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.

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.

Efficient solution of two-sided nonlinear space-fractional diffusion equations using fast Poisson preconditioners

We develop a fast Poisson preconditioner for the efficient numerical solution of a class of two-sided nonlinear space fractional diffusion equations in one and two dimensions using the method of lines. Using the shifted Gr¨unwald finite difference formulas to approximate the two-sided(i.e. the left and right Riemann-Liouville) fractional derivatives, the resulting semi-discrete nonlinear systems have dense Jacobian matrices owing to the non-local property of fractional derivatives. We employ a modern initial value problem solver utilising backward differentiation formulas and Jacobian-free Newton-Krylov methods to solve these systems. For efficient performance of the Jacobianfree Newton-Krylov method it is essential to apply an effective preconditioner to accelerate the convergence of the linear iterative solver. The key contribution of our work is to generalise the fast Poisson preconditioner, widely used for integer-order diffusion equations, so that it applies to the two-sided space fractional diffusion equation. A number of numerical experiments are presented to demonstrate the effectiveness of the preconditioner and the overall solution strategy.

A preconditioned Lanczos method for space-fractional reaction-diffusion equations on two-dimensional unstructured meshes

We consider a two-dimensional space-fractional reaction diffusion equation with a fractional Laplacian operator and homogeneous Neumann boundary conditions. The finite volume method is used with the matrix transfer technique of Ilić et al. (2006) to discretise in space, yielding a system of equations that requires the action of a matrix function to solve at each timestep. Rather than form this matrix function explicitly, we use Krylov subspace techniques to approximate the action of this matrix function. Specifically, we apply the Lanczos method, after a suitable transformation of the problem to recover symmetry. To improve the convergence of this method, we utilise a preconditioner that deflates the smallest eigenvalues from the spectrum. We demonstrate the efficiency of our approach for a fractional Fisher’s equation on the unit disk.

A sustainable tourism development in Alacati, Turkey : (Re)invention of public space with clean energy

Although there is an increasing recognition of the impacts of climate change on communities, residents often resist changing their lifestyle to reduce the effects of the problem. By using a landscape architectural design medium, this paper argues that public space, when designed as an ecological system, has the capacity to create social and environmental change and to increase the quality of the human environment. At the same time, this ecological system can engage residents, enrich the local economy, and increase the social network. Through methods of design, research and case study analysis, an alternative master plan is proposed for a sustainable tourism development in Alacati, Turkey. Our master plan uses local geographical, economic and social information within a sustainable landscape architectural design scheme that addresses the key issues of ecology, employment, public space and community cohesion. A preliminary community empowerment model (CEM) is proposed to manage the designs. The designs address: the coexistence of local agricultural and sustainable energy generation; state of the art water management; and the functional and sustainable social and economic interrelationship of inhabitants, NGOs, and local government.

A computationally effective alternating direction method for the space and time fractional Bloch–Torrey equation in 3-D

The space and time fractional Bloch–Torrey equation (ST-FBTE) has been used to study anomalous diffusion in the human brain. Numerical methods for solving ST-FBTE in three-dimensions are computationally demanding. In this paper, we propose a computationally effective fractional alternating direction method (FADM) to overcome this problem. We consider ST-FBTE on a finite domain where the time and space derivatives are replaced by the Caputo–Djrbashian and the sequential Riesz fractional derivatives, respectively. The stability and convergence properties of the FADM are discussed. Finally, some numerical results for ST-FBTE are given to confirm our theoretical findings.

Australia - taxation of trusts – the problem of aligning concepts of income

The Australian government is currently considering options for the rewrite and reform of the current provisions which apply to the taxation of trust income. This article provides a discussion of the current regime and the proposed reforms. It is suggested that a major revamp of taxation of trust income in Australia is problematic and a simpler approach may be to leave the law as is, with modification where necessary to address key issues as and when they arise.

