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.

Mathematical modelling of chronic wound healing

Chronicwounds fail to proceed through an orderly process to produce anatomic and functional integrity and are a significant socioeconomic problem. There is much debate about the best way to treat these wounds. In this thesis we review earlier mathematical models of angiogenesis and wound healing. Many of these models assume a chemotactic response of endothelial cells, the primary cell type involved in angiogenesis. Modelling this chemotactic response leads to a system of advection-dominated partial differential equations and we review numerical methods to solve these equations and argue that the finite volume method with flux limiting is best-suited to these problems. One treatment of chronic wounds that is shrouded with controversy is hyperbaric oxygen therapy (HBOT). There is currently no conclusive data showing that HBOT can assist chronic wound healing, but there has been some clinical success. In this thesis we use several mathematical models of wound healing to investigate the use of hyperbaric oxygen therapy to assist the healing process - a novel threespecies model and a more complex six-species model. The second model accounts formore of the biological phenomena but does not lend itself tomathematical analysis. Bothmodels are then used tomake predictions about the efficacy of hyperbaric oxygen therapy and the optimal treatment protocol. Based on our modelling, we are able to make several predictions including that intermittent HBOT will assist chronic wound healing while normobaric oxygen is ineffective in treating such wounds, treatment should continue until healing is complete and finding the right protocol for an individual patient is crucial if HBOT is to be effective. Analysis of the models allows us to derive constraints for the range of HBOT protocols that will stimulate healing, which enables us to predict which patients are more likely to have a positive response to HBOT and thus has the potential to assist in improving both the success rate and thus the cost-effectiveness of this therapy.

Effects of different practice task constraints on fluctuations of player heart rate in small-sided football games

This paper analyzes effects of different practice task constraints on heart rate (HR) variability during 4v4 smallsided football games. Participants were sixteen football players divided into two age groups (U13, Mean age: 12.4±0.5 yrs; U15: 14.6±0.5). The task consisted of a 4v4 sub-phase without goalkeepers, on a 25x15 m field, of 15 minutes duration with an active recovery period of 6 minutes between each condition. We recorded players’ heart rates using heart rate monitors (Polar Team System, Polar Electro, Kempele, Finland) as scoring mode was manipulated (line goal: scoring by dribbling past an extended line; double goal: scoring in either of two lateral goals; and central goal: scoring only in one goal). Subsequently, %HR reserve was calculated with the Karvonen formula. We performed a time-series analysis of HR for each individual in each condition. Mean data for intra-participant variability showed that autocorrelation function was associated with more short-range dependence processes in the “line goal” condition, compared to other conditions, demonstrating that the “line goal” constraint induced more randomness in HR response. Relative to inter-individual variability, line goal constraints demonstrated lower %CV and %RMSD (U13: 9% and 19%; U15: 10% and 19%) compared with double goal (U13: 12% and 21%; U15: 12% and 21%) and central goal (U13: 14% and 24%; U15: 13% and 24%) task constraints, respectively. Results suggested that line goal constraints imposed more randomness on cardiovascular stimulation of each individual and lower inter-individual variability than double goal and central goal constraints.

Effect of heating rate on the crystal composition of ferroelectric lithium niobate crystallites

A series of lithium niobate powders were synthesized by the combustion method at different heating rates. The effect of heating rate on the crystal composition of lithium niobate powders was investigated by powder X-ray diffraction measurements. It has been found that the lithium content in the as-synthesized lithium niobate powders increases with decreasing the heating rate. On the basis of the existed structure-property relationship of lithium niobate single crystals, it was concluded that high quality lithium niobate powders can be effectively synthesized at a lower heating rate (in the range of 5-10 C/min) by the combustion method.

Multiscale analysis of protein functions and stochastic modelling of gene transcriptional regulatory networks

Genomic and proteomic analyses have attracted a great deal of interests in biological research in recent years. Many methods have been applied to discover useful information contained in the enormous databases of genomic sequences and amino acid sequences. The results of these investigations inspire further research in biological fields in return. These biological sequences, which may be considered as multiscale sequences, have some specific features which need further efforts to characterise using more refined methods. This project aims to study some of these biological challenges with multiscale analysis methods and stochastic modelling approach. The first part of the thesis aims to cluster some unknown proteins, and classify their families as well as their structural classes. A development in proteomic analysis is concerned with the determination of protein functions. The first step in this development is to classify proteins and predict their families. This motives us to study some unknown proteins from specific families, and to cluster them into families and structural classes. We select a large number of proteins from the same families or superfamilies, and link them to simulate some unknown large proteins from these families. We use multifractal analysis and the wavelet method to capture the characteristics of these linked proteins. The simulation results show that the method is valid for the classification of large proteins. The second part of the thesis aims to explore the relationship of proteins based on a layered comparison with their components. Many methods are based on homology of proteins because the resemblance at the protein sequence level normally indicates the similarity of functions and structures. However, some proteins may have similar functions with low sequential identity. We consider protein sequences at detail level to investigate the problem of comparison of proteins. The comparison is based on the empirical mode decomposition (EMD), and protein sequences are detected with the intrinsic mode functions. A measure of similarity is introduced with a new cross-correlation formula. The similarity results show that the EMD is useful for detection of functional relationships of proteins. The third part of the thesis aims to investigate the transcriptional regulatory network of yeast cell cycle via stochastic differential equations. As the investigation of genome-wide gene expressions has become a focus in genomic analysis, researchers have tried to understand the mechanisms of the yeast genome for many years. How cells control gene expressions still needs further investigation. We use a stochastic differential equation to model the expression profile of a target gene. We modify the model with a Gaussian membership function. For each target gene, a transcriptional rate is obtained, and the estimated transcriptional rate is also calculated with the information from five possible transcriptional regulators. Some regulators of these target genes are verified with the related references. With these results, we construct a transcriptional regulatory network for the genes from the yeast Saccharomyces cerevisiae. The construction of transcriptional regulatory network is useful for detecting more mechanisms of the yeast cell cycle.

