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.

Numerical simulation of a fractional mathematical model for epidermal wound healing

A number of mathematical models investigating certain aspects of the complicated process of wound healing are reported in the literature in recent years. However, effective numerical methods and supporting error analysis for the fractional equations which describe the process of wound healing are still limited. In this paper, we consider numerical simulation of fractional model based on the coupled advection-diffusion equations for cell and chemical concentration in a polar coordinate system. The space fractional derivatives are defined in the Left and Right Riemann-Liouville sense. Fractional orders in advection and diffusion terms belong to the intervals (0; 1) or (1; 2], respectively. Some numerical techniques will be used. Firstly, the coupled advection-diffusion equations are decoupled to a single space fractional advection-diffusion equation in a polar coordinate system. Secondly, we propose a new implicit difference method for simulating this equation by using the equivalent of the Riemann-Liouville and Gr¨unwald-Letnikov fractional derivative definitions. Thirdly, its stability and convergence are discussed, respectively. Finally, some numerical results are given to demonstrate the 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.

Space, place and power : the spatial turn in literacy research

Place matters to literacy because the meanings of our language and actions are always materially and socially placed in the world (Scollon & Scollon, 2003). We cannot interpret signs, whether an icon, symbol, gesture, word, or action, without taking into account their associations with other meanings and objects in places. This chapter maps an emergent strand of literacy research that foregrounds place and space as constitutive, rather than a backdrop for the real action. Space and place are seen as relational and dynamic, not as fixed and unchanging. Space and place are socially produced, and hence, can be contested, re-imagined and re-made. In bringing space and place into the frame of literacy studies we see a subtle shift – a rebalancing of the semiotic with the materiality of lived, embodied, and situated experience. ...

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.

Their space, my space, our space? Report of the 2009 Redland City Council Youth and Public Space Survey

This report analyses data collected through the Redland City Council’s Young People and Public Space Survey of 2148 high school students aged 12-19. The survey conducted in 2009 explored their sense of safety and experiences in public spaces across the City, and views on what Council could do to improve these. It is apparent they base their assessment of a space as ‘public’ on their ‘use’ of a space alone or with friends, and where strangers may be present, rather than on a type of ownership of a space (public/ private). The findings of the survey are summarised according to the themes of safety, community attitudes towards young people being in public spaces, young people and authorities, young people’s views of what is needed, and understanding different young people’s experiences of public space.

Cyclic Imide and Open-Chain Amide Carboxylic Acid Derivatives from the Facile Reaction of cis-Cyclohexane-1,2-dicarboxylic Anhydride with Three Substituted Anilines

The structures of the compounds from the reaction of cis-cyclohexane-1,2-dicarboxylic anhydride with 4-chloroaniline [rac-N-(4-chlorophenyl)-2-carboxycycloclohexane-1-carboxamide] (1), 4-bromoaniline [2-(4-bromophenyl)-perhydroisoindolyl-1,3-dione] (2) and 3-hydroxy-4-carboxyaniline (5-aminosalicylic acid) [2-(3-hydroxy-4-carboxyphenyl)-perhydroisoindolyl-1,3-dione] (3) have been determined at 200 K. Crystals of the open-chain amide carboxylic acid 1 are orthorhombic, space group Pbcn, with unit cell dimensions a = 20.1753(10), b = 8.6267(4), c = 15.9940(9) Å, and Z = 8. Compounds 2 and 3 are cyclic imides, with 1 monoclinic having space group P21 and cell dimensions a = 11.5321(3), b = 6.7095(2), c = 17.2040(5) Å, β = 102.527(3)o. Compound 3 is orthorhombic with cell dimensions a = 6.4642(3), b = 12.8196(5), c = 16.4197(7) Å. Molecules of 1 form hydrogen-bonded cyclic dimers which are extended into a two-dimensional layered structure through amide-group associations: 3 forms into one-dimensional zigzag chains through carboxylic acid…imide O-atom hydrogen bonds, while compound 2 is essentially unassociated. With both cyclic imides 2 and 3, disorder is found which involves the presence of partial enantiomeric replacement of the cis-cyclohexane-1,2-substituted ring systems.

Orthogonality defect and reduced search-space size for solving integer least-squares problems

In the context of ambiguity resolution (AR) of Global Navigation Satellite Systems (GNSS), decorrelation among entries of an ambiguity vector, integer ambiguity search and ambiguity validations are three standard procedures for solving integer least-squares problems. This paper contributes to AR issues from three aspects. Firstly, the orthogonality defect is introduced as a new measure of the performance of ambiguity decorrelation methods, and compared with the decorrelation number and with the condition number which are currently used as the judging criterion to measure the correlation of ambiguity variance-covariance matrix. Numerically, the orthogonality defect demonstrates slightly better performance as a measure of the correlation between decorrelation impact and computational efficiency than the condition number measure. Secondly, the paper examines the relationship of the decorrelation number, the condition number, the orthogonality defect and the size of the ambiguity search space with the ambiguity search candidates and search nodes. The size of the ambiguity search space can be properly estimated if the ambiguity matrix is decorrelated well, which is shown to be a significant parameter in the ambiguity search progress. Thirdly, a new ambiguity resolution scheme is proposed to improve ambiguity search efficiency through the control of the size of the ambiguity search space. The new AR scheme combines the LAMBDA search and validation procedures together, which results in a much smaller size of the search space and higher computational efficiency while retaining the same AR validation outcomes. In fact, the new scheme can deal with the case there are only one candidate, while the existing search methods require at least two candidates. If there are more than one candidate, the new scheme turns to the usual ratio-test procedure. Experimental results indicate that this combined method can indeed improve ambiguity search efficiency for both the single constellation and dual constellations respectively, showing the potential for processing high dimension integer parameters in multi-GNSS environment.