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.

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.

Building Large Scale Traversability Maps Using Vehicle Experience

Traversability maps are a global spatial representation of the relative difficulty in driving through a local region. These maps support simple optimisation of robot paths and have been very popular in path planning techniques. Despite the popularity of these maps, the methods for generating global traversability maps have been limited to using a-priori information. This paper explores the construction of large scale traversability maps for a vehicle performing a repeated activity in a bounded working environment, such as a repeated delivery task.We evaluate the use of vehicle power consumption, longitudinal slip, lateral slip and vehicle orientation to classify the traversability and incorporate this into a map generated from sparse information.

Relating vehicle generated pollutants to urban stormwater quality

Rapid urbanisation and resulting continuous increase in traffic has been recognised as key factors in the contribution of increased pollutant loads to urban stormwater and in turn to receiving waters. Urbanisation primarily increases anthropogenic activities and the percentage of impervious surfaces in urban areas. These processes are collectively responsible for urban stormwater pollution. In this regard, urban traffic and land use related activities have been recognised as the primary pollutant sources. This is primarily due to the generation of a range of key pollutants such as solids, heavy metals and PAHs. Appropriate treatment system design is the most viable approach to mitigate stormwater pollution. However, limited understanding of the pollutant process and transport pathways constrains effective treatment design. This highlights necessity for the detailed understanding of traffic and other land use related pollutants processes and pathways in relation to urban stormwater pollution. This study has created new knowledge in relation to pollutant processes and transport pathways encompassing atmospheric pollutants, atmospheric deposition and build-up on ground surfaces of traffic generated key pollutants. The research study was primarily based on in-depth experimental investigations. This thesis describes the extensive knowledge created relating to the processes of atmospheric pollutant build-up, atmospheric deposition and road surface build-up and establishing their relationships as a chain of processes. The analysis of atmospheric deposition revealed that both traffic and land use related sources contribute total suspended particulate matter (TSP) to the atmosphere. Traffic sources become dominant during weekdays whereas land use related sources become dominant during weekends due to the reduction in traffic sources. The analysis further concluded that atmospheric TSP, polycyclic aromatic hydrocarbons (PAHs) and heavy metals (HMs) concentrations are highly influenced by total average daily heavy duty traffic, traffic congestion and the fraction of commercial and industrial land uses. A set of mathematical equation were developed to predict TSP, PAHs and HMs concentrations in the atmosphere based on the influential traffic and land use related parameters. Dry deposition samples were collected for different antecedent dry days and wet deposition samples were collected immediately after rainfall events. The dry deposition was found to increase with the antecedent dry days and consisted of relatively coarser particles (greater than 1.4 ìm) when compared to wet deposition. The wet deposition showed a strong affinity to rainfall depth, but was not related to the antecedent dry period. It was also found that smaller size particles (less than 1.4 ìm) travel much longer distances from the source and deposit mainly with the wet deposition. Pollutants in wet deposition are less sensitive to the source characteristics compared to dry deposition. Atmospheric deposition of HMs is not directly influenced by land use but rather by proximity to high emission sources such as highways. Therefore, it is important to consider atmospheric deposition as a key pollutant source to urban stormwater in the vicinity of these types of sources. Build-up was analysed for five different particle size fractions, namely, <1 ìm, 1-75 ìm, 75-150 ìm, 150-300 ìm and >300 ìm for solids, PAHs and HMs. The outcomes of the study indicated that PAHs and HMs in the <75 ìm size fraction are generated mainly by traffic related activities whereas the > 150 ìm size fraction is generated by both traffic and land use related sources. Atmospheric deposition is an important source for HMs build-up on roads, whereas the contribution of PAHs from atmospheric sources is limited. A comprehensive approach was developed to predict traffic and other land use related pollutants in urban stormwater based on traffic and other land use characteristics. This approach primarily included the development of a set of mathematical equations to predict traffic generated pollutants by linking traffic and land use characteristics to stormwater quality through mathematical modelling. The outcomes of this research will contribute to the design of appropriate treatment systems to safeguard urban receiving water quality for future traffic growth scenarios. The „real world. application of knowledge generated was demonstrated through mathematical modelling of solids in urban stormwater, accounting for the variability in traffic and land use characteristics.

Fusing loop detector and probe vehicle data to estimate travel time statistics on signalized urban networks

This article presents a methodology that integrates cumulative plots with probe vehicle data for estimation of travel time statistics (average, quartile) on urban networks. The integration reduces relative deviation among the cumulative plots so that the classical analytical procedure of defining the area between the plots as the total travel time can be applied. For quartile estimation, a slicing technique is proposed. The methodology is validated with real data from Lucerne, Switzerland and it is concluded that the travel time estimates from the proposed methodology are statistically equivalent to the observed values.