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.

Analytical and numerical solutions for the time and space-symmetric fractional diffusion equation

We consider a time and space-symmetric fractional diffusion equation (TSS-FDE) under homogeneous Dirichlet conditions and homogeneous Neumann conditions. The TSS-FDE is obtained from the standard diffusion equation by replacing the first-order time derivative by a Caputo fractional derivative, and the second order space derivative by a symmetric fractional derivative. First, a method of separating variables expresses the analytical solution of the TSS-FDE in terms of the Mittag--Leffler function. Second, we propose two numerical methods to approximate the Caputo time fractional derivative: the finite difference method; and the Laplace transform method. The symmetric space fractional derivative is approximated using the matrix transform method. Finally, numerical results demonstrate the effectiveness of the numerical methods and to confirm the theoretical claims.

Analytical and numerical solutions for the time and space-symmetric fractional diffusion equation

We consider a time and space-symmetric fractional diffusion equation (TSS-FDE) under homogeneous Dirichlet conditions and homogeneous Neumann conditions. The TSS-FDE is obtained from the standard diffusion equation by replacing the first-order time derivative by the Caputo fractional derivative and the second order space derivative by the symmetric fractional derivative. Firstly, a method of separating variables is used to express the analytical solution of the tss-fde in terms of the Mittag–Leffler function. Secondly, we propose two numerical methods to approximate the Caputo time fractional derivative, namely, the finite difference method and the Laplace transform method. The symmetric space fractional derivative is approximated using the matrix transform method. Finally, numerical results are presented to demonstrate the effectiveness of the numerical methods and to confirm the theoretical claims.

A new method to quantify the application thickness of sunscreen on skin

Proper application of sunscreen is essential as an effective public health strategy for skin cancer prevention. Insufficient application is common among sunbathers, results in decreased sun protection and may therefore lead to increased UV damage of the skin. However, no objective measure of sunscreen application thickness (SAT) is currently available for field-based use. We present a method to detect SAT on human skin for determining the amount of sunscreen applied and thus enabling comparisons to manufacturer recommendations. Using a skin swabbing method and subsequent spectrophotometric analysis, we were able to determine SAT on human skin. A swabbing method was used to derive SAT on skin (in mg sunscreen per cm2 of skin area) through the concentration–absorption relationship of sunscreen determined in laboratory experiments. Analysis differentiated SATs between 0.25 and 4 mg cm−2 and showed a small but significant decrease in concentration over time postapplication. A field study was performed, in which the heterogeneity of sunscreen application could be investigated. The proposed method is a low cost, noninvasive method for the determination of SAT on skin and it can be used as a valid tool in field- and population-based studies.

Semantic oscillations : encoding context and structure in complex valued holographic vectors

In computational linguistics, information retrieval and applied cognition, words and concepts are often represented as vectors in high dimensional spaces computed from a corpus of text. These high dimensional spaces are often referred to as Semantic Spaces. We describe a novel and efficient approach to computing these semantic spaces via the use of complex valued vector representations. We report on the practical implementation of the proposed method and some associated experiments. We also briefly discuss how the proposed system relates to previous theoretical work in Information Retrieval and Quantum Mechanics and how the notions of probability, logic and geometry are integrated within a single Hilbert space representation. In this sense the proposed system has more general application and gives rise to a variety of opportunities for future research.

The ghrelin receptor isoforms (GHS-R1a and GHS-R1b) and GPR39 : an investigation into receptor dimerisation

