High strong order methods for non-commutative stochastic ordinary differential equation systems and the Magnus formula

In recent years considerable attention has been paid to the numerical solution of stochastic ordinary differential equations (SODEs), as SODEs are often more appropriate than their deterministic counterparts in many modelling situations. However, unlike the deterministic case numerical methods for SODEs are considerably less sophisticated due to the difficulty in representing the (possibly large number of) random variable approximations to the stochastic integrals. Although Burrage and Burrage [High strong order explicit Runge-Kutta methods for stochastic ordinary differential equations, Applied Numerical Mathematics 22 (1996) 81-101] were able to construct strong local order 1.5 stochastic Runge-Kutta methods for certain cases, it is known that all extant stochastic Runge-Kutta methods suffer an order reduction down to strong order 0.5 if there is non-commutativity between the functions associated with the multiple Wiener processes. This order reduction down to that of the Euler-Maruyama method imposes severe difficulties in obtaining meaningful solutions in a reasonable time frame and this paper attempts to circumvent these difficulties by some new techniques. An additional difficulty in solving SODEs arises even in the Linear case since it is not possible to write the solution analytically in terms of matrix exponentials unless there is a commutativity property between the functions associated with the multiple Wiener processes. Thus in this present paper first the work of Magnus [On the exponential solution of differential equations for a linear operator, Communications on Pure and Applied Mathematics 7 (1954) 649-673] (applied to deterministic non-commutative Linear problems) will be applied to non-commutative linear SODEs and methods of strong order 1.5 for arbitrary, linear, non-commutative SODE systems will be constructed - hence giving an accurate approximation to the general linear problem. Secondly, for general nonlinear non-commutative systems with an arbitrary number (d) of Wiener processes it is shown that strong local order I Runge-Kutta methods with d + 1 stages can be constructed by evaluated a set of Lie brackets as well as the standard function evaluations. A method is then constructed which can be efficiently implemented in a parallel environment for this arbitrary number of Wiener processes. Finally some numerical results are presented which illustrate the efficacy of these approaches. (C) 1999 Elsevier Science B.V. All rights reserved.

Heuristics and building price forecasting

The construction industry contains two types of estimators the contractors' estimator and the designers' price forecaster. Each has two models of the building in which to systemize his procedures - the production model and the design model. The use of these models is discussed in the light of the industry's particular problems of complexity and uncertainty together with the pressures of the market. It is argued that estimators and forecasters, in order to function effectively in these conditions, are forced to exercise a high degree of subjective judgment. Means of eliciting good heuristics involved in judgment making are considered by reference to the artificial intelligence and construction literature and a methodology is proposed based on these findings. The results of two early trials of the method with students are given, indicating the usefulness of the approach.

Numerical methods of the variable-order Rayleigh-Stokes problem for a heated generalized second grade fluid with fractional derivative

Rayleigh–Stokes problems have in recent years received much attention due to their importance in physics. In this article, we focus on the variable-order Rayleigh–Stokes problem for a heated generalized second grade fluid with fractional derivative. Implicit and explicit numerical methods are developed to solve the problem. The convergence, stability of the numerical methods and solvability of the implicit numerical method are discussed via Fourier analysis. Moreover, a numerical example is given and the results support the effectiveness of the theoretical analysis.

Mathematical modelling of controlled drug release from polymer micro-spheres : incorporating the effects of swelling, diffusion and dissolution via moving boundary problems

