Numerical methods for solving a two-dimensional variable-order anomalous subdiffusion equation

Anomalous subdiffusion equations have in recent years received much attention. In this paper, we consider a two-dimensional variable-order anomalous subdiffusion equation. Two numerical methods (the implicit and explicit methods) are developed to solve the equation. Their stability, convergence and solvability are investigated by the Fourier method. Moreover, the effectiveness of our theoretical analysis is demonstrated by some numerical examples. © 2011 American Mathematical Society.

Analytical and numerical solutions of the space and time fractional bloch-torrey equation

Fractional order dynamics in physics, particularly when applied to diffusion, leads to an extension of the concept of Brown-ian motion through a generalization of the Gaussian probability function to what is termed anomalous diffusion. As MRI is applied with increasing temporal and spatial resolution, the spin dynamics are being examined more closely; such examinations extend our knowledge of biological materials through a detailed analysis of relaxation time distribution and water diffusion heterogeneity. Here the dynamic models become more complex as they attempt to correlate new data with a multiplicity of tissue compartments where processes are often anisotropic. Anomalous diffusion in the human brain using fractional order calculus has been investigated. Recently, a new diffusion model was proposed by solving the Bloch-Torrey equation using fractional order calculus with respect to time and space (see R.L. Magin et al., J. Magnetic Resonance, 190 (2008) 255-270). However effective numerical methods and supporting error analyses for the fractional Bloch-Torrey equation are still limited. In this paper, the space and time fractional Bloch-Torrey equation (ST-FBTE) is considered. The time and space derivatives in the ST-FBTE are replaced by the Caputo and the sequential Riesz fractional derivatives, respectively. Firstly, we derive an analytical solution for the ST-FBTE with initial and boundary conditions on a finite domain. Secondly, we propose an implicit numerical method (INM) for the ST-FBTE, and the stability and convergence of the INM are investigated. We prove that the implicit numerical method for the ST-FBTE is unconditionally stable and convergent. Finally, we present some numerical results that support our theoretical analysis.

Embedding human expert cognition and real-time trajectory planning in autonomous UAS

This thesis presents a new approach to compute and optimize feasible three dimensional (3D) flight trajectories using aspects of Human Decision Making (HDM) strategies, for fixed wing Unmanned Aircraft (UA) operating in low altitude environments in the presence of real time planning deadlines. The underlying trajectory generation strategy involves the application of Manoeuvre Automaton (MA) theory to create sets of candidate flight manoeuvres which implicitly incorporate platform dynamic constraints. Feasible trajectories are formed through the concatenation of predefined flight manoeuvres in an optimized manner. During typical UAS operations, multiple objectives may exist, therefore the use of multi-objective optimization can potentially allow for convergence to a solution which better reflects overall mission requirements and HDM preferences. A GUI interface was developed to allow for knowledge capture from a human expert during simulated mission scenarios. The expert decision data captured is converted into value functions and corresponding criteria weightings using UTilite Additive (UTA) theory. The inclusion of preferences elicited from HDM decision data within an Automated Decision System (ADS) allows for the generation of trajectories which more closely represent the candidate HDM’s decision strategies. A novel Computationally Adaptive Trajectory Decision optimization System (CATDS) has been developed and implemented in simulation to dynamically manage, calculate and schedule system execution parameters to ensure that the trajectory solution search can generate a feasible solution, if one exists, within a given length of time. The inclusion of the CATDS potentially increases overall mission efficiency and may allow for the implementation of the system on different UAS platforms with varying onboard computational capabilities. These approaches have been demonstrated in simulation using a fixed wing UAS operating in low altitude environments with obstacles present.

CT and X-Ray analysis of sagittal curve changes following thoracoscopic anterior scoliosis surgery

Normal thoracic kyphosis Cobb angle for T5-T12 is most commonly reported as a range of 20-40º [1]. Patients with adolescent idiopathic scoliosis (AIS) exhibit a reduced thoracic kyphosis or hypokyphosis [2] accompanying the coronal and rotary distortion components. As a result, surgical restoration of the thoracic kyphosis while maintaining lumbar lordosis and overall sagittal balance is a critical aspect of achieving good clinical outcomes in AIS patients. Previous studies report an increase in thoracic kyphosis after anterior surgical approaches [3] and a flattening of sagittal contours following posterior approaches [4]. Difficulties with measuring sagittal parameters on radiographs are avoided with reformatted sagittal CT reconstructions due to the superior endplate clarity afforded by this imaging modality and are the subject of analysis in this study.

The persistence of phase-separation in LiFePO4 with two-dimensional Li+ transport : the Cahn-Hilliard-reaction equation and the role of defects

We examine the solution of the two-dimensional Cahn-Hilliard-reaction (CHR) equation in the xy plane as a model of Li+ intercalation into LiFePO4 material. We validate our numerical solution against the solution of the depth-averaged equation, which has been used to model intercalation in the limit of highly orthotropic diffusivity and gradient penalty tensors. We then examine the phase-change behaviour in the full CHR system as these parameters become more isotropic, and find that as the Li+ diffusivity is increased in the x direction, phase separation persists at high currents, even in small crystals with averaged coherency strain included. The resulting voltage curves decrease monotonically, which has previously been considered a hallmark of crystals that fill homogeneously.