Prostate cancer is the second most common cause of cancer-related deaths in Western males. Current diagnostic, prognostic and treatment approaches are not ideal and advanced metastatic prostate cancer is incurable. There is an urgent need for improved adjunctive therapies and markers for this disease. GPCRs are likely to play a significant role in the initiation and progression of prostate cancer. Over the last decade, it has emerged that G protein coupled receptors (GPCRs) are likely to function as homodimers and heterodimers. Heterodimerisation between GPCRs can result in the formation of novel pharmacological receptors with altered functional outcomes, and a number of GPCR heterodimers have been implicated in the pathogenesis of human disease. Importantly, novel GPCR heterodimers represent potential new targets for the development of more specific therapeutic drugs. Ghrelin is a 28 amino acid peptide hormone which has a unique n-octanoic acid post-translational modification. Ghrelin has a number of important physiological roles, including roles in appetite regulation and the stimulation of growth hormone release. The ghrelin receptor is the growth hormone secretagogue receptor type 1a, GHS-R1a, a seven transmembrane domain GPCR, and GHS-R1b is a C-terminally truncated isoform of the ghrelin receptor, consisting of five transmembrane domains. Growing evidence suggests that ghrelin and the ghrelin receptor isoforms, GHS-R1a and GHS-R1b, may have a role in the progression of a number of cancers, including prostate cancer. Previous studies by our research group have shown that the truncated ghrelin receptor isoform, GHS-R1b, is not expressed in normal prostate, however, it is expressed in prostate cancer. The altered expression of this truncated isoform may reflect a difference between a normal and cancerous state. A number of mutant GPCRs have been shown to regulate the function of their corresponding wild-type receptors. Therefore, we investigated the potential role of interactions between GHS-R1a and GHS-R1b, which are co-expressed in prostate cancer and aimed to investigate the function of this potentially new pharmacological receptor. In 2005, obestatin, a 23 amino acid C-terminally amidated peptide derived from preproghrelin was identified and was described as opposing the stimulating effects of ghrelin on appetite and food intake. GPR39, an orphan GPCR which is closely related to the ghrelin receptor, was identified as the endogenous receptor for obestatin. Recently, however, the ability of obestatin to oppose the effects of ghrelin on appetite and food intake has been questioned, and furthermore, it appears that GPR39 may in fact not be the obestatin receptor. The role of GPR39 in the prostate is of interest, however, as it is a zinc receptor. Zinc has a unique role in the biology of the prostate, where it is normally accumulated at high levels, and zinc accumulation is altered in the development of prostate malignancy. Ghrelin and zinc have important roles in prostate cancer and dimerisation of their receptors may have novel roles in malignant prostate cells. The aim of the current study, therefore, was to demonstrate the formation of GHS-R1a/GHS-R1b and GHS-R1a/GPR39 heterodimers and to investigate potential functions of these heterodimers in prostate cancer cell lines. To demonstrate dimerisation we first employed a classical co-immunoprecipitation technique. Using cells co-overexpressing FLAG- and Myc- tagged GHS-R1a, GHS-R1b and GPR39, we were able to co-immunoprecipitate these receptors. Significantly, however, the receptors formed high molecular weight aggregates. A number of questions have been raised over the propensity of GPCRs to aggregate during co-immunoprecipitation as a result of their hydrophobic nature and this may be misinterpreted as receptor dimerisation. As we observed significant receptor aggregation in this study, we used additional methods to confirm the specificity of these putative GPCR interactions. We used two different resonance energy transfer (RET) methods; bioluminescence resonance energy transfer (BRET) and fluorescence resonance energy transfer (FRET), to investigate interactions between the ghrelin receptor isoforms and GPR39. RET is the transfer of energy from a donor fluorophore to an acceptor fluorophore when they are in close proximity, and RET methods are, therefore, applicable to the observation of specific protein-protein interactions. Extensive studies using the second generation bioluminescence resonance energy transfer (BRET2) technology were performed, however, a number of technical limitations were observed. The substrate used during BRET2 studies, coelenterazine 400a, has a low quantum yield and rapid signal decay. This study highlighted the requirement for the expression of donor and acceptor tagged receptors at high levels so that a BRET ratio can be determined. After performing a number of BRET2 experimental controls, our BRET2 data did not fit the predicted results for a specific interaction between these receptors. The interactions that we observed may in fact represent ‘bystander BRET’ resulting from high levels of expression, forcing the donor and acceptor into close proximity. Our FRET studies employed two different FRET techniques, acceptor photobleaching FRET and sensitised emission FRET measured by flow cytometry. We were unable to observe any significant FRET, or FRET values that were likely to result from specific receptor dimerisation between GHS-R1a, GHS-R1b and GPR39. While we were unable to conclusively demonstrate direct dimerisation between GHS-R1a, GHS-R1b and GPR39 using several methods, our findings do not exclude the possibility that these receptors interact. We aimed to investigate if co-expression of combinations of these receptors had functional effects in prostate cancers cells. It has previously been demonstrated that ghrelin stimulates cell proliferation in prostate cancer cell lines, through ERK1/2 activation, and GPR39 can stimulate ERK1/2 signalling in response to zinc treatments. Additionally, both GHS-R1a and GPR39 display a high level of constitutive signalling and these constitutively active receptors can attenuate apoptosis when overexpressed individually in some cell types. We, therefore, investigated ERK1/2 and AKT signalling and cell survival in prostate cancer the potential modulation of these functions by dimerisation between GHS-R1a, GHS-R1b and GPR39. Expression of these receptors in the PC-3 prostate cancer cell line, either alone or in combination, did not alter constitutive ERK1/2 or AKT signalling, basal apoptosis or tunicamycin-stimulated apoptosis, compared to controls. In summary, the potential interactions between the ghrelin receptor isoforms, GHS-R1a and GHS-R1b, and the related zinc receptor, GPR39, and the potential for functional outcomes in prostate cancer were investigated using a number of independent methods. We did not definitively demonstrate the formation of these dimers using a number of state of the art methods to directly demonstrate receptor-receptor interactions. We investigated a number of potential functions of GPR39 and GHS-R1a in the prostate and did not observe altered function in response to co-expression of these receptors. The technical questions raised by this study highlight the requirement for the application of extensive controls when using current methods for the demonstration of GPCR dimerisation. Similar findings in this field reflect the current controversy surrounding the investigation of GPCR dimerisation. Although GHS-R1a/GHS-R1b or GHS-R1a/GPR39 heterodimerisation was not clearly demonstrated, this study provides a basis for future investigations of these receptors in prostate cancer. Additionally, the results presented in this study and growing evidence in the literature highlight the requirement for an extensive understanding of the experimental method and the performance of a range of controls to avoid the spurious interpretation of data gained from artificial expression systems. The future development of more robust techniques for investigating GPCR dimerisation is clearly required and will enable us to elucidate whether GHS-R1a, GHS-R1b and GPR39 form physiologically relevant dimers.