A sustainable tourism development in Alacati, Turkey: (Re) invention of public space with clean energy

Although there is an increasing recognition of the impacts of climate change on communities, residents often resist changing their lifestyle to reduce the effects of the problem. By using a landscape architectural design medium, this paper argues that public space, when designed as an ecological system, has the capacity to create social and environmental change and to increase the quality of the human environment. At the same time, this ecological system can engage residents, enrich the local economy, and increase the social network. Through methods of design, research and case study analysis, an alternative master plan is proposed for a sustainable tourism development in Alacati, Turkey. Our master plan uses local geographical, economic and social information within a sustainable landscape architectural design scheme that addresses the key issues of ecology, employment, public space and community cohesion. A preliminary community empowerment model (CEM) is proposed to manage the designs. The designs address: the coexistence of local agricultural and sustainable energy generation; state of the art water management; and the functional and sustainable social and economic interrelationship of inhabitants, NGOs, and local government.

Public Space and the Marginalisation of Children and Young People

Throughout much of the world, urban and rural public spaces may be said to be under attack by property developers, commercial interests and also attempts by civic authorities to regulate, restrict, reframe and rebrand these spaces. A consequence of the increasingly security driven, privatised, commercial and surveilled nature of public space is the exclusion and displacement of those considered ‘flawed’ and unwelcome in the ‘spectacular’ consumption spaces of many major urban centres. In the name of urban regeneration, processes of securitisation, ‘gentrification’ and creative cities initiatives can act to refashion public space as sites of selective inclusion and exclusion. The use of surveillance and other control technologies as deployed in and around the UK ‘Riots’ of 2011 may help to promote and encourage a passing sense of personal safety and confidence in using public space. Through systems of social sorting, the same surveillance assemblages can also further the physical, emotional and psychological exclusion of certain groups and individuals, deemed to be both ‘out of time and out of place’ in major zones of urban, conspicuous, consumption. In this harsh environment of monitoring and control procedures, children and young people’s use of public spaces and places in parks, neighbourhoods, shopping malls and streets is often viewed as a threat to social order, requiring various forms of punitive and/or remedial action. Much of this civic action actively excludes some children and young people from participation and as a consequence, their trust in local processes and communities is eroded. This paper discusses worldwide developments in the surveillance, governance and control of the public space environments used by children and young people in particular and the capacity for their displacement and marginality, diminishing their sense of belonging, wellbeing and rights to public space as an expression of their social, political and civil citizenship(s).

A finite volume scheme with preconditioned Lanczos method for two-dimensional space-fractional reaction–diffusion equations

Fractional differential equations have been increasingly used as a powerful tool to model the non-locality and spatial heterogeneity inherent in many real-world problems. However, a constant challenge faced by researchers in this area is the high computational expense of obtaining numerical solutions of these fractional models, owing to the non-local nature of fractional derivatives. In this paper, we introduce a finite volume scheme with preconditioned Lanczos method as an attractive and high-efficiency approach for solving two-dimensional space-fractional reaction–diffusion equations. The computational heart of this approach is the efficient computation of a matrix-function-vector product f(A)bf(A)b, where A A is the matrix representation of the Laplacian obtained from the finite volume method and is non-symmetric. A key aspect of our proposed approach is that the popular Lanczos method for symmetric matrices is applied to this non-symmetric problem, after a suitable transformation. Furthermore, the convergence of the Lanczos method is greatly improved by incorporating a preconditioner. Our approach is show-cased by solving the fractional Fisher equation including a validation of the solution and an analysis of the behaviour of the model.

Introduction : Understanding Connections between Time, Space and the Body

The experiences and constructs of time, space and bodies saturate human discourse—naturally enough, since they are fundamental to existence—yet there has long been a tendency for the terms to be approached somewhat independently, belying the depth of their interconnections. It was a desire to address that apparent shortcoming that inspired this book, and the interdisciplinary meetings from which it was born, the 1st Global Conferences on ‘Time, Space and the Body’ and ‘Body Horror’ held in Sydney in February 2013. Following the lively, often provocative, exchange of ideas throughout those meetings, the writing here crosses conventional boundaries inhabiting everyday life and liminal experiences, across cultures, life circumstances, and bodily states. Through numerous theoretical frameworks and with reference to a variety of media, the authors problematize or deconstruct commonplace assumptions to reveal challenging new perspectives on the diverse cultures and communities which make our world. If there is an overarching theme of this collection it is diversity itself. The writers here come from numerous academic fields, but a good number of them also draw on first-hand cultural production in the arts: photography, sculpture and fine art instillation, for example. Of course, however laudable it might be, there is a potential problem in such diversity: does it produce fruitful dialogue moving toward creative, workable syntheses or simply a cacophony of competing, incomprehensible, barely comprehending voices? To a large degree this depends upon the intellectual, existential ambitions as well as the old-fashioned goodnatured tolerance of both writers and readers. But we hope three unifying characteristics are discernable in the following chapters viewed as a whole: firstly, a genuine concern for the world humans inhabit and the communities they form as bodies in space and time; secondly, an emphasis upon the experience of the human subject, exemplified perhaps by the number of chapters drawing on phenomenology; thirdly, an adventurous, explorative impulse associated with an underlying sense that being, since it is inseparable from the body’s temporality, is always becoming, and here the presence of poststructuralist influences is unmistakable, often explicit. Our challenge as editors has been to present the enormous variety of subjects and views in a way that would render the book coherent and at the same time encourage readers to make explorations themselves into realms they might usually consider beyond their field of interest. To that end we have divided the book into six sections around loosely defined themes, each offering different angles on how time and/or space unfold in and around bodies.

