Efficient preconditioning of the method of lines for solving nonlinear two-sided space-fractional diffusion equations

A standard method for the numerical solution of partial differential equations (PDEs) is the method of lines. In this approach the PDE is discretised in space using �finite di�fferences or similar techniques, and the resulting semidiscrete problem in time is integrated using an initial value problem solver. A significant challenge when applying the method of lines to fractional PDEs is that the non-local nature of the fractional derivatives results in a discretised system where each equation involves contributions from many (possibly every) spatial node(s). This has important consequences for the effi�ciency of the numerical solver. First, 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. Second, since the Jacobian matrix of the system is dense (partially or fully), methods that avoid the need to form and factorise this matrix are preferred. In this paper, we consider a nonlinear two-sided space-fractional di�ffusion equation in one spatial dimension. A key contribution of this paper is to demonstrate how an eff�ective preconditioner is crucial for improving the effi�ciency of the method of lines for solving this equation. In particular, we 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.

Corporate social and environmental responsibility in relation to the agricultural sector and the food supply chain : a Case Study of Waitrose

Waitrose has a strong commitment to organic farming but also uses products from 'conventional' farms. At the production stage, Waitrose own-label products are fully traceable, GM-free and all suppliers undergo a detailed assessment programme based on current best practice. Crop suppliers to Waitrose operate an authenticity programme to certify that each assignment is GM-free and produce is screened for pesticide residues. Waitrose sources conventional crops grown from 'Integrated Crop Management Systems' (ICMS) using best horticultural practices. The 'Assured Product' scheme regulates all UK produce to ICMS standards and these audits are being extended worldwide. Business is withdrawn from suppliers who fail the audit. In relation to this, Waitrose has increased its Fairtrade range as in its view 'Buying these products provides direct additional benefit to workers in the developing countries where they are produced and assists marginal producers by giving them access to markets they would not otherwise have'. Currently, Waitrose is developing its own sustainable timber assessment criteria. For livestock, protocols are in place to ensure that animals are reared under the 'most natural conditions possible' and free range produce is offered where animals have access to open space although some produce is not from free-range animals. Waitrose also use a 'Hazards Analysis Critical Points' system to identify food safety hazards that occur at any stage from production to point of sale and to ensure that full measures are in place to control them. In addition, mechanisms have been implemented to reduce fuel use and hence reduce CO2 emissions in the transport of products and staff, and to increase the energy use efficiency of refrigeration systems which account for approximately 60% of Waitrose energy use.

Discussions in space : interactive urban screens for enhancing citizen engagement

Local governments struggle to engage time poor and seemingly apathetic citizens, as well as the city's young digital natives, the digital locals. Capturing the attention of this digitally literate community who are technology and socially savvy adds a new quality to the challenge of community engagement for urban planning. This project developed and tested a lightweight design intervention towards removing the hierarchy between those who plan the city and those who use it. The aim is to narrow this gap by enhancing people's experience of physical spaces with digital, civic technologies that are directly accessible within that space. The study's research informed the development of a public screen system called Discussions In Space (DIS). It facilitates a feedback platform about specific topics, e.g., a concrete urban planning project, and encourages direct, in-situ, real-time user responses via SMS and Twitter. The thesis presents the findings of deploying and integrating DIS in a wide range of public and urban environments, including the iconic urban screen at Federation Square in Melbourne, to explore the Human-Computer Interaction (HCI) related challenges and implications. It was also deployed in conjunction with a major urban planning project in Brisbane to explore the system's opportunities and challenges of better engaging with Australia's new digital locals. Finally, the merits of the short-texted and ephemeral data generated by the system were evaluated in three focus groups with professional urban planners. DIS offers additional benefits for civic participation as it gives voice to residents who otherwise would not be easily heard. It also promotes a positive attitude towards local governments and gathers complementary information that is different than that captured by more traditional public engagement tools.

