Stability and convergence of an effective numerical method for the time-space fractional Fokker-Planck equation with a nonlinear source term

Fractional Fokker-Planck equations (FFPEs) have gained much interest recently for describing transport dynamics in complex systems that are governed by anomalous diffusion and nonexponential relaxation patterns. However, effective numerical methods and analytic techniques for the FFPE are still in their embryonic state. In this paper, we consider a class of time-space fractional Fokker-Planck equations with a nonlinear source term (TSFFPE-NST), which involve the Caputo time fractional derivative (CTFD) of order α ∈ (0, 1) and the symmetric Riesz space fractional derivative (RSFD) of order μ ∈ (1, 2). Approximating the CTFD and RSFD using the L1-algorithm and shifted Grunwald method, respectively, a computationally effective numerical method is presented to solve the TSFFPE-NST. The stability and convergence of the proposed numerical method are investigated. Finally, numerical experiments are carried out to support the theoretical claims.

Computationally efficient numerical methods for time- and space-fractional Fokker–Planck equations

Fractional Fokker–Planck equations have been used to model several physical situations that present anomalous diffusion. In this paper, a class of time- and space-fractional Fokker–Planck equations (TSFFPE), which involve the Riemann–Liouville time-fractional derivative of order 1-α (α(0, 1)) and the Riesz space-fractional derivative (RSFD) of order μ(1, 2), are considered. The solution of TSFFPE is important for describing the competition between subdiffusion and Lévy flights. However, effective numerical methods for solving TSFFPE are still in their infancy. We present three computationally efficient numerical methods to deal with the RSFD, and approximate the Riemann–Liouville time-fractional derivative using the Grünwald method. The TSFFPE is then transformed into a system of ordinary differential equations (ODE), which is solved by the fractional implicit trapezoidal method (FITM). Finally, numerical results are given to demonstrate the effectiveness of these methods. These techniques can also be applied to solve other types of fractional partial differential equations.

Analytical and numerical solutions for the time and space-symmetric fractional diffusion equation

We consider a time and space-symmetric fractional diffusion equation (TSS-FDE) under homogeneous Dirichlet conditions and homogeneous Neumann conditions. The TSS-FDE is obtained from the standard diffusion equation by replacing the first-order time derivative by the Caputo fractional derivative and the second order space derivative by the symmetric fractional derivative. Firstly, a method of separating variables is used to express the analytical solution of the tss-fde in terms of the Mittag–Leffler function. Secondly, we propose two numerical methods to approximate the Caputo time fractional derivative, namely, the finite difference method and the Laplace transform method. The symmetric space fractional derivative is approximated using the matrix transform method. Finally, numerical results are presented to demonstrate the effectiveness of the numerical methods and to confirm the theoretical claims.

Some problems with a behavioristic account of early group pretense

In normal child development, both individual and group pretense first emerges at approximately two years of age. The metarepresentational account of pretense holds that children already have the concept PRETEND when they first engage in early group pretense. A behavioristic account suggests that early group pretense is analogous to early beliefs or desires and thus require no mental state concepts. I argue that a behavioral account does not explain the actual behavior observed in children and it cannot explain how children come to understand that a specific action is one of pretense versus one of belief. I conclude that a mentalistic explanation of pretense best explains the behavior under consideration.

Group segmentation during object tracking using optical flow discontinuities

Within a surveillance video, occlusions are commonplace, and accurately resolving these occlusions is key when seeking to accurately track objects. The challenge of accurately segmenting objects is further complicated by the fact that within many real-world surveillance environments, the objects appear very similar. For example, footage of pedestrians in a city environment will consist of many people wearing dark suits. In this paper, we propose a novel technique to segment groups and resolve occlusions using optical flow discontinuities. We demonstrate that the ratio of continuous to discontinuous pixels within a region can be used to locate the overlapping edges, and incorporate this into an object tracking framework. Results on a portion of the ETISEO database show that the proposed algorithm results in improved tracking performance overall, and improved tracking within occlusions.

Civic engagement and associationalism : the impact of group membership scope versus intensity of participation

The relationship between participation in civic and political activities and membership of voluntary associations is now well established. What is less clear is the relative impacts of how much time people spend on group activities (associational intensity), and the number and type of groups that individuals are involved with (associational scope). Does it matter in terms of civic engagement, for example, whether one is a member of a quilting-circle or trade union? Does it matter whether association ‘membership’ is simply an annual payment or a major commitment of time and energy? In this article, we use a large survey to explore these questions empirically by focusing on the membership patterns and civic engagement practices of 4,001 citizens drawn from eight suburbs across Greater Melbourne, Australia. Our findings indicate that, while associational intensity is positively related to civic engagement, associational scope (the number of group memberships per person), is a more influential determinant of the level of civic and political participation. The results also suggest that while all forms of associationalism are important in terms of fostering greater levels of civic activity, not all forms have the same impact.