Controlled drug delivery is a key topic in modern pharmacotherapy, where controlled drug delivery devices are required to prolong the period of release, maintain a constant release rate, or release the drug with a predetermined release profile. In the pharmaceutical industry, the development process of a controlled drug delivery device may be facilitated enormously by the mathematical modelling of drug release mechanisms, directly decreasing the number of necessary experiments. Such mathematical modelling is difficult because several mechanisms are involved during the drug release process. The main drug release mechanisms of a controlled release device are based on the device’s physiochemical properties, and include diffusion, swelling and erosion. In this thesis, four controlled drug delivery models are investigated. These four models selectively involve the solvent penetration into the polymeric device, the swelling of the polymer, the polymer erosion and the drug diffusion out of the device but all share two common key features. The first is that the solvent penetration into the polymer causes the transition of the polymer from a glassy state into a rubbery state. The interface between the two states of the polymer is modelled as a moving boundary and the speed of this interface is governed by a kinetic law. The second feature is that drug diffusion only happens in the rubbery region of the polymer, with a nonlinear diffusion coefficient which is dependent on the concentration of solvent. These models are analysed by using both formal asymptotics and numerical computation, where front-fixing methods and the method of lines with finite difference approximations are used to solve these models numerically. This numerical scheme is conservative, accurate and easily implemented to the moving boundary problems and is thoroughly explained in Section 3.2. From the small time asymptotic analysis in Sections 5.3.1, 6.3.1 and 7.2.1, these models exhibit the non-Fickian behaviour referred to as Case II diffusion, and an initial constant rate of drug release which is appealing to the pharmaceutical industry because this indicates zeroorder release. The numerical results of the models qualitatively confirms the experimental behaviour identified in the literature. The knowledge obtained from investigating these models can help to develop more complex multi-layered drug delivery devices in order to achieve sophisticated drug release profiles. A multi-layer matrix tablet, which consists of a number of polymer layers designed to provide sustainable and constant drug release or bimodal drug release, is also discussed in this research. The moving boundary problem describing the solvent penetration into the polymer also arises in melting and freezing problems which have been modelled as the classical onephase Stefan problem. The classical one-phase Stefan problem has unrealistic singularities existed in the problem at the complete melting time. Hence we investigate the effect of including the kinetic undercooling to the melting problem and this problem is called the one-phase Stefan problem with kinetic undercooling. Interestingly we discover the unrealistic singularities existed in the classical one-phase Stefan problem at the complete melting time are regularised and also find out the small time behaviour of the one-phase Stefan problem with kinetic undercooling is different to the classical one-phase Stefan problem from the small time asymptotic analysis in Section 3.3. In the case of melting very small particles, it is known that surface tension effects are important. The effect of including the surface tension to the melting problem for nanoparticles (no kinetic undercooling) has been investigated in the past, however the one-phase Stefan problem with surface tension exhibits finite-time blow-up. Therefore we investigate the effect of including both the surface tension and kinetic undercooling to the melting problem for nanoparticles and find out the the solution continues to exist until complete melting. The investigation of including kinetic undercooling and surface tension to the melting problems reveals more insight into the regularisations of unphysical singularities in the classical one-phase Stefan problem. This investigation gives a better understanding of melting a particle, and contributes to the current body of knowledge related to melting and freezing due to heat conduction.

A nanometre-scale non-periodic structural variation in high temperature superconducting ceramics and the implications for properties

Non-periodic structural variation has been found in the high Tc cuprates, YBa2Cu3O7-x and Hg0.67Pb0.33Ba2Ca2Cu 3O8+δ, by image analysis of high resolution transmission electron microscope (HRTEM) images. We use two methods for analysis of the HRTEM images. The first method is a means for measuring the bending of lattice fringes at twin planes. The second method is a low-pass filter technique which enhances information contained by diffuse-scattered electrons and reveals what appears to be an interference effect between domains of differing lattice parameter in the top and bottom of the thin foil. We believe that these methods of image analysis could be usefully applied to the many thousands of HRTEM images that have been collected by other workers in the high temperature superconductor field. This work provides direct structural evidence for phase separation in high Tc cuprates, and gives support to recent stripes models that have been proposed to explain various angle resolved photoelectron spectroscopy and nuclear magnetic resonance data. We believe that the structural variation is a response to an opening of an electronic solubility gap where holes are not uniformly distributed in the material but are confined to metallic stripes. Optimum doping may occur as a consequence of the diffuse boundaries between stripes which arise from spinodal decomposition. Theoretical ideas about the high Tc cuprates which treat the cuprates as homogeneous may need to be modified in order to take account of this type of structural variation.

Shadow boxing with an imaginary enemy : a response to "Law and Order Blues"

In our rejoinder to Don Weatherburn's paper,"Law and Order Blues", we do not take issue with his advocacy of the need to take crime seriously and to foster a more rational approach to the problems it poses. Where differences do emerge is (1) with his claim that he is willing to do so whilst we (in our different ways) are not; and (2) on the question of what this involves. Of particular concern is the way in which his argument proceeds by a combination of simple misrepresentation of the positions it seeks to disparage, and silence concerning issues of real substance where intellectual debate and exchange would be welcome and useful. Our paper challenges, in turn, the misrepresentation of Indermaur's analysis of trends in violent crime, the misrepresentation of Hogg and Brown's Rethinking Law and Order, the misrepresentation of the findings of some of the research into the effectiveness of punitive policies and the silence on sexual assault in "Law and Order Blues". We suggest that his silence on sexual assault reflects a more widespread unwillingness to acknowledge the methodological problems that arise in the measurement of crime because such problems severely limit the extent to which confident assertions can be made about prevalence and trends.

Shadow boxing with an imaginary enemy : a response to “law and order blues”

In our rejoinder to Don Weatherburn's paper, “Law and Order Blues”, we do not take issue with his advocacy of the need to take crime seriously and to foster a more rational approach to the problems it poses. Where differences do emerge is (1) with his claim that he is willing to do so whilst we (in our different ways) are not; and (2) on the question of what this involves. Of particular concern is the way in which his argument proceeds by a combination of simple misrepresentation of the positions it seeks to disparage, and silence concerning issues of real substance where intellectual debate and exchange would be welcome and useful. Our paper challenges, in turn, the misrepresentation of Indermaur's analysis of trends in violent crime, the misrepresentation of Hogg and Brown's Rethinking Law and Order, the misrepresentation of the findings of some of the research into the effectiveness of punitive policies and the silence on sexual assault in “Law and Order Blues”. We suggest that his silence on sexual assault reflects a more widespread unwillingness to acknowledge the methodological problems that arise in the measurement of crime because such problems severely limit the extent to which confident assertions can be made about prevalence and trends.

A sub‒domain smoothed Galerkin method for solid mechanics problems

A sub‒domain smoothed Galerkin method is proposed to integrate the advantages of mesh‒free Galerkin method and FEM. Arbitrarily shaped sub‒domains are predefined in problems domain with mesh‒free nodes. In each sub‒domain, based on mesh‒free Galerkin weak formulation, the local discrete equation can be obtained by using the moving Kriging interpolation, which is similar to the discretization of the high‒order finite elements. Strain smoothing technique is subsequently applied to the nodal integration of sub‒domain by dividing the sub‒domain into several smoothing cells. Moreover, condensation of DOF can also be introduced into the local discrete equations to improve the computational efficiency. The global governing equations of present method are obtained on the basis of the scheme of FEM by assembling all local discrete equations of the sub‒domains. The mesh‒free properties of Galerkin method are retained in each sub‒domain. Several 2D elastic problems have been solved on the basis of this newly proposed method to validate its computational performance. These numerical examples proved that the newly proposed sub‒domain smoothed Galerkin method is a robust technique to solve solid mechanics problems based on its characteristics of high computational efficiency, good accuracy, and convergence.

Working with Behavioural and Emotional Problems in Young People

This chapter outlines specific issues relating to behavioural and emotional problems in Aboriginal and Torres Strait Islander young people. It describes the most common disorders and their consequences, and how young Aboriginal people are at higher risk for developing such problems than other young Australians. The chapter also discusses the importance of psychosocial, cultural and environmental issues that need to be recognised in assessing and treating Aboriginal young people with behavioural and emotional problems. Issues concerning the delivery of both universal and culturally responsive prevention and intervention programs to address social and emotional wellbeing and mental health are discussed and possible interventions to enhance student engagement at school are provided. Finally, a range of mental health services for Aboriginal families which offer a culturally responsive approach to mental health treatment are listed.