Negative space and positive environment : mapping opportunities for urban resilience

Cities have long held a fascination for people – as they grow and develop, there is a desire to know and understand the intricate interplay of elements that makes cities ‘live’. In part, this is a need for even greater efficiency in urban centres, yet the underlying quest is for a sustainable urban form. In order to make sense of the complex entities that we recognise cities to be, they have been compared to buildings, organisms and more recently machines. However the search for better and more elegant urban centres is hardly new, healthier and more efficient settlements were the aim of Modernism’s rational sub-division of functions, which has been translated into horizontal distribution through zoning, or vertical organisation thought highrise developments. However both of these approaches have been found to be unsustainable, as too many resources are required to maintain this kind or urbanisation and social consequences of either horizontal or vertical isolation must also be considered. From being absolute consumers of resources, of energy and of technology, cities need to change, to become sustainable in order to be more resilient and more efficient in supporting culture, society as well as economy. Our urban centres need to be re-imagined, re-conceptualised and re-defined, to match our changing society. One approach is to re-examine the compartmentalised, mono-functional approach of urban Modernism and to begin to investigate cities like ecologies, where every element supports and incorporates another, fulfilling more than just one function. This manner of seeing the city suggests a framework to guide the re-mixing of urban settlements. Beginning to understand the relationships between supporting elements and the nature of the connecting ‘web’ offers an invitation to investigate the often ignored, remnant spaces of cities. This ‘negative space’ is the residual from which space and place are carved out in the Contemporary city, providing the link between elements of urban settlement. Like all successful ecosystems, cities need to evolve and change over time in order to effectively respond to different lifestyles, development in culture and society as well as to meet environmental challenges. This paper seeks to investigate the role that negative space could have in the reorganisation of the re-mixed city. The space ‘in-between’ is analysed as an opportunity for infill development or re-development which provides to the urban settlement the variety that is a pre-requisite for ecosystem resilience. An analysis of the urban form is suggested as an empirical tool to map the opportunities already present in the urban environment and negative space is evaluated as a key element in achieving a positive development able to distribute diverse environmental and social facilities in the city.

Implications of lipid biology for the pathogenesis of schizophrenia

Objective: Preclinical and clinical data suggest that lipid biology is integral to brain development and neurodegeneration. Both aspects are proposed as being important in the pathogenesis of schizophrenia. The purpose of this paper is to examine the implications of lipid biology, in particular the role of essential fatty acids (EFA), for schizophrenia. Methods: Medline databases were searched from 1966 to 2001 followed by the crosschecking of references. Results: Most studies investigating lipids in schizophrenia described reduced EFA, altered glycerophospholipids and an increased activity of a calcium-independent phospholipase A2 in blood cells and in post-mortem brain tissue. Additionally, in vivo brain phosphorus-31 Magnetic Resonance Spectroscopy (31P-MRS) demonstrated lower phosphomonoesters (implying reduced membrane precursors) in first- and multi-episode patients. In contrast, phosphodiesters were elevated mainly in first-episode patients (implying increased membrane breakdown products), whereas inconclusive results were found in chronic patients. EFA supplementation trials in chronic patient populations with residual symptoms have demonstrated conflicting results. More consistent results were observed in the early and symptomatic stages of illness, especially if EFA with a high proportion of eicosapentaenoic acid was used. Conclusion: Peripheral blood cell, brain necropsy and 31P-MRS analysis reveal a disturbed lipid biology, suggesting generalized membrane alterations in schizophrenia. 31P-MRS data suggest increased membrane turnover at illness onset and persisting membrane abnormalities in established schizophrenia. Cellular processes regulating membrane lipid metabolism are potential new targets for antipsychotic drugs and might explain the mechanism of action of treatments such as eicosapentaenoic acid.

Alternating direction implicit numerical method for the space and time fractional Bloch-Torrey equation in 3-D

Recently, some authors have considered a new diffusion model–space and time fractional Bloch-Torrey equation (ST-FBTE). Magin et al. (2008) have derived analytical solutions with fractional order dynamics in space (i.e., _ = 1, β an arbitrary real number, 1 < β ≤ 2) and time (i.e., 0 < α < 1, and β = 2), respectively. Yu et al. (2011) have derived an analytical solution and an effective implicit numerical method for solving ST-FBTEs, and also discussed the stability and convergence of the implicit numerical method. However, due to the computational overheads necessary to perform the simulations for nuclear magnetic resonance (NMR) in three dimensions, they present a study based on a two-dimensional example to confirm their theoretical analysis. Alternating direction implicit (ADI) schemes have been proposed for the numerical simulations of classic differential equations. The ADI schemes will reduce a multidimensional problem to a series of independent one-dimensional problems and are thus computationally efficient. In this paper, we consider the numerical solution of a ST-FBTE on a finite domain. The time and space derivatives in the ST-FBTE are replaced by the Caputo and the sequential Riesz fractional derivatives, respectively. A fractional alternating direction implicit scheme (FADIS) for the ST-FBTE in 3-D is proposed. Stability and convergence properties of the FADIS are discussed. Finally, some numerical results for ST-FBTE are given.

