Analytical and numerical solutions of the space and time fractional bloch-torrey equation

Fractional order dynamics in physics, particularly when applied to diffusion, leads to an extension of the concept of Brown-ian motion through a generalization of the Gaussian probability function to what is termed anomalous diffusion. As MRI is applied with increasing temporal and spatial resolution, the spin dynamics are being examined more closely; such examinations extend our knowledge of biological materials through a detailed analysis of relaxation time distribution and water diffusion heterogeneity. Here the dynamic models become more complex as they attempt to correlate new data with a multiplicity of tissue compartments where processes are often anisotropic. Anomalous diffusion in the human brain using fractional order calculus has been investigated. Recently, a new diffusion model was proposed by solving the Bloch-Torrey equation using fractional order calculus with respect to time and space (see R.L. Magin et al., J. Magnetic Resonance, 190 (2008) 255-270). However effective numerical methods and supporting error analyses for the fractional Bloch-Torrey equation are still limited. In this paper, the space and time fractional Bloch-Torrey equation (ST-FBTE) is considered. The time and space derivatives in the ST-FBTE are replaced by the Caputo and the sequential Riesz fractional derivatives, respectively. Firstly, we derive an analytical solution for the ST-FBTE with initial and boundary conditions on a finite domain. Secondly, we propose an implicit numerical method (INM) for the ST-FBTE, and the stability and convergence of the INM are investigated. We prove that the implicit numerical method for the ST-FBTE is unconditionally stable and convergent. Finally, we present some numerical results that support our theoretical analysis.

The persistence of phase-separation in LiFePO4 with two-dimensional Li+ transport : the Cahn-Hilliard-reaction equation and the role of defects

We examine the solution of the two-dimensional Cahn-Hilliard-reaction (CHR) equation in the xy plane as a model of Li+ intercalation into LiFePO4 material. We validate our numerical solution against the solution of the depth-averaged equation, which has been used to model intercalation in the limit of highly orthotropic diffusivity and gradient penalty tensors. We then examine the phase-change behaviour in the full CHR system as these parameters become more isotropic, and find that as the Li+ diffusivity is increased in the x direction, phase separation persists at high currents, even in small crystals with averaged coherency strain included. The resulting voltage curves decrease monotonically, which has previously been considered a hallmark of crystals that fill homogeneously.

Family violence or woman abuse? Putting gender back into the Canadian research equation

Research on violence against women has been among the most scrutinized areas in social science. From the beginning, efforts to empirically document the prevalence, incidence, and characteristics of violence against women have been hotly debated (DeKeseredy, 2011; Dragiewicz & DeKeseredy, forthcoming; Minaker & Snider, 2006). Objections that violence against women was rare have given way to acknowledgement that it is more common than once thought. Research on the outcomes of woman abuse has documented the serious ramifications of this type of violence for individual victims and the broader community. However, violence against women was not simply “discovered” by scholars in the 1960s, leading to a progressive growth of the literature. Knowledge production around violence against women has been fiercely contested, and feminist insights in particular have always been met with backlash(Gotell, 2007; Minkaer & Snider, 2006; Randall, 1989; Sinclair, 2003)...

Food for thought : designing for critical reflection on food practices

This workshop is a continuation and extension to the successful past workshops including [4, 5, 6]. The workshop addresses the opportunities and challenges for the design of digital interactive systems that engage individuals in critical reflection on their everyday food practices - including designing for engagement in more environmentally aware, socially inclusive, and healthier behaviour. These three themes represent the focus of much recent HCI work related to food. The workshop aims to further the conversation on these themes through understanding specifically how the process of critical reflection can be encouraged by interactive technology. While the focus will be on food as an application area, the intention is to also explore, more generally, how the process of critical reflection can be facilitated through interactive technology. The workshop provides a unique forum to discuss existing theoretical and pragmatic approaches, and to envision novel ways to design technology that encourages sustained critical reflection.