A comparative review of quantitative VFM methodology for PPP infrastructure project in Indonesia and Australia

The Government of Indonesia (GoI) increasingly relies on the private sector financing to build and operate infrastructures through public private partnership (PPP) schemes. However, PPP does not automatically provide the solution for the financing scheme due to value for money (VFM) issues. The procurement authority must show whether a PPP proposal is the optimal solution that provides best VFM outcome. The paper presents a literature review of comparing quantitative VFM methodology for PPP infrastructure project procurement in Indonesia and Australia. Public Sector Comparator (PSC) is used to assess the potential project VFM quantitatively in Australia. In Indonesia, the PSC has not been applied, where the PPP procurement authority tends to utilize a common project evaluation method that ignores the issues of risk. Unlike the conventional price bid evaluation, the PSC enables a financial comparison including costs/gains and risks. Since the construction of PSC is primarily on risk management approach, it can facilitate risk negotiation processes between the involved parties. The study indicates that the quantitative VFM methodology of PSC is potentially applicable in Indonesia for water supply sector. Various supporting regulations are available that emphasize the importance of VFM and risk management in infrastructure investment. However, the study also reveals a number of challenges that need to be anticipated, such as the need of a more comprehensive PPP policy at both central and local government level, a more specific legal instrument for bidding evaluation method and the issue of institutional capacity development in PPP Units at the local level.

Hydrodynamic and thruster model validation for autonomous underwater vehicles

From Pontryagin’s Maximum Principle to the Duke Kahanamoku Aquatic Complex; we develop the theory and generate implementable time efficient trajectories for a test-bed autonomous underwater vehicle (AUV). This paper is the beginning of the journey from theory to implementation. We begin by considering pure motion trajectories and move into a rectangular trajectory which is a concatenation of pure surge and pure sway. These trajectories are tested using our numerical model and demonstrated by our AUV in the pool. In this paper we demonstrate that the above motions are realizable through our method, and we gain confidence in our numerical model. We conclude that using our current techniques, implementation of time efficient trajectories is likely to succeed.

An analytical method to solve a general class of nonlinear reactive transport models

Three recent papers published in Chemical Engineering Journal studied the solution of a model of diffusion and nonlinear reaction using three different methods. Two of these studies obtained series solutions using specialized mathematical methods, known as the Adomian decomposition method and the homotopy analysis method. Subsequently it was shown that the solution of the same particular model could be written in terms of a transcendental function called Gauss’ hypergeometric function. These three previous approaches focused on one particular reactive transport model. This particular model ignored advective transport and considered one specific reaction term only. Here we generalize these previous approaches and develop an exact analytical solution for a general class of steady state reactive transport models that incorporate (i) combined advective and diffusive transport, and (ii) any sufficiently differentiable reaction term R(C). The new solution is a convergent Maclaurin series. The Maclaurin series solution can be derived without any specialized mathematical methods nor does it necessarily involve the computation of any transcendental function. Applying the Maclaurin series solution to certain case studies shows that the previously published solutions are particular cases of the more general solution outlined here. We also demonstrate the accuracy of the Maclaurin series solution by comparing with numerical solutions for particular cases.