A new implicit numerical method for the space and time fractional Bloch-Torrey equation

In recent years, it has been found that many phenomena in engineering, physics, chemistry and other sciences can be described very successfully by models using mathematical tools from fractional calculus. Recently, noted a new space and time fractional Bloch-Torrey equation (ST-FBTE) has been proposed (see Magin et al. (2008)), and successfully applied to analyse diffusion images of human brain tissues to provide new insights for further investigations of tissue structures. In this paper, we consider the ST-FBTE on a finite domain. The time and space derivatives in the ST-FBTE are replaced by the Caputo and the sequential Riesz fractional derivatives, respectively. Firstly, we propose a new effective implicit numerical method (INM) for the STFBTE whereby we discretize the Riesz fractional derivative using a fractional centered difference. Secondly, we prove that the implicit numerical method for the ST-FBTE is unconditionally stable and convergent, and the order of convergence of the implicit numerical method is ( T2 - α + h2 x + h2 y + h2 z ). Finally, some numerical results are presented to support our theoretical analysis.

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.

A second-order accurate numerical approximation for the Riesz space fractional advection-dispersion equation

In this paper, we consider a space Riesz fractional advection-dispersion equation. The equation is obtained from the standard advection-diffusion equation by replacing the ¯rst-order and second-order space derivatives by the Riesz fractional derivatives of order β 1 Є (0; 1) and β2 Є(1; 2], respectively. Riesz fractional advection and dispersion terms are approximated by using two fractional centered difference schemes, respectively. A new weighted Riesz fractional ¯nite difference approximation scheme is proposed. When the weighting factor Ѳ = 1/2, a second- order accurate numerical approximation scheme for the Riesz fractional advection-dispersion equation is obtained. Stability, consistency and convergence of the numerical approximation scheme are discussed. A numerical example is given to show that the numerical results are in good agreement with our theoretical analysis.

Analytical solutions for the multi-term time-space Caputo-Riesz fractional advection-diffusion equations on a finite domain

Generalized fractional partial differential equations have now found wide application for describing important physical phenomena, such as subdiffusive and superdiffusive processes. However, studies of generalized multi-term time and space fractional partial differential equations are still under development. In this paper, the multi-term time-space Caputo-Riesz fractional advection diffusion equations (MT-TSCR-FADE) with Dirichlet nonhomogeneous boundary conditions are considered. The multi-term time-fractional derivatives are defined in the Caputo sense, whose orders belong to the intervals [0, 1], [1, 2] and [0, 2], respectively. These are called respectively the multi-term time-fractional diffusion terms, the multi-term time-fractional wave terms and the multi-term time-fractional mixed diffusion-wave terms. The space fractional derivatives are defined as Riesz fractional derivatives. Analytical solutions of three types of the MT-TSCR-FADE are derived with Dirichlet boundary conditions. By using Luchko's Theorem (Acta Math. Vietnam., 1999), we proposed some new techniques, such as a spectral representation of the fractional Laplacian operator and the equivalent relationship between fractional Laplacian operator and Riesz fractional derivative, that enabled the derivation of the analytical solutions for the multi-term time-space Caputo-Riesz fractional advection-diffusion equations. © 2012.

Educational markets in space : gamekeeping professionals across Australian communities

This paper argues that the logic of neoliberal choice policy is typically blind to considerations of space and place, but inevitably impacts on rural and remote locations in the way that middle class professionals view the opportunities available in their local educational markets. The paper considers the value of middle class professionals’ educational capitals in regional communities and their problematic distribution, given that class fraction’s particular investment in choice strategies to ensure their children’s future. It then profiles the educational market in six communities along a transect between a major regional centre and a remote ‘outback’ town, using publicly available data from the Australian government’s ‘My School’ website. Comparison of the local markets shows how educational outcomes are distributed across the local markets and how dimensions of ‘choice’ thin out over the transect. Interview data offers insights into how professional families in these localities engage selectively with these local educational markets, or plan to transcend them. The discussion reflects on the growing importance of educational choices as a marker of place in the competition between localities to attract and retain professionals to staff vital human services in their communities.