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.

Addressing accuracy and precision issues in iTRAQ quantitation

iTRAQ (isobaric tags for relative or absolute quantitation) is a mass spectrometry technology that allows quantitative comparison of protein abundance by measuring peak intensities of reporter ions released from iTRAQ-tagged peptides by fragmentation during MS/MS. However, current data analysis techniques for iTRAQ struggle to report reliable relative protein abundance estimates and suffer with problems of precision and accuracy. The precision of the data is affected by variance heterogeneity: low signal data have higher relative variability; however, low abundance peptides dominate data sets. Accuracy is compromised as ratios are compressed toward 1, leading to underestimation of the ratio. This study investigated both issues and proposed a methodology that combines the peptide measurements to give a robust protein estimate even when the data for the protein are sparse or at low intensity. Our data indicated that ratio compression arises from contamination during precursor ion selection, which occurs at a consistent proportion within an experiment and thus results in a linear relationship between expected and observed ratios. We proposed that a correction factor can be calculated from spiked proteins at known ratios. Then we demonstrated that variance heterogeneity is present in iTRAQ data sets irrespective of the analytical packages, LC-MS/MS instrumentation, and iTRAQ labeling kit (4-plex or 8-plex) used. We proposed using an additive-multiplicative error model for peak intensities in MS/MS quantitation and demonstrated that a variance-stabilizing normalization is able to address the error structure and stabilize the variance across the entire intensity range. The resulting uniform variance structure simplifies the downstream analysis. Heterogeneity of variance consistent with an additive-multiplicative model has been reported in other MS-based quantitation including fields outside of proteomics; consequently the variance-stabilizing normalization methodology has the potential to increase the capabilities of MS in quantitation across diverse areas of biology and chemistry.

Neurological consequences of diabetic ketoacidosis at initial presentation of type 1 diabetes in a prospective cohort study of children

OBJECTIVE To investigate the impact of new-onset diabetic ketoacidosis (DKA) during child- hood on brain morphology and function. RESEARCH DESIGN AND METHODS Patients aged 6–18 years with and without DKA at diagnosis were studied at four time points: <48 h, 5 days, 28 days, and 6 months postdiagnosis. Patients under- went magnetic resonance imaging (MRI) and spectroscopy with cognitive assess- ment at each time point. Relationships between clinical characteristics at presentation and MRI and neurologic outcomes were examined using multiple linear regression, repeated-measures, and ANCOVA analyses. RESULTS Thirty-six DKA and 59 non-DKA patients were recruited between 2004 and 2009. With DKA, cerebral white matter showed the greatest alterations with increased total white matter volume and higher mean diffusivity in the frontal, temporal, and parietal white matter. Total white matter volume decreased over the first 6 months. For gray matter in DKA patients, total volume was lower at baseline and increased over 6 months. Lower levels of N-acetylaspartate were noted at base- line in the frontal gray matter and basal ganglia. Mental state scores were lower at baseline and at 5 days. Of note, although changes in total and regional brain volumes over the first 5 days resolved, they were associated with poorer delayed memory recall and poorer sustained and divided attention at 6 months. Age at time of presentation and pH level were predictors of neuroimaging and functional outcomes. CONCLUSIONS DKA at type 1 diabetes diagnosis results in morphologic and functional brain changes. These changes are associated with adverse neurocognitive outcomes in the medium term.

The continuing decline of science and mathematics enrolments in Australian high schools

Is there a crisis in Australian science and mathematics education? Declining enrolments in upper secondary Science and Mathematics courses have gained much attention from the media, politicians and high-profile scientists over the last few years, yet there is no consensus amongst stakeholders about either the nature or the magnitude of the changes. We have collected raw enrolment data from the education departments of each of the Australian states and territories from 1992 to 2012 and analysed the trends for Biology, Chemistry, Physics, two composite subject groups (Earth Sciences and Multidisciplinary Sciences), as well as entry, intermediate and advanced Mathematics. The results of these analyses are discussed in terms of participation rates, raw enrolments and gender balance. We have found that the total number of students in Year 12 increased by around 16% from 1992 to 2012 while the participation rates for most Science and Mathematics subjects, as a proportion of the total Year 12 cohort, fell (Biology (-10%), Chemistry (-5%), Physics (-7%), Multidisciplinary Science (-5%), intermediate Mathematics (-11%), advanced Mathematics (-7%) in the same period. There were increased participation rates in Earth Sciences (+0.3%) and entry Mathematics (+11%). In each case the greatest rates of change occurred prior to 2001 and have been slower and steadier since. We propose that the broadening of curriculum offerings, further driven by students' self-perception of ability and perceptions of subject difficulty and usefulness, are the most likely cause of the changes in participation. While these continuing declines may not amount to a crisis, there is undoubtedly serious cause for concern.