Novel analytical and numerical methods for solving fractional dynamical systems

During the past three decades, the subject of fractional calculus (that is, calculus of integrals and derivatives of arbitrary order) has gained considerable popularity and importance, mainly due to its demonstrated applications in numerous diverse and widespread fields in science and engineering. For example, fractional calculus has been successfully applied to problems in system biology, physics, chemistry and biochemistry, hydrology, medicine, and finance. In many cases these new fractional-order models are more adequate than the previously used integer-order models, because fractional derivatives and integrals enable the description of the memory and hereditary properties inherent in various materials and processes that are governed by anomalous diffusion. Hence, there is a growing need to find the solution behaviour of these fractional differential equations. However, the analytic solutions of most fractional differential equations generally cannot be obtained. As a consequence, approximate and numerical techniques are playing an important role in identifying the solution behaviour of such fractional equations and exploring their applications. The main objective of this thesis is to develop new effective numerical methods and supporting analysis, based on the finite difference and finite element methods, for solving time, space and time-space fractional dynamical systems involving fractional derivatives in one and two spatial dimensions. A series of five published papers and one manuscript in preparation will be presented on the solution of the space fractional diffusion equation, space fractional advectiondispersion equation, time and space fractional diffusion equation, time and space fractional Fokker-Planck equation with a linear or non-linear source term, and fractional cable equation involving two time fractional derivatives, respectively. One important contribution of this thesis is the demonstration of how to choose different approximation techniques for different fractional derivatives. Special attention has been paid to the Riesz space fractional derivative, due to its important application in the field of groundwater flow, system biology and finance. We present three numerical methods to approximate the Riesz space fractional derivative, namely the L1/ L2-approximation method, the standard/shifted Gr¨unwald method, and the matrix transform method (MTM). The first two methods are based on the finite difference method, while the MTM allows discretisation in space using either the finite difference or finite element methods. Furthermore, we prove the equivalence of the Riesz fractional derivative and the fractional Laplacian operator under homogeneous Dirichlet boundary conditions – a result that had not previously been established. This result justifies the aforementioned use of the MTM to approximate the Riesz fractional derivative. After spatial discretisation, the time-space fractional partial differential equation is transformed into a system of fractional-in-time differential equations. We then investigate numerical methods to handle time fractional derivatives, be they Caputo type or Riemann-Liouville type. This leads to new methods utilising either finite difference strategies or the Laplace transform method for advancing the solution in time. The stability and convergence of our proposed numerical methods are also investigated. Numerical experiments are carried out in support of our theoretical analysis. We also emphasise that the numerical methods we develop are applicable for many other types of fractional partial differential equations.