Reflection in Integrated Problem Based Learning : 20 years of continuing application to the teaching of architecture

In 1984 the School of Architecture and Built Environment within the University of Newcastle, Australia introduced an integrated program based on real design projects and using Integrated Problem Based Learning (IPBL) as the teaching method. Since 1984 there have been multiple changes arising from the expectations of the architectural fraternity, enrolling students, lecturers, available facilities, accreditation authorities and many others. These challenges have been successfully accommodated whilst maintaining the original purposes and principles of IPBL. The Architecture program has a combined two-degree structure consisting of a first degree, Bachelor of Science (Architecture), followed by a second degree, Bachelor of Architecture. The program is designed to simulate the problem-solving situations that face a working architect in every day practice. This paper will present the degree structure where each student is enrolled in a single course per semester incorporating design integration and study areas in design studies, professional studies, historical studies, technical studies, environmental studies and communication skills. Each year the design problems increase in complexity and duration set around an annual theme. With 20 years of successful delivery of any program there are highlights and challenges along the way and this paper will discuss some of the successes and barriers experienced within the School of Architecture and Built Environment in delivering IPBL. In addition, the reflective process investigates the currency of IPBL as an appropriate vehicle for delivering the curriculum in 2004 and any additional administrative or staff considerations required to enhance the continuing application of IPBL.

The pedagogical balancing act : teaching reflection in higher education

Despite the common use of the term reflection in higher education assessment tasks, learners are not often taught how to communicate their disciplinary knowledge through reflection. This paper argues that students can and should be taught how to reflect in deep and transformative ways. It highlights the reflexive pedagogical balancing act of attending to different levels of reflection as a way to stimulate focused, thoughtful and reasoned reflections that show evidence of new ways of thinking and doing. The paper uses data from a current project to illustrate the effects of focusing on particular levels of reflection in the pedagogical strategies used, and argues that while the goal of academic or professional reflection is generally to move students to the highest level of reflection to transform their learning/practice, unless higher education teachers attend to every level of reflection, there are specific, observable gaps in the reflections that students produce.

Exploration and reflection in interactive art : glass pond

Glass Pond is an interactive artwork designed to engender exploration and reflection through an intuitive, tangible interface and a simulation agent. It is being developed using iterative methods. A study has been conducted with the aim of illuminating user experience, interface, design, and performance issues.The paper describes the study methodology and process of data analysis including coding schemes for cognitive states and movements. Analysis reveals that exploration and reflection occurred as well as composing behaviours (unexpected). Results also show that participants interacted to varying degrees. Design discussion includes the artwork's (novel) interface and configuration.

Some reflection of research 'Schools' and geographies

Reflecting on real and perceived differences between European and North American research cultures, I challenge views that ‘European’ research is under appreciated or discriminated against, and caution against isolationist European positions. Instead, I argue that although no distinctive and coherent European tradition or culture really exists, there may be elements of the prevalent research culture that can be turned into an advantage for Europe-based and/or European-trained researchers in helping to influence and improve one, global research conversation. Of course, a range of sub-communities and sub-conversations will and should exist, but there is no reason for these to be based on geography.

Food for thought : designing for critical reflection on food practices

With increasing demands on our time, everyday behaviors such as food purchasing, preparation, and consumption have become habitual and unconscious. Indeed, modern food values are focused on conve- nience and effortlessness, overshad- owing other values such as environ- mental sustainability, health, and pleasure. The rethinking of how we approach everyday food behaviors appears to be a particularly timely concern. In this special section, we explore work carried out and dis- cussed during the recent workshop “Food for Thought: Designing for Critical Reflection on Food Practices,” at the 2012 Designing Interactive Systems Conference in Newcastle upon Tyne, U.K.

