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.

Regulating IVF and pre-implantation tissue-typing for the creation of "saviour siblings" : a harm analysis

Scientific discoveries, developments in medicine and health issues are the constant focus of media attention and the principles surrounding the creation of so called ‘saviour siblings’ are of no exception. The development in the field of reproductive techniques has provided the ability to genetically analyse embryos created in the laboratory to enable parents to implant selected embryos to create a tissue-matched child who may be able to cure an existing sick child. The research undertaken in this thesis examines the regulatory frameworks overseeing the delivery of assisted reproductive technologies (ART) in Australia and the United Kingdom and considers how those frameworks impact on the accessibility of in vitro fertilisation (IVF) procedures for the creation of ‘saviour siblings’. In some jurisdictions, the accessibility of such techniques is limited by statutory requirements. The limitations and restrictions imposed by the state in relation to the technology are analysed in order to establish whether such restrictions are justified. The analysis is conducted on the basis of a harm framework. The framework seeks to establish whether those affected by the use of the technology (including the child who will be created) are harmed. In order to undertake such evaluation, the concept of harm is considered under the scope of John Stuart Mill’s liberal theory and the Harm Principle is used as a normative tool to judge whether the level of harm that may result, justifies state intervention or restriction with the reproductive decision-making of parents in this context. The harm analysis conducted in this thesis seeks to determine an appropriate regulatory response in relation to the use of pre-implantation tissue-typing for the creation of ‘saviour siblings’. The proposals outlined in the last part of this thesis seek to address the concern that harm may result from the practice of pre-implantation tissue-typing. The current regulatory frameworks in place are also analysed on the basis of the harm framework established in this thesis. The material referred to in this thesis reflects the law and policy in place in Australia and the UK at the time the thesis was submitted for examination (December 2009).

Systematic variations in polymer gel dosimeter calibration due to container influence and deviations from water equivalence

There are a number of gel dosimeter calibration methods in contemporary usage. The present study is a detailed Monte Carlo investigation into the accuracy of several calibration techniques. Results show that for most arrangements the dose to gel accurately reflects the dose to water, with the most accurate method involving the use of a large diameter flask of gel into which multiple small fields of varying dose are directed. The least accurate method was found to be that of a long test tube in a water phantom, coaxial with the beam. The large flask method is also the most straightforward and least likely to introduce errors during setup, though, to its detriment, the volume of gel required is much more than other methods.

VO2max in overweight and obese adults

Purpose: The purpose of this study was to determine whether adiposity affects the attainment of VO2max. Methods: Sixty-seven male and 68 female overweight (body mass index (BMI) = 25-29.9 kg·m-2) and obese (BMI >= 30 kg·m-2) participants undertook a graded treadmill test to volitional exhaustion (phase 1) followed by a verification test (phase 2) to determine the proportion who could achieve a plateau in VO2 and other "maximal" markers (RER, lactate, HR, RPE). Results: At the end of phase 1, 46% of the participants reached a plateau in VO2, 83% increased HR to within 11 beats of age-predicted maximum, 89% reached an RER of >=1.15, 70% reached a blood lactate concentration of >=8 mmol·L-1, and 74% reached an RPE of >=18. No significant differences between genders and between BMI groups were found with the exception of blood lactate concentration (males = 84% vs females = 56%, P < 0.05). Neither gender nor fatness predicted the number of other markers attained, and attainment of other markers did not differentiate whether a VO2 plateau was achieved. The verification test (phase 2) revealed that an additional 52 individuals (39%) who did not exhibit a plateau in V·O2 in phase 1 had no further increase in VO2 in phase 2 despite an increase in workload. Conclusions: These findings indicate that the absence of a plateau in VO2 alone is not indicative of a failure to reach a true maximal VO2 and that individuals with excessive body fat are no less likely than "normal-weight" individuals to exhibit a plateau in VO2 provided that the protocol is appropriate and encouragement to exercise to maximal exertion is provided.