Young people, participation and public space

Young people are major users of public space, White (1990,1998) Loader (1996). * Young people are constructed as ‘problem’ and ‘non citizens’, Brown (1998). * Young people report feeling unconsulted, not part of community life, Measor & Squires (2000), Article 12 (2000), Tyler et al (1998). * Young people and citizenship/participation is a major issue. * Public Space issues affect others also such as the homeless, aged, women, Indigenous people and people of middle eastern appearance.

Young people and the struggle for public space in Australia

Public Space is important to the overall health and wellbeing of children and young people in allowing them to explore their local and wider community, meet up with friends, get some exercise and feel included in the society in which they live. A problem exists in the capacity of modern, urban public space to genuinely accommodate children and young people’s need to experience excitement and fun in what has been termed “unprogrammed space” (Lynch 1977:71), or simply to ‘hang out’ in unstructured social space, with control by civic authorities a key concern. For many children and young people, their experiences of attempting to use public space are sometimes marred by the denial of everyday rights and courtesies, in youth ‘unfriendly’ spaces and this is often the case in shopping centres in Australia as expolored in this paper.

A Novel High Order Space-Time Spectral Method for the Time Fractional Fokker--Planck Equation

The fractional Fokker-Planck equation is an important physical model for simulating anomalous diffusions with external forces. Because of the non-local property of the fractional derivative an interesting problem is to explore high accuracy numerical methods for fractional differential equations. In this paper, a space-time spectral method is presented for the numerical solution of the time fractional Fokker-Planck initial-boundary value problem. The proposed method employs the Jacobi polynomials for the temporal discretization and Fourier-like basis functions for the spatial discretization. Due to the diagonalizable trait of the Fourier-like basis functions, this leads to a reduced representation of the inner product in the Galerkin analysis. We prove that the time fractional Fokker-Planck equation attains the same approximation order as the time fractional diffusion equation developed in [23] by using the present method. That indicates an exponential decay may be achieved if the exact solution is sufficiently smooth. Finally, some numerical results are given to demonstrate the high order accuracy and efficiency of the new numerical scheme. The results show that the errors of the numerical solutions obtained by the space-time spectral method decay exponentially.

Space Station Design Report 2007

Design Proposal for the Blue Lunar Support Hub The conceptual design of a space station is one of the most challenging tasks in aerospace engineering. The history of the space station Mir and the assembly of the International Space Station demonstrate that even within the assembly phase quick solutions have to be found to cope with budget and technical problems or changing objectives. This report is the outcome of the conceptual design of the Space Station Design Workshop (SSDW) 2007, which took place as an international design project from the 16th to the 21st of July 2007 at the Australian Centre for Field Robotics (ACFR), University of Sydney, Australia. The participants were tasked to design a human-tended space station in low lunar orbit (LLO) focusing on supporting future missions to the moon in a programmatic context of space exploration beyond low Earth orbit (LEO). The design included incorporating elements from systems engineering to interior architecture. The customised, intuitive, rapid-turnaround software tools enabled the team to successfully tackle the complex problem of conceptual design of crewed space systems. A strong emphasis was put on improving the integration of the human crew, as it is the major contributor to mission success, while always respecting the boundary conditions imposed by the challenging environment of space. This report documents the methodology, tools and outcomes of the Space Station Design Workshop during the SSDW 2007. The design results produced by Team Blue are presented.