Stability and convergence of a finite volume method for the space fractional advection-dispersion equation

We consider the space fractional advection–dispersion equation, which is obtained from the classical advection–diffusion equation by replacing the spatial derivatives with a generalised derivative of fractional order. We derive a finite volume method that utilises fractionally-shifted Grünwald formulae for the discretisation of the fractional derivative, to numerically solve the equation on a finite domain with homogeneous Dirichlet boundary conditions. We prove that the method is stable and convergent when coupled with an implicit timestepping strategy. Results of numerical experiments are presented that support the theoretical analysis.

Reliable and valid survey design for structural equation modelling

My quantitative study asks how Chinese Australians’ “Chineseness” and their various resources influence their Chinese language proficiency, using online survey and snowball sampling. ‘Operationalization’ is a challenging process which ensures that the survey design talks back to the informing theory and forwards to the analysis model. It requires the attention to two core methodological concerns, namely ‘validity’ and ‘reliability’. Construction of a high-quality questionnaire is critical to the achievement of valid and reliable operationalization. A series of strategies were chosen to ensure the quality of the questions, and thus the eventual data. These strategies enable the use of structural equation modelling to examine how well the data fits the theoretical framework, which was constructed in light of Bourdieu’s theory of habitus, capital and field.

Learning and teaching reflection : developing capacities for lifelong learning

Reflection is an essential part of being an effective learner and working as a productive teacher. It enables the learner or teacher to deliberate about the factors that lead to successful learning and teaching for them and/or their students, in a particular place and for a specific purpose, so they can make reasoned and effective choices. This chapter introduces important frameworks that cover a century of thinking around reflection in education, and illustrates how preservice teachers can use these ideas across three phases. First, becoming a reflective learner as a university student to enhance learning and assessment outcomes; second, becoming a reflective teacher to improve classroom teaching and learning outcomes; and third, developing the reflective capacities of primary students so they can enhance their skills for lifelong learning.

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.

General order conditions for stochastic Runge-Kutta methods for both commuting and non-commuting stochastic ordinary differential equation systems

In many modeling situations in which parameter values can only be estimated or are subject to noise, the appropriate mathematical representation is a stochastic ordinary differential equation (SODE). However, unlike the deterministic case in which there are suites of sophisticated numerical methods, numerical methods for SODEs are much less sophisticated. Until a recent paper by K. Burrage and P.M. Burrage (1996), the highest strong order of a stochastic Runge-Kutta method was one. But K. Burrage and P.M. Burrage (1996) showed that by including additional random variable terms representing approximations to the higher order Stratonovich (or Ito) integrals, higher order methods could be constructed. However, this analysis applied only to the one Wiener process case. In this paper, it will be shown that in the multiple Wiener process case all known stochastic Runge-Kutta methods can suffer a severe order reduction if there is non-commutativity between the functions associated with the Wiener processes. Importantly, however, it is also suggested how this order can be repaired if certain commutator operators are included in the Runge-Kutta formulation. (C) 1998 Elsevier Science B.V. and IMACS. All rights reserved.

A banded preconditioner for the two-sided, nonlinear space-fractional diffusion equation

The method of lines is a standard method for advancing the solution of partial differential equations (PDEs) in time. In one sense, the method applies equally well to space-fractional PDEs as it does to integer-order PDEs. However, there is a significant challenge when solving space-fractional PDEs in this way, owing to the non-local nature of the fractional derivatives. Each equation in the resulting semi-discrete system involves contributions from every spatial node in the domain. This has important consequences for the efficiency of the numerical solver, especially when the system is large. First, the Jacobian matrix of the system is dense, and hence methods that avoid the need to form and factorise this matrix are preferred. Second, since the cost of evaluating the discrete equations is high, it is essential to minimise the number of evaluations required to advance the solution in time. In this paper, we show how an effective preconditioner is essential for improving the efficiency of the method of lines for solving a quite general two-sided, nonlinear space-fractional diffusion equation. A key contribution is to show, how to construct suitable banded approximations to the system Jacobian for preconditioning purposes that permit high orders and large stepsizes to be used in the temporal integration, without requiring dense matrices to be formed. The results of numerical experiments are presented that demonstrate the effectiveness of this approach.