A finite volume method for solving the two-sided time-space fractional advection-dispersion equation

The field of fractional differential equations provides a means for modelling transport processes within complex media which are governed by anomalous transport. Indeed, the application to anomalous transport has been a significant driving force behind the rapid growth and expansion of the literature in the field of fractional calculus. In this paper, we present a finite volume method to solve the time-space two-sided fractional advection dispersion equation on a one-dimensional domain. Such an equation allows modelling different flow regime impacts from either side. The finite volume formulation provides a natural way to handle fractional advection-dispersion equations written in conservative form. The novel spatial discretisation employs fractionally-shifted Gr¨unwald formulas to discretise the Riemann-Liouville fractional derivatives at control volume faces in terms of function values at the nodes, while the L1-algorithm is used to discretise the Caputo time fractional derivative. Results of numerical experiments are presented to demonstrate the effectiveness of the approach.

An exploratory study of the potential of resurfacing articular cartilage with synthetic phospholipids

This thesis is aimed at further understanding the uppermost lipid-filled membranous layer (i.e. surface amorphous layer (SAL)) of articular cartilage and to develop a scientific framework for re-introducing lipids onto the surface of lipid-depleted articular cartilage (i.e. "resurfacing"). The outcome will potentially contribute to knowledge that will facilitate the repair of the articular surface of cartilage where degradation is limited to the loss of the lipids of the SAL only. The surface amorphous layer is of utmost importance to the effective load-spreading, lubrication, and semipermeability (which controls its fluid management, nutrient transport and waste removal) of articular cartilage in the mammalian joints. However, because this uppermost layer of cartilage is often in contact during physiological function, it is prone to wear and tear, and thus, is the site for damage initiation that can lead to the early stages of joint condition like osteoarthritis, and related conditions that cause pain and discomfort leading to low quality of life in patients. It is therefore imperative to conduct a study which offers insight into remedying this problem. It is hypothesized that restoration (resurfacing) of the surface amorphous layer can be achieved by re-introducing synthetic surface-active phospholipids (SAPL) into the joint space. This hypothesis was tested in this thesis by exposing cartilage samples whose surface lipids had been depleted to individual and mixtures of synthetic saturated and unsaturated phospholipids. The surfaces of normal, delipidized, and relipidized samples of cartilage were characterized for their structural integrity and functionality using atomic force microscope (AFM), confocal microscope (COFM), Raman spectroscopy, magnetic resonance imaging (MRI) with image processing in the MATLAB® environment and mechanical loading experiments. The results from AFM imaging, confocal microscopy, and Raman spectroscopy revealed a successful deposition of new surface layer on delipidized cartilage when incubated in synthetic phospholipids. The relipidization resulted in a significant improvement in the surface nanostructure of the artificially degraded cartilage, with the complete SAPL mixture providing better outcomes in comparison to those created with the single SAPL components (palmitoyl-oleoyl-phosphatidylcholine, POPC and dipalmitoyl-phosphatidylcholine, DPPC). MRI analysis revealed that the surface created with the complete mixture of synthetic lipids was capable of providing semipermeability to the surface layer of the treated cartilage samples relative to the normal intact surface. Furthermore, deformation energy analysis revealed that the treated samples were capable of delivering the elastic properties required for load bearing and recovery of the tissue relative to the normal intact samples, with this capability closer between the normal and the samples incubated in the complete lipid mixture. In conclusion, this thesis has established that it is possible to deposit/create a potentially viable layer on the surface of cartilage following degradation/lipid loss through incubation in synthetic lipid solutions. However, further studies will be required to advance the ideas developed in this thesis, for the development of synthetic lipid-based injections/drugs for treatment of osteoarthritis and other related joint conditions.

A finite volume method for solving the two-sided time-space fractional advection-dispersion equation

We present a finite volume method to solve the time-space two-sided fractional advection-dispersion equation on a one-dimensional domain. The spatial discretisation employs fractionally-shifted Grünwald formulas to discretise the Riemann-Liouville fractional derivatives at control volume faces in terms of function values at the nodes. We demonstrate how the finite volume formulation provides a natural, convenient and accurate means of discretising this equation in conservative form, compared to using a conventional finite difference approach. Results of numerical experiments are presented to demonstrate the effectiveness of the approach.