Analysis of steel structures for first- and second-order element (displacement and force) solutions

Finite element method (FEM) relies on an approximate function to fit into a governing equation and minimizes the residual error in the integral sense in order to generate solutions for the boundary value problems (nodal solutions). Because of this FEM does not show simultaneous capacities for accurate displacement and force solutions at node and along an element, especially when under the element loads, which is of much ubiquity. If the displacement and force solutions are strictly confined to an element’s or member’s ends (nodal response), the structural safety along an element (member) is inevitably ignored, which can definitely hinder the design of a structure for both serviceability and ultimate limit states. Although the continuous element deflection and force solutions can be transformed into the discrete nodal solutions by mesh refinement of an element (member), this setback can also hinder the effective and efficient structural assessment as well as the whole-domain accuracy for structural safety of a structure. To this end, this paper presents an effective, robust, applicable and innovative approach to generate accurate nodal and element solutions in both fields of displacement and force, in which the salient and unique features embodies its versatility in applications for the structures to account for the accurate linear and second-order elastic displacement and force solutions along an element continuously as well as at its nodes. The significance of this paper is on shifting the nodal responses (robust global system analysis) into both nodal and element responses (sophisticated element formulation).

Amplitude equations and asymptotic expansions for multi-scale problems

In this paper we introduce a new technique to obtain the slow-motion dynamics in nonequilibrium and singularly perturbed problems characterized by multiple scales. Our method is based on a straightforward asymptotic reduction of the order of the governing differential equation and leads to amplitude equations that describe the slowly-varying envelope variation of a uniformly valid asymptotic expansion. This may constitute a simpler and in certain cases a more general approach toward the derivation of asymptotic expansions, compared to other mainstream methods such as the method of Multiple Scales or Matched Asymptotic expansions because of its relation with the Renormalization Group. We illustrate our method with a number of singularly perturbed problems for ordinary and partial differential equations and recover certain results from the literature as special cases. © 2010 - IOS Press and the authors. All rights reserved.

Adaptive finite element analysis of elliptic problems based on bubble-type local mesh generation

A new mesh adaptivity algorithm that combines a posteriori error estimation with bubble-type local mesh generation (BLMG) strategy for elliptic differential equations is proposed. The size function used in the BLMG is defined on each vertex during the adaptive process based on the obtained error estimator. In order to avoid the excessive coarsening and refining in each iterative step, two factor thresholds are introduced in the size function. The advantages of the BLMG-based adaptive finite element method, compared with other known methods, are given as follows: the refining and coarsening are obtained fluently in the same framework; the local a posteriori error estimation is easy to implement through the adjacency list of the BLMG method; at all levels of refinement, the updated triangles remain very well shaped, even if the mesh size at any particular refinement level varies by several orders of magnitude. Several numerical examples with singularities for the elliptic problems, where the explicit error estimators are used, verify the efficiency of the algorithm. The analysis for the parameters introduced in the size function shows that the algorithm has good flexibility.

Model-based analysis and optimization of bioreactor for hematopoietic stem cell cultivation

One of the problems to be solved in attaining the full potentials of hematopoietic stem cell (HSC) applications is the limited availability of the cells. Growing HSCs in a bioreactor offers an alternative solution to this problem. Besides, it also offers the advantages of eliminating labour intensive process as well as the possible contamination involved in the periodic nutrient replenishments in the traditional T-flask stem cell cultivation. In spite of this, the optimization of HSC cultivation in a bioreactor has been barely explored. This manuscript discusses the development of a mathematical model to describe the dynamics in nutrient distribution and cell concentration of an ex vivo HSC cultivation in a microchannel perfusion bioreactor. The model was further used to optimize the cultivation by proposing three alternative feeding strategies in order to prevent the occurrence of nutrient limitation in the bioreactor. The evaluation of these strategies, the periodic step change increase in the inlet oxygen concentration, the periodic step change increase in the media inflow, and the feedback control of media inflow, shows that these strategies can successfully improve the cell yield of the bioreactor. In general, the developed model is useful for the design and optimization of bioreactor operation.

A Fixed Order Model Matching Approach for the Design of Robust Power System Oscillation Damper

This book focuses on how evolutionary computing techniques benefit engineering research and development tasks by converting practical problems of growing complexities into simple formulations, thus largely reducing development efforts. This book begins with an overview of the optimization theory and modern evolutionary computing techniques, and goes on to cover specific applications of evolutionary computing to power system optimization and control problems.