Extracting deck trend to plan descent trajectory for safe landing of UAVs

This paper outlines a feasible scheme to extract deck trend when a rotary-wing unmanned aerial vehicle (RUAV)approaches an oscillating deck. An extended Kalman filter (EKF) is de- veloped to fuse measurements from multiple sensors for effective estimation of the unknown deck heave motion. Also, a recursive Prony Analysis (PA) procedure is proposed to implement online curve-fitting of the estimated heave mo- tion. The proposed PA constructs an appropriate model with parameters identified using the forgetting factor recursive least square (FFRLS)method. The deck trend is then extracted by separating dominant modes. Performance of the proposed procedure is evaluated using real ship motion data, and simulation results justify the suitability of the proposed method into safe landing of RUAVs operating in a maritime environment.

Family violence or woman abuse? Putting gender back into the Canadian research equation

Research on violence against women has been among the most scrutinized areas in social science. From the beginning, efforts to empirically document the prevalence, incidence, and characteristics of violence against women have been hotly debated (DeKeseredy, 2011; Dragiewicz & DeKeseredy, forthcoming; Minaker & Snider, 2006). Objections that violence against women was rare have given way to acknowledgement that it is more common than once thought. Research on the outcomes of woman abuse has documented the serious ramifications of this type of violence for individual victims and the broader community. However, violence against women was not simply “discovered” by scholars in the 1960s, leading to a progressive growth of the literature. Knowledge production around violence against women has been fiercely contested, and feminist insights in particular have always been met with backlash(Gotell, 2007; Minkaer & Snider, 2006; Randall, 1989; Sinclair, 2003)...

Pore formation during dehydration of polycrystalline gypsum observed and quantified in a time-series synchrotron radiation based X-ray micro-tomography experiment

We conducted an in-situ X-ray micro-computed tomography heating experiment at the Advanced Photon Source (USA) to dehydrate an unconfined 2.3 mm diameter cylinder of Volterra Gypsum. We used a purpose-built X-ray transparent furnace to heat the sample to 388 K for a total of 310 min to acquire a three-dimensional time-series tomography dataset comprising nine time steps. The voxel size of 2.2 μm3 proved sufficient to pinpoint reaction initiation and the organization of drainage architecture in space and time. We observed that dehydration commences across a narrow front, which propagates from the margins to the centre of the sample in more than four hours. The advance of this front can be fitted with a square-root function, implying that the initiation of the reaction in the sample can be described as a diffusion process. Novel parallelized computer codes allow quantifying the geometry of the porosity and the drainage architecture from the very large tomographic datasets (20483 voxels) in unprecedented detail. We determined position, volume, shape and orientation of each resolvable pore and tracked these properties over the duration of the experiment. We found that the pore-size distribution follows a power law. Pores tend to be anisotropic but rarely crack-shaped and have a preferred orientation, likely controlled by a pre-existing fabric in the sample. With on-going dehydration, pores coalesce into a single interconnected pore cluster that is connected to the surface of the sample cylinder and provides an effective drainage pathway. Our observations can be summarized in a model in which gypsum is stabilized by thermal expansion stresses and locally increased pore fluid pressures until the dehydration front approaches to within about 100 μm. Then, the internal stresses are released and dehydration happens efficiently, resulting in new pore space. Pressure release, the production of pores and the advance of the front are coupled in a feedback loop.

Stability and convergence of a finite volume method for the space fractional advection-dispersion equation

We consider the space fractional advection–dispersion equation, which is obtained from the classical advection–diffusion equation by replacing the spatial derivatives with a generalised derivative of fractional order. We derive a finite volume method that utilises fractionally-shifted Grünwald formulae for the discretisation of the fractional derivative, to numerically solve the equation on a finite domain with homogeneous Dirichlet boundary conditions. We prove that the method is stable and convergent when coupled with an implicit timestepping strategy. Results of numerical experiments are presented that support the theoretical analysis.

Reliable and valid survey design for structural equation modelling

My quantitative study asks how Chinese Australians’ “Chineseness” and their various resources influence their Chinese language proficiency, using online survey and snowball sampling. ‘Operationalization’ is a challenging process which ensures that the survey design talks back to the informing theory and forwards to the analysis model. It requires the attention to two core methodological concerns, namely ‘validity’ and ‘reliability’. Construction of a high-quality questionnaire is critical to the achievement of valid and reliable operationalization. A series of strategies were chosen to ensure the quality of the questions, and thus the eventual data. These strategies enable the use of structural equation modelling to examine how well the data fits the theoretical framework, which was constructed in light of Bourdieu’s theory of habitus, capital and field.