A comparison of finite difference and finite volume methods for solving the space-fractional advection-dispersion equation with variable coefficients

Transport processes within heterogeneous media may exhibit non- classical diffusion or dispersion which is not adequately described by the classical theory of Brownian motion and Fick’s law. We consider a space-fractional advection-dispersion equation based on a fractional Fick’s law. Zhang et al. [Water Resources Research, 43(5)(2007)] considered such an equation with variable coefficients, which they dis- cretised using the finite difference method proposed by Meerschaert and Tadjeran [Journal of Computational and Applied Mathematics, 172(1):65-77 (2004)]. For this method the presence of variable coef- ficients necessitates applying the product rule before discretising the Riemann–Liouville fractional derivatives using standard and shifted Gru ̈nwald formulas, depending on the fractional order. As an alternative, we propose using a finite volume method that deals directly with the equation in conservative form. Fractionally-shifted Gru ̈nwald formulas are used to discretise the Riemann–Liouville fractional derivatives at control volume faces, eliminating the need for product rule expansions. We compare the two methods for several case studies, highlighting the convenience of the finite volume approach.

A fast semi-implicit difference method for a nonlinear two-sided space-fractional diffusion equation with variable diffusivity coefficients

In this paper, we derive a new nonlinear two-sided space-fractional diffusion equation with variable coefficients from the fractional Fick’s law. A semi-implicit difference method (SIDM) for this equation is proposed. The stability and convergence of the SIDM are discussed. For the implementation, we develop a fast accurate iterative method for the SIDM by decomposing the dense coefficient matrix into a combination of Toeplitz-like matrices. This fast iterative method significantly reduces the storage requirement of O(n2)O(n2) and computational cost of O(n3)O(n3) down to n and O(nlogn)O(nlogn), where n is the number of grid points. The method retains the same accuracy as the underlying SIDM solved with Gaussian elimination. Finally, some numerical results are shown to verify the accuracy and efficiency of the new method.

Engaging Students in the Implementation of Criterion Referenced Assessment in First Year Law

The Queensland University of Technology (QUT) University Academic Board approved a new QUT Assessment Policy in September 2003, which requires a criterion-referenced approach as opposed to a norm-referenced approach to assessment across the university(QUT,MOPP,2003). In 2004, the QUT Law School embarked upon a process of awareness raising about criterion-referenced assessment amongst staff and from 2004 – 2005 staggered the implementation of criterion-referenced assessment in all first year core undergraduate law units. This paper will briefly discuss the benefits and potential pitfalls of criterion referenced assessment and the context for implementing it in the first year law program, report on student’s feedback on the introduction of criterion referenced assessment and the strategies adopted in 2005 to engage students more fully in criterion referenced assessment processes to enhance their learning outcomes.

Increasing relevance and interest : teaching environmental and planning law to built environment students

A new approach was taken to delivering a challenging "stewarship of land" unit to over 350 predominantly first year built environment students stewardship. The new approach involved incorporating environmental and planning law into the syllabus, exposing students to a wide range of statutes, selecting legal cases according to a et of criteria and revisiting the material using different modes of delivery and teaching resources. To evaluate the effectiveness of the new approach, the students were surveyed to elicit their learning experience and preferences. The survey found that most students perceived learning about environmental and planning law, including legal cases, worthwhile.----- Areas identified by the surcey for improvement included the perception by some students that: environmenatl and planning law is irrelevant to their discipline and future caree; studying law is dull and sometimes daunting; and the prescribed reading could be omitted.----- To address student perceptions, it is proposed to reorder the topics commencing with local, charismatic topics, while explanding international content and cases, to enlarge and enhance the repertoire of video clips to include sites of legal cawses and development projects, and to reformat the online weekly quizzes to promote reading of primary material.----- Overall, the approach to teaching environmental and planning law to built environment students, including the criteria for selecting legal cases, described in this paper, was found to be effective.

Contemporary issues in Australian family law: Do we need a more unified and interventionist judicial model

Recent decisions of the Family Court of Australian reflect concerns over the adversarial nature of the legal process. The processes and procedures of the judicial system militate against a detailed examination of the issues and rights of the parties in dispute. The limitations of the family law framework are particularly demonstrated in disputes over the custody of children where the Court has tended to neglect the rights and interests of the primary carer. An alternative "unified family court" framework will be examined in which the Court pursues a more active and interventionist approach in the determination of family law disputes.