An overview of physical growth and maturation

An understanding of physical growth and maturation is relevant to many disciplines, including exercise and sport science, anthropology, human biology, medicine, psychology and education. Growth and maturation is governed by a complex interaction between genetic and environmental factors. There is increasing evidence that physical activity plays an important role in normal growth, development, health and well-being of children and youth, however, caution is required in the activity setting so that growth and maturation is not jeopardised. To appreciate the impact of physical activity and/or exercise on growth and maturation, a thorough understanding of the general principles of auxology is useful. Following an introduction to terminology, an overview of physical growth and development is provided in the context of morphological changes. Detailed information is provided regarding individual variability in growth and development along with sexual dimorphism. A small degree of sexual dimorphism exists at birth however striking differences develop during the pubertal years. Sexual dimorphism in body composition is largely regulated by endocrine factors with critical roles played by growth hormone and gonadal steroids.

Regular exercise participation mediates the affective response to acute bouts of vigorous exercise

Physical inactivity is a leading factor associated with cardiovascular disease and a major contributor to the global burden of disease in developed countries. Subjective mood states associated with acute exercise are likely to influence future exercise adherence and warrant further investigation. The present study examined the effects of a single bout of vigorous exercise on mood and anxiety between individuals with substantially different exercise participation histories. Mood and anxiety were assessed one day before an exercise test (baseline), 5 minutes before (pre-test) and again 10 and 25 minutes post-exercise. Participants were 31 university students (16 males, 15 females; Age M = 20), with 16 participants reporting a history of regular exercise with the remaining 15 reporting to not exercise regularly. Each participant completed an incremental exercise test on a Monark cycle ergometer to volitional exhaustion. Regular exercisers reported significant post-exercise improvements in mood and reductions in state anxiety. By contrast, non-regular exercisers reported an initial decline in post-exercise mood and increased anxiety, followed by an improvement in mood and reduction in anxiety back to pre-exercise levels. Our findings suggest that previous exercise participation mediates affective responses to acute bouts of vigorous exercise. We suggest that to maximise positive mood changes following exercise, practitioners should carefully consider the individual’s exercise participation history before prescribing new regimes.

What the experts think : fast bowling expertise acquisition and talent development

For applied sport scientists charged with developing talented performers an essential requirement is to identify components contributing to the development and maintenance of expertise. Previous qualitative analysis has revealed several psychological (e.g., mental focus, goal-setting and selfevaluation), socio-cultural (e.g. community and family support, cultural influence), physical (e.g., strength, height) and environmental (e.g., access to facilities and climate) constraints on successful Olympian development (Abbott et al., 2005). Open-ended interviews with expert athletes and/or expert coaches have been used to reveal competencies of elite performers to derive factors associated with success (Durand-Bush et al., 2002). However, the influence of these factors is likely to be sport-specific due to different task constraints and the changing nature of the performer-environment relationship through practice, coaching and competing (Vaeyens et al., 2008). So far, only one study on expertise acquisition in cricket has been undertaken. Weissensteiner, et al. (2009) found that development of expertise in cricket batting in Australia may be facilitated by early unstructured play (i.e. ‘backyard cricket’), a wide range of sport experience during development, and early exposure to playing with seniors.

Quantifying variability within technique and performance in elite fast bowlers : is technical variability dysfunctional or functional?

In fast bowling, cricketers are expected to produce a range of delivery lines and lengths while maximising ball speed. From a coaching perspective, technique consistency has been typically associated with superior performance in these areas. However, although bowlers are required to bowl consistently, at the elite level they must also be able to vary line, length and speed to adapt to opposition batters’ strengths and weaknesses. The relationship between technique and performance variability (and consistency) has not been investigated in previous fast bowling research. Consequently, the aim of this study was to quantify both technique (bowling action and coordination) and performance variability in elite fast bowlers from Australian Junior and National Pace Squads. Technique variability was analysed to investigate whether it could be classified as functional or dysfunctional in relation to speed and accuracy.

Replication of the Frank-Starling response in a Mock Circulation loop

Mock circulation loops (MCLs) are used to evaluate cardiovascular devices prior to in-vivo trials; however they lack the vital autoregulatory responses that occur in humans. This study aimed to develop and implement a left and right ventricular Frank-Starling response in a MCL. A proportional controller based on ventricular end diastolic volume was used to control the driving pressure of the MCL’s pneumatically operated ventricles. Ventricular pressure-volume loops and end systolic pressure-volume relationships were produced for a variety of healthy and pathological conditions and compared with human data to validate the simulated Frank-Starling response. The non-linear Frank-Starling response produced in this study successfully altered left and right ventricular contractility with changing preload and was validated with previously reported data. This improvement to an already detailed MCL has resulted in a test rig capable of further refining cardiovascular devices and reducing the number of in-vivo trials.