Design education online : learning delivery and evaluation

Online learning has been recognised as an effective pedagogical method and tool, and is broadly integrated into various types of teaching and learning strategies in higher education. In practice, the use of Virtual Learning Environment (VLE) in higher education has become an integral strategy for quality education. The field of design education however has not been researched extensively in regard to online learning, delivery and evaluation. This paper discusses design education from an online learning perspective. It proposes an integrated framework with three key components for online learning via VLE including an interactive delivery structure, communication channels, and learning evaluation. Additionally, the paper describes and evaluates how VLE sites for two design units were built based on an integrated framework and student learning experiences. The results indicate that online design education should be integrated with various educational values and functional features in a systematic manner, and requires designing learning evaluation protocols as part of learning activities and communicative forms within online-based learning sites.

Media play : new (and old) pedagogies with young children

Modern technologies mean that the principles of quality arts education are the same (as they ever were) and different. Discussion in this paper is based on a small research project that used art as pedagogy, art as research method and, for the young children participants, celebrated art for art's sake. The project was designed with two aims. Firstly, the authors were interested in how young children engage with media as a strand of the arts. This also informed some of their thinking around the debates over Information and Communication Technology (ICT) as a process for the production of a media text. Secondly, they were interested in the extent to which digital media could enable young children to make their learning visible.

Multi-step ahead direct prediction for the machine condition prognosis using regression trees and neuro-fuzzy systems

This paper presents an approach to predict the operating conditions of machine based on classification and regression trees (CART) and adaptive neuro-fuzzy inference system (ANFIS) in association with direct prediction strategy for multi-step ahead prediction of time series techniques. In this study, the number of available observations and the number of predicted steps are initially determined by using false nearest neighbor method and auto mutual information technique, respectively. These values are subsequently utilized as inputs for prediction models to forecast the future values of the machines’ operating conditions. The performance of the proposed approach is then evaluated by using real trending data of low methane compressor. A comparative study of the predicted results obtained from CART and ANFIS models is also carried out to appraise the prediction capability of these models. The results show that the ANFIS prediction model can track the change in machine conditions and has the potential for using as a tool to machine fault prognosis.

Silylation of layered double hydroxides via an induced hydrolysis method

A series of layered double hydroxides (LDHs) based composites were synthesized by using induced hydrolysis silylation method (IHS), surfactant precursor method, in-situ coprecipitation method, and direct silylation method. Their structures, morphologies, bonding modes and thermal stabilities can be readily adjusted by changing the parameters during preparation and drying processing of the LDHs. The characterization results show that the direct silylation reaction cannot occur between the dried LDHs and 3-aminopropyltriethoxysilane (APS) in an ethanol medium. However, the condensation reaction can proceed with heating process between adsorbed APS and LDHs plates. While using wet state substrates with and without surfactant and ethanol as the solvent, the silylation process can be induced by hydrolysis of APS on the surface of LDHs plates. Surfactants improve the hydrophobicity of the LDHs during the process of nucleation and crystallization, resulting in fluffy shaped crystals; meanwhile, they occupy the surface –OH positions and leave less “free –OH” available for the silylation reaction, favoring formation of silylated products with a higher population in the hydrolyzed bidentate (T2) and tridentate (T3) bonding forms. These bonding characteristics lead to spherical aggregates and tightly bonded particles. All silylated products show higher thermal stability than those of pristine LDHs.

Natural convection flow with combined buoyancy effects due to thermal and mass diffusion in a thermally stratified media

We present here a numerical study of laminar doubly diffusive free convection flows adjacent to a vertical surface in a stable thermally stratified medium. The governing equations of mass, momentum, energy and species are non-dimensionalized. These equations have been solved by using an implicit finite difference method and local non-similarity method. The results show many interesting aspects of complex interaction of the two buoyant mechanisms that have been shown in both the tabular as well as graphical form.

Natural convection adjacent to an inclined flat plate and in an attic space under various thermal forcing conditions

The effect of viscous dissipation on natural convection from a vertical plate placed in a thermally stratified environment has been investigated numerically. The reduced equations are integrated by employing the implicit finite difference scheme or Ke1ler-box method and obtained the effect of heat due to viscous dissipation on the local skin-friction and loca1 Nusselt number at various stratification levels, for fluids having Prandtl number equals 10, 50, and 100. Solutions are also obtained using the perturbation technique for small values of viscous dissipation parameters and compared with the Finite Difference solutions. Effect of the heat transfer due to viscous dissipation and the temperature stratification are also shown on the velocity and temperature distributions in the boundary layer region. A numerical study of laminar doubly diffusive free convection flows adjacent to a vertical surface in a stable thermally stratified medium is also considered for this study. Solutions are obtained using the implicit Finite Difference method and compared with the local non-similarity method. The velocity and temperature distributions for different values of stratification parameter are shown graphically. The results show many interesting aspects of complex interaction of the two buoyant mechanisms.