Optimism about safety and group-serving interpretations of safety among pedestrians and cyclists in relation to road use in general and under low light conditions

Drivers are known to be optimistic about their risk of crash involvement, believing that they are less likely to be involved in a crash than other drivers. However, little comparative research has been conducted among other road users. In addition, optimism about crash risk is conceptualised as applying only to an individual’s assessment of his or her personal risk of crash involvement. The possibility that the self-serving nature of optimism about safety might be generalised to the group-level as a cyclist or a pedestrian, i.e., becoming group-serving rather than self-serving, has been overlooked in relation to road safety. This study analysed a subset of data collected as part of a larger research project on the visibility of pedestrians, cyclists and road workers, focusing on a set of questionnaire items administered to 406 pedestrians, 838 cyclists and 622 drivers. The items related to safety in various scenarios involving drivers, pedestrians and cyclists, allowing predictions to be derived about group differences in agreement with items based on the assumption that the results would exhibit group-serving bias. Analysis of the responses indicated that specific hypotheses about group-serving interpretations of safety and responsibility were supported in 22 of the 26 comparisons. When the nine comparisons relevant to low lighting conditions were considered separately, seven were found to be supported. The findings of the research have implications for public education and for the likely acceptance of messages which are inconsistent with current assumptions and expectations of pedestrians and cyclists. They also suggest that research into group-serving interpretations of safety, even for temporary roles rather than enduring groups, could be fruitful. Further, there is an implication that gains in safety can be made by better educating road users about the limitations of their visibility and the ramifications of this for their own road safety, particularly in low light.

David Hicks and Cathie Holden (eds.) (2007). Teaching The Global Dimension. London: Routledge, Taylor & Francis Group, 212 pages.

Teaching The Global Dimension (2007) is intended for primary and secondary teachers, pre-service teachers and educators interested in fostering global concerns in the education system. It aims at linking theory and practice and is structured as follows. Part 1, the global dimension, proposes an educational framework for understanding global concerns. Individual chapters in this section deal with some educational responses to global issues and the ways in which young people might become, in Hick’s terms, more “world-minded”. In the first two chapters, Hicks presents first, some educational responses to global issues that have emerged in recent decades, and second, an outline of the evolution of global education as a specific field. As with all the chapters in this book, most of the examples are drawn from the United Kingdom. Young people’s concerns, student teachers’ views and the teaching of controversial issues, comprise the other chapters in this section. Taken collectively, the chapters in Part 2 articulate the conceptual framework for developing, teaching and evaluating a global dimension across the curriculum. Individual chapters in this section, written by a range of authors, explore eight key concepts considered necessary to underpin appropriate learning experiences in the classroom. These are conflict, social justice, values and perceptions, sustainability, interdependence, human rights, diversity and citizenship. These chapters are engaging and well structured. Their common format consists of a succinct introduction, reference to positive action for change, and examples of recent effective classroom practice. Two chapters comprise the final section of this book and suggest different ways in which the global dimension can be achieved in the primary and the secondary classroom.

Effect of group formation on performance, task management and social loafing

Engineering graduates of today are required to adapt to a rapidly changing work environment. In particular, they are expected to demonstrate enhanced capabilities in both mono-disciplinary and multi-disciplinary teamwork environments. Engineering education needs, as a result, to further focus on developing group work capabilities amongst engineering graduates. Over the last two years, the authors trialed various group work strategies across two engineering disciplines. In particular, the effect of group formation on students' performance, task management, and social loafing was analyzed. A recently developed online teamwork management tool, Teamworker, was used to collect students' experience of the group work. Analysis showed that students who were allowed to freely allocate to any group were less likely to report loafing from other team members, than students who were pre-allocated to a group. It also showed that performance was more affected by the presence or absence of a leader in pre-allocated rather than free-allocated groups.

