related to the aspect ratio of the agents. Our work provides a physical connection between modeling collective cell spreading and the use of either the linear diffusion equation or the pme to represent cell density profiles. Results suggest that when using continuum models to represent cell population spreading, we should take care to account for variations in the cell aspect ratio because different aspect ratios lead to different continuum models.

Variable structure control for power systems stabilization

The application of variable structure control (VSC) for power systems stabilization is studied in this paper. It is the application, aspects and constraints of VSC which are of particular interest. A variable structure control methodology has been proposed for power systems stabilization. The method is implemented using thyristor controlled series compensators. A three machine power system is stabilized using a switching line control for large disturbances which becomes a sliding control as the disturbance becomes smaller. The results demonstrate the effectiveness of the methodology proposed as an useful tool to suppress the oscillations in power systems.

Calibration and operational analysis of variable speed limits for high flow condition

Variable Speed Limits (VSL) is a control tool of Intelligent Transportation Systems (ITS) which can enhance traffic safety and which has the potential to contribute to traffic efficiency. This study presents the results of a calibration and operational analysis of a candidate VSL algorithm for high flow conditions on an urban motorway of Queensland, Australia. The analysis was done using a framework consisting of a microscopic simulation model combined with runtime API and a proposed efficiency index. The operational analysis includes impacts on speed-flow curve, travel time, speed deviation, fuel consumption and emission.

Numerical simulation for the variable-order Galilei invariant advection diffusion equation with a nonlinear source term

In this paper, we consider the variable-order Galilei advection diffusion equation with a nonlinear source term. A numerical scheme with first order temporal accuracy and second order spatial accuracy is developed to simulate the equation. The stability and convergence of the numerical scheme are analyzed. Besides, another numerical scheme for improving temporal accuracy is also developed. Finally, some numerical examples are given and the results demonstrate the effectiveness of theoretical analysis. Keywords: The variable-order Galilei invariant advection diffusion equation with a nonlinear source term; The variable-order Riemann–Liouville fractional partial derivative; Stability; Convergence; Numerical scheme improving temporal accuracy

Queue protection parameters’ fine-tuning for variable speed limits

Any incident on motorways potentially can be followed by secondary crashes. Rear-end crashes also could happen as a result of queue formation downstream of high speed platoons. To decrease the occurrence of secondary crashes and rear-end crashes, Variable Speed Limits (VSL) can be applied to protect queue formed downstream. This paper focuses on fine tuning the Queue Protection algorithm of VSL. Three performance indicators: activation time, deactivation time and number of false alarms are selected to optimise the Queue Protection algorithm. A calibrated microscopic traffic simulation model of Pacific Motorway in Brisbane is used for the optimisation. Performance of VSL during an incident and heavy congestion and the benefit of VSL will be presented in the paper.

Learning from episodes of degradation and recovery in variable Australian rangelands

Land-change science emphasizes the intimate linkages between the human and environmental components of land management systems. Recent theoretical developments in drylands identify a small set of key principles that can guide the understanding of these linkages. Using these principles, a detailed study of seven major degradation episodes over the past century in Australian grazed rangelands was reanalyzed to show a common set of events: (i) good climatic and economic conditions for a period, leading to local and regional social responses of increasing stocking rates, setting the preconditions for rapid environmental collapse, followed by (ii) a major drought coupled with a fall in the market making destocking financially unattractive, further exacerbating the pressure on the environment; then (iii) permanent or temporary declines in grazing productivity, depending on follow-up seasons coupled again with market and social conditions. The analysis supports recent theoretical developments but shows that the establishment of environmental knowledge that is strictly local may be insufficient on its own for sustainable management. Learning systems based in a wider community are needed that combine local knowledge, formal research, and institutional support. It also illustrates how natural variability in the state of both ecological and social systems can interact to precipitate nonequilibrial change in each other, so that planning cannot be based only on average conditions. Indeed, it is this variability in both environment and social subsystems that hinders the local learning required to prevent collapse.

Variable Speed Limits : conceptual design for Queensland practice

Variable Speed Limits (VSL) is an Intelligent Transportation Systems (ITS) control tool which can enhance traffic safety and which has the potential to contribute to traffic efficiency. Queensland's motorways experience a large volume of commuter traffic in peak periods, leading to heavy recurrent congestion and a high frequency of incidents. Consequently, Queensland's Department of Transport and Main Roads have considered deploying VSL to improve safety and efficiency. This paper identifies three types of VSL and three applicable conditions for activating VSL on for Queensland motorways: high flow, queuing and adverse weather. The design objectives and methodology for each condition are analysed, and micro-simulation results are presented to demonstrate the effectiveness of VSL.