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.

A simultaneous equations model of crash frequency by collision type for rural intersections

Safety at roadway intersections is of significant interest to transportation professionals due to the large number of intersections in transportation networks, the complexity of traffic movements at these locations that leads to large numbers of conflicts, and the wide variety of geometric and operational features that define them. A variety of collision types including head-on, sideswipe, rear-end, and angle crashes occur at intersections. While intersection crash totals may not reveal a site deficiency, over exposure of a specific crash type may reveal otherwise undetected deficiencies. Thus, there is a need to be able to model the expected frequency of crashes by collision type at intersections to enable the detection of problems and the implementation of effective design strategies and countermeasures. Statistically, it is important to consider modeling collision type frequencies simultaneously to account for the possibility of common unobserved factors affecting crash frequencies across crash types. In this paper, a simultaneous equations model of crash frequencies by collision type is developed and presented using crash data for rural intersections in Georgia. The model estimation results support the notion of the presence of significant common unobserved factors across crash types, although the impact of these factors on parameter estimates is found to be rather modest.

A parallel computation of power system equations

Streaming SIMD Extensions (SSE) is a unique feature embedded in the Pentium III and P4 classes of microprocessors. By fully exploiting SSE, parallel algorithms can be implemented on a standard personal computer and a theoretical speedup of four can be achieved. In this paper, we demonstrate the implementation of a parallel LU matrix decomposition algorithm for solving power systems network equations with SSE and discuss advantages and disadvantages of this approach.

SSE based parallel solution for power systems network equations

Streaming SIMD Extensions (SSE) is a unique feature embedded in the Pentium III class of microprocessors. By fully exploiting SSE, parallel algorithms can be implemented on a standard personal computer and a theoretical speedup of four can be achieved. In this paper, we demonstrate the implementation of a parallel LU matrix decomposition algorithm for solving power systems network equations with SSE and discuss advantages and disadvantages of this approach.

Comparing charge transport predictions for a ternary electrolyte using the Maxwell–Stefan and Nernst–Planck equations

In this work, we investigate and compare the Maxwell–Stefan and Nernst–Planck equations for modeling multicomponent charge transport in liquid electrolytes. Specifically, we consider charge transport in the Li+/I−/I3−/ACN ternary electrolyte originally found in dye-sensitized solar cells. We employ molecular dynamics simulations to obtain the Maxwell–Stefan diffusivities for this electrolyte. These simulated diffusion coefficients are used in a multicomponent charge transport model based on the Maxwell– Stefan equations, and this is compared to a Nernst–Planck based model which employs binary diffusion coefficients sourced from the literature. We show that significant differences between the electrolyte concentrations at electrode interfaces, as predicted by the Maxwell–Stefan and Nernst–Planck models, can occur. We find that these differences are driven by a pressure term that appears in the Maxwell–Stefan equations. We also investigate what effects the Maxwell–Stefan diffusivities have on the simulated charge transport. By incorporating binary diffusivities found in the literature into the Maxwell–Stefan framework, we show that the simulated transient concentration profiles depend on the diffusivities; however, the simulated equilibrium profiles remain unaffected.