The differential impact of holocaust trauma across three generations

In the current thesis, the reasons for the differential impact of Holocaust trauma on Holocaust survivors, and the differential intergenerational transmission of this trauma to survivors’ children and grandchildren were explored. A model specifically related to Holocaust trauma and its transmission was developed based on trauma, family systems and attachment theories as well as theoretical and anecdotal conjecture in the Holocaust literature. The Model of the Differential Impact of Holocaust Trauma across Three Generations was tested firstly by extensive meta-analyses of the literature pertaining to the psychological health of Holocaust survivors and their descendants and secondly via analysis of empirical study data. The meta-analyses reported in this thesis represent the first conducted with research pertaining to Holocaust survivors and grandchildren of Holocaust survivors. The meta-analysis of research conducted with children of survivors is the first to include both published and unpublished research. Meta-analytic techniques such as meta-regression and sub-set meta-analyses provided new information regarding the influence of a number of unmeasured demographic variables on the psychological health of Holocaust survivors and descendants. Based on the results of the meta-analyses it was concluded that Holocaust survivors and their children and grandchildren suffer from a statistically significantly higher level or greater severity of psychological symptoms than the general population. However it was also concluded that there is statistically significant variation in psychological health within the Holocaust survivor and descendant populations. Demographic variables which may explain a substantial amount of this variation have been largely under-assessed in the literature and so an empirical study was needed to clarify the role of demographics in determining survivor and descendant mental health. A total of 124 participants took part in the empirical study conducted for this thesis with 27 Holocaust survivors, 69 children of survivors and 28 grandchildren of survivors. A worldwide recruitment process was used to obtain these participants. Among the demographic variables assessed in the empirical study, aspects of the survivors’ Holocaust trauma (namely the exact nature of their Holocaust experiences, the extent of family bereavement and their country of origin) were found to be particularly potent predictors of not only their own psychological health but continue to be strongly influential in determining the psychological health of their descendants. Further highlighting the continuing influence of the Holocaust was the finding that number of Holocaust affected ancestors was the strongest demographic predictor of grandchild of survivor psychological health. Apart from demographic variables, the current thesis considered family environment dimensions which have been hypothesised to play a role in the transmission of the traumatic impact of the Holocaust from survivors to their descendants. Within the empirical study, parent-child attachment was found to be a key determinant in the transmission of Holocaust trauma from survivors to their children and insecure parent-child attachment continues to reverberate through the generations. In addition, survivors’ communication about the Holocaust and their Holocaust experiences to their children was found to be more influential than general communication within the family. Ten case studies (derived from the empirical study data set) are also provided; five Holocaust survivors, three children of survivors and two grandchildren of survivors. These cases add further to the picture of heterogeneity of the survivor and descendant populations in both experiences and adaptations. It is concluded that the legacy of the Holocaust continues to leave its mark on both its direct survivors and their descendants. Even two generations removed, the direct and indirect effects of the Holocaust have yet to be completely nullified. Research with Holocaust survivor families serves to highlight the differential impacts of state-based trauma and the ways in which its effects continue to be felt for generations. The revised and empirically tested Model of the Differential Impact of Holocaust Trauma across Three Generations presented at the conclusion of this thesis represents a further clarification of existing trauma theories as well as the first attempt at determining the relative importance of both cognitive, interpersonal/interfamilial interaction processes and demographic variables in post-trauma psychological health and transmission of traumatic impact.

Differential roles of the pRb and Arf/p53 pathways in murine naevus and melanoma genesis

We report on a systematic analysis of genotype-specific melanocyte (MC) UVR responses in transgenic mouse melanoma models along with tumour penetrance and comparative histopathology. pRb or p53 pathway mutations cooperated with NrasQ61K to transform MCs. We previously reported that MCs migrate from the follicular outer root sheath into the epidermis after neonatal UVR. Here, we found that Arf or p53 loss markedly diminished this response. Despite this, mice carrying these mutations developed melanoma with very early age of onset after neonatal UVR. Cdk4R24C did not affect the MC migration. Instead, independent of UVR exposure, interfollicular dermal MCs were more prevalent in Cdk4R24C mice. Subsequently, in adulthood, these mutants developed dermal MC proliferations reminiscent of superficial congenital naevi. Two types of melanoma were observed in this model. The location and growth pattern of the first was consistent with derivation from the naevi, while the second appeared to be of deep dermal origin. In animals carrying the Arf or p53 defects, no naevi were detected, with all tumours ostensibly skipping the benign precursor stage in progression.

Discrimination or Differential Involvement? A Review of the Research Exploring the Impact of Indigenous Status on Sentencing

Existing court data suggest that adult Indigenous offenders are more likely than non-Indigenous defendants to be sentenced to prison but once imprisoned generally receive shorter terms. Using findings from international and Australian multivariate statistical analyses, this paper reviews the three key hypotheses advanced as plausible explanations for these differences: 1) differential involvement, 2) negative discrimination, 3) positive discrimination. Overall, prior research shows strong support for the differential involvement thesis, some support for positive discrimination and little foundation for negative discrimination in the sentencing of Indigenous defendants. Where discrimination is found, we argue that this may be explained by the lack of a more complete set of control variables in researchers’ multivariate models.