Strategies to engage engineering students in group project work

Project focused group work is significant in developing social and personal skills as well as extending the ability to identify, formulate and solve engineering problems. As a result of increasing undergraduate class sizes, along with the requirement for many students to work part-time, group projects, peer and collaborative learning are seen as a fundamental part of engineering education. Group formation, connection to learning objectives and fairness of assessment has been widely reported as major issues that leave students dissatisfied with group project based units. Several strategies were trialled including a study of formation of groups by different methods across two engineering disciplines over the past 2 years. Other strategies involved a more structured approach to assessment practices of civil and electrical engineering disciplines design units. A confidential online teamwork management tool was used to collect and collate student self and peer assessment ratings and used for both formative feedback as well as assessment purposes. Student satisfaction and overall academic results in these subjects have improved since the introduction of these interventions. Both student and staff feedback highlight this approach as enhancing student engagement and satisfaction, improved student understanding of group roles, reducing number of dysfunctional groups whilst requiring less commitment of academic resources.

Numerical evaluation of pollutant dispersion at a toll plaza based on system dynamics and Computational Fluid Dynamics models

The health of tollbooth workers is seriously threatened by long-term exposure to polluted air from vehicle exhausts. Using traffic data collected at a toll plaza, vehicle movements were simulated by a system dynamics model with different traffic volumes and toll collection procedures. This allowed the average travel time of vehicles to be calculated. A three-dimension Computational Fluid Dynamics (CFD) model was used with a k–ε turbulence model to simulate pollutant dispersion at the toll plaza for different traffic volumes and toll collection procedures. It was shown that pollutant concentration around tollbooths increases as traffic volume increases. Whether traffic volume is low or high (1500 vehicles/h or 2500 vehicles/h), pollutant concentration decreases if electronic toll collection (ETC) is adopted. In addition, pollutant concentration around tollbooths decreases as the proportion of ETC-equipped vehicles increases. However, if the proportion of ETC-equipped vehicles is very low and the traffic volume is not heavy, then pollutant concentration increases as the number of ETC lanes increases.

Executive function effects and numerical development in children : behavioural and ERP evidence from a numerical Stroop paradigm

Most research on numerical development in children is behavioural, focusing on accuracy and response time in different problem formats. However, Temple and Posner (1998) used ERPs and the numerical distance task with 5-year-olds to show that the development of numerical representations is difficult to disentangle from the development of the executive components of response organization and execution. Here we use the numerical Stroop paradigm (NSP) and ERPs to study possible executive interference in numerical processing tasks in 6–8-year-old children. In the NSP, the numerical magnitude of the digits is task-relevant and the physical size of the digits is task-irrelevant. We show that younger children are highly susceptible to interference from irrelevant physical information such as digit size, but that access to the numerical representation is almost as fast in young children as in adults. We argue that the developmental trajectories for executive function and numerical processing may act together to determine numerical development in young children.

Lie group symmetry analysis for granular media stress equations

The Airy stress function, although frequently employed in classical linear elasticity, does not receive similar usage for granular media problems. For plane strain quasi-static deformations of a cohesionless Coulomb–Mohr granular solid, a single nonlinear partial differential equation is formulated for the Airy stress function by combining the equilibrium equations with the yield condition. This has certain advantages from the usual approach, in which two stress invariants and a stress angle are introduced, and a system of two partial differential equations is needed to describe the flow. In the present study, the symmetry analysis of differential equations is utilised for our single partial differential equation, and by computing an optimal system of one-dimensional Lie algebras, a complete set of group-invariant solutions is derived. By this it is meant that any group-invariant solution of the governing partial differential equation (provided it can be derived via the classical symmetries method) may be obtained as a member of this set by a suitable group transformation. For general values of the parameters (angle of internal friction and gravity g) it is found there are three distinct classes of solutions which correspond to granular flows considered previously in the literature. For the two limiting cases of high angle of internal friction and zero gravity, the governing partial differential equation admit larger families of Lie point symmetries, and from these symmetries, further solutions are derived, many of which are new. Furthermore, the majority of these solutions are exact, which is rare for granular flow, especially in the case of gravity driven flows.

Atomistic numerical investigation of single-crystal copper nanowire with surface defects

Based on the embedded atom method (EAM), a molecular dynamics (MD) simulation is performed to study the single-crystal copper nanowire with surface defects through tension. The tension simulations for nanowire without defect are first carried out under different temperatures, strain rates and time steps and then surface defect effects for nanowire are investigated. The stress-strain curves obtained by the MD simulations of various strain rates show a rate below 1 x 10(9) s-1 will exert less effect on the yield strength and yield point, and the Young's modulus is independent of strain rate. a time step below 5 fs is recommend for the atomic model during the MD simulation. It is observed that high temperature leads to low Young's modulus, as well as the yield strength. The surface defects on nanowires are systematically studied in considering different defect orientations. It is found that the surface defect serves as a dislocation source, and the yield strength shows 34.20% decresse with 45 degree surface defect. Both yield strength and yield point are significantly influenced by the surface defects, except the Young's modulus.