Thermal decomposition of hydrotalcite with hexacyanoferrite(II) and hexacyanoferrate(III) anions in the interlayer : a controlled rate thermal analysis study

The mechanism for the decomposition of hydrotalcite remains unsolved. Controlled rate thermal analysis enables this decomposition pathway to be explored. The thermal decomposition of hydrotalcites with hexacyanoferrite(II) and hexacyanoferrate(III) in the interlayer has been studied using controlled rate thermal analysis technology. X-ray diffraction shows the hydrotalcites studied have a d(003) spacing of 11.1 and 10.9 Å which compares with a d-spacing of 7.9 and 7.98 Å for the hydrotalcite with carbonate or sulphate in the interlayer. Calculations based upon CRTA measurements show that 7 moles of water is lost, proving the formula of hexacyanoferrite(II) intercalated hydrotalcite is Mg6Al2(OH)16[Fe(CN)6]0.5 .7 H2O and for the hexacyanoferrate(III) intercalated hydrotalcite is Mg6Al2(OH)16[Fe(CN)6]0.66 * 9 H2O. Dehydroxylation combined with CN unit loss occurs in three steps between a) 310 and 367°C b) 367 and 390°C and c) between 390 and 428°C for both the hexacyanoferrite(II) and hexacyanoferrate(III) intercalated hydrotalcite.

Hall-petch relationship and strain rate sensitivity of nanocrystalline Mg - 5wt% A1 alloy

This study investigated the grain size dependence of mechanical properties and deformation mechanisms of microcrystalline (mc) and nanocrystalline (nc: grain size below 100 nm) Mg-5wt% Al alloys. The Hall-Petch relationship was investigated by both instrumented indentation tests and compression tests. The test results from the indentation tests and compression tests match well with each other. The breakdown of Hall-Petch relationship and the elevated strain rate sensitivity (SRS) of present Mg-5wt% Al alloys when the grain size was reduced below 58nm indicated the more significant role of GB mediated mechanisms in plastic deformation process. However, the relatively smaller SRS values compared to GB sliding and coble creep process suggested the plastic deformation in the current study is still dislocation mediated mechanism dominant.

Multiple evolutionary rate classes in animal genome evolution

The proportion of functional sequence in the human genome is currently a subject of debate. The most widely accepted figure is that approximately 5% is under purifying selection. In Drosophila, estimates are an order of magnitude higher, though this corresponds to a similar quantity of sequence. These estimates depend on the difference between the distribution of genomewide evolutionary rates and that observed in a subset of sequences presumed to be neutrally evolving. Motivated by the widening gap between these estimates and experimental evidence of genome function, especially in mammals, we developed a sensitive technique for evaluating such distributions and found that they are much more complex than previously apparent. We found strong evidence for at least nine well-resolved evolutionary rate classes in an alignment of four Drosophila species and at least seven classes in an alignment of four mammals, including human. We also identified at least three rate classes in human ancestral repeats. By positing that the largest of these ancestral repeat classes is neutrally evolving, we estimate that the proportion of nonneutrally evolving sequence is 30% of human ancestral repeats and 45% of the aligned portion of the genome. However, we also question whether any of the classes represent neutrally evolving sequences and argue that a plausible alternative is that they reflect variable structure-function constraints operating throughout the genomes of complex organisms.

Numerical simulation for the variable-order Galilei invariant advection diffusion equation with a nonlinear source term

In this paper, we consider the variable-order Galilei advection diffusion equation with a nonlinear source term. A numerical scheme with first order temporal accuracy and second order spatial accuracy is developed to simulate the equation. The stability and convergence of the numerical scheme are analyzed. Besides, another numerical scheme for improving temporal accuracy is also developed. Finally, some numerical examples are given and the results demonstrate the effectiveness of theoretical analysis. Keywords: The variable-order Galilei invariant advection diffusion equation with a nonlinear source term; The variable-order Riemann–Liouville fractional partial derivative; Stability; Convergence; Numerical scheme improving temporal accuracy