An advanced implicit meshless approach for fractional partial differential equation in computational mechanics

Recently, the numerical modelling and simulation for fractional partial differential equations (FPDE), which have been found with widely applications in modern engineering and sciences, are attracting increased attentions. The current dominant numerical method for modelling of FPDE is the explicit Finite Difference Method (FDM), which is based on a pre-defined grid leading to inherited issues or shortcomings. This paper aims to develop an implicit meshless approach based on the radial basis functions (RBF) for numerical simulation of time fractional diffusion equations. The discrete system of equations is obtained by using the RBF meshless shape functions and the strong-forms. The stability and convergence of this meshless approach are then discussed and theoretically proven. Several numerical examples with different problem domains are used to validate and investigate accuracy and efficiency of the newly developed meshless formulation. The results obtained by the meshless formations are also compared with those obtained by FDM in terms of their accuracy and efficiency. It is concluded that the present meshless formulation is very effective for the modelling and simulation for FPDE.

Optimization problems for controlled mechanical systems : bridging the gap between theory and application

Mechanical control systems have become a part of our everyday life. Systems such as automobiles, robot manipulators, mobile robots, satellites, buildings with active vibration controllers and air conditioning systems, make life easier and safer, as well as help us explore the world we live in and exploit it’s available resources. In this chapter, we examine a specific example of a mechanical control system; the Autonomous Underwater Vehicle (AUV). Our contribution to the advancement of AUV research is in the area of guidance and control. We present innovative techniques to design and implement control strategies that consider the optimization of time and/or energy consumption. Recent advances in robotics, control theory, portable energy sources and automation increase our ability to create more intelligent robots, and allows us to conduct more explorations by use of autonomous vehicles. This facilitates access to higher risk areas, longer time underwater, and more efficient exploration as compared to human occupied vehicles. The use of underwater vehicles is expanding in every area of ocean science. Such vehicles are used by oceanographers, archaeologists, geologists, ocean engineers, and many others. These vehicles are designed to be agile, versatile and robust, and thus, their usage has gone from novelty to necessity for any ocean expedition.

A geometrical analysis of trajectory design for underwater vehicles

Designing trajectories for a submerged rigid body motivates this paper. Two approaches are addressed: the time optimal approach and the motion planning ap- proach using concatenation of kinematic motions. We focus on the structure of singular extremals and their relation to the existence of rank-one kinematic reduc- tions; thereby linking the optimization problem to the inherent geometric frame- work. Using these kinematic reductions, we provide a solution to the motion plan- ning problem in the under-actuated scenario, or equivalently, in the case of actuator failures. We finish the paper comparing a time optimal trajectory to one formed by concatenation of pure motions.

Efficient control of an autonomous underwater vehicle while accounting for thruster failure

This paper is concerned with the design and implementation of control strategies onto a test-bed vehicle with six degrees-of-freedom. We design our trajectories to be efficient in time and in power consumption. Moreover, we also consider cases when actuator failure can arise and discuss alternate control strategies in this situation. Our calculations are supplemented by experimental results.

A first extension of geometric control theory to underwater vehicles

This paper serves as a first study on the implementation of control strategies developed using a kinematic reduction onto test bed autonomous underwater vehicles (AUVs). The equations of motion are presented in the framework of differential geometry, including external dissipative forces, as a forced affine connection control system. We show that the hydrodynamic drag forces can be included in the affine connection, resulting in an affine connection control system. The definitions of kinematic reduction and decoupling vector field are thus extended from the ideal fluid scenario. Control strategies are computed using this new extension and are reformulated for implementation onto a test-bed AUV. We compare these geometrically computed controls to time and energy optimal controls for the same trajectory which are computed using a previously developed algorithm. Through this comparison we are able to validate our theoretical results based on the experiments conducted using the time and energy efficient strategies.

Design and implementation of time efficient trajectories for an underwater vehicle

This paper discusses control strategies adapted for practical implementation and efficient motion of underwater vehicles. These trajectories are piecewise constant thrust arcs with few actuator switchings. We provide the numerical algorithm which computes the time efficient trajectories parameterized by the switching times. We discuss both the theoretical analysis and experimental implementation results.