Identification of factors contributing to recurrence of venous leg ulcers

Background and Significance Venous leg ulcers are a significant cause of chronic ill-health for 1–3% of those aged over 60 years, increasing in incidence with age. The condition is difficult and costly to heal, consuming 1–2.5% of total health budgets in developed countries and up to 50% of community nursing time. Unfortunately after healing, there is a recurrence rate of 60 to 70%, frequently within the first 12 months after heaing. Although some risk factors associated with higher recurrence rates have been identified (e.g. prolonged ulcer duration, deep vein thrombosis), in general there is limited evidence on treatments to effectively prevent recurrence. Patients are generally advised to undertake activities which aim to improve the impaired venous return (e.g. compression therapy, leg elevation, exercise). However, only compression therapy has some evidence to support its effectiveness in prevention and problems with adherence to this strategy are well documented. Aim The aim of this research was to identify factors associated with recurrence by determining relationships between recurrence and demographic factors, health, physical activity, psychosocial factors and self-care activities to prevent recurrence. Methods Two studies were undertaken: a retrospective study of participants diagnosed with a venous leg ulcer which healed 12 to 36 months prior to the study (n=122); and a prospective longitudinal study of participants recruited as their ulcer healed and data collected for 12 months following healing (n=80). Data were collected from medical records on demographics, medical history and ulcer history and treatments; and from self-report questionnaires on physical activity, nutrition, psychosocial measures, ulcer history, compression and other self-care activities. Follow-up data for the prospective study were collected every three months for 12 months after healing. For the retrospective study, a logistic regression model determined the independent influences of variables on recurrence. For the prospective study, median time to recurrence was calculated using the Kaplan-Meier method and a Cox proportional-hazards regression model was used to adjust for potential confounders and determine effects of preventive strategies and psychosocial factors on recurrence. Results In total, 68% of participants in the retrospective study and 44% of participants in the prospective study suffered a recurrence. After mutual adjustment for all variables in multivariable regression models, leg elevation, compression therapy, self efficacy and physical activity were found to be consistently related to recurrence in both studies. In the retrospective study, leg elevation, wearing Class 2 or 3 compression hosiery, the level of physical activity, cardiac disease and self efficacy scores remained significantly associated (p<0.05) with recurrence. The model was significant (p <0.001); with a R2 equivalent of 0.62. Examination of relationships between psychosocial factors and adherence to wearing compression hosiery found wearing compression hosiery was significantly positively associated with participants’ knowledge of the cause of their condition (p=0.002), higher self-efficacy scores (p=0.026) and lower depression scores (p=0.009). Analysis of data from the prospective study found there were 35 recurrences (44%) in the 12 months following healing and median time to recurrence was 27 weeks. After adjustment for potential confounders, a Cox proportional hazards regression model found that at least an hour/day of leg elevation, six or more days/week in Class 2 (20–25mmHg) or 3 (30–40mmHg) compression hosiery, higher social support scale scores and higher General Self-Efficacy scores remained significantly associated (p<0.05) with a lower risk of recurrence, while male gender and a history of DVT remained significant risk factors for recurrence. Overall the model was significant (p <0.001); with an R2 equivalent 0.72. Conclusions The high rates of recurrence found in the studies highlight the urgent need for further information in this area to support development of effective strategies for prevention. Overall, results indicate leg elevation, physical activity, compression hosiery and strategies to improve self-efficacy are likely to prevent recurrence. In addition, optimal management of depression and strategies to improve patient knowledge and self-efficacy may positively influence adherence to compression therapy. This research provides important information for development of strategies to prevent recurrence of venous leg ulcers, with the potential to improve health and decrease health care costs in this population.

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.

Processing load and the use of concrete representations and strategies for solving linear equations

This paper is a report of students' responses to instruction which was based on the use of concrete representations to solve linear equations. The sample consisted of 21 Grade 8 students from a middle-class suburban state secondary school with a reputation for high academic standards and innovative mathematics teaching. The students were interviewed before and after instruction. Interviews and classroom interactions were observed and videotaped. A qualitative analysis of the responses revealed that students did not use the materials in solving problems. The increased processing load caused by concrete representations is hypothesised as a reason.

Communicating assessment criteria is not sufficient for influencing students' approaches to assessment tasks – perspectives from a differential equations class

This report presents the findings of an exploratory study into the perceptions held by students regarding the use of criterion-referenced assessment in an undergraduate differential equations class. Students in the class were largely unaware of the concept of criterion referencing and of the various interpretations that this concept has among mathematics educators. Our primary goal was to investigate whether explicitly presenting assessment criteria to students was useful to them and guided them in responding to assessment tasks. Quantitative data and qualitative feedback from students indicates that while students found the criteria easy to understand and useful in informing them as to how they would be graded, the manner in which they actually approached the assessment activity was not altered as a result of the use of explicitly communicated grading criteria.

Relationships between preventive activities, psychosocial factors and recurrence of venous leg ulcers : a prospective study

Aim To identify relationships between preventive activities, psychosocial factors and leg ulcer recurrence in patients with chronic venous leg ulcers. Background Chronic venous leg ulcers are slow to heal and frequently recur, resulting in years of suffering and intensive use of health care resources. Methods A prospective longitudinal study was undertaken with a sample of 80 patients with a venous leg ulcer recruited when their ulcer healed. Data were collected from 2006–2009 from medical records on demographics, medical history and ulcer history; and from self-report questionnaires on physical activity, nutrition, preventive activities and psychosocial measures. Follow-up data were collected via questionnaires every three months for 12 months after healing. Median time to recurrence was calculated using the Kaplan-Meier method. A Cox proportional-hazards regression model was used to adjust for potential confounders and determine effects of preventive strategies and psychosocial factors on recurrence. Results: There were 35 recurrences in a sample of 80 participants. Median time to recurrence was 27 weeks. After adjustment for potential confounders, a Cox proportional hazards regression model found that at least an hour/day of leg elevation, six or more days/week in Class 2 (20–25mmHg) or 3 (30–40mmHg) compression hosiery, higher social support scale scores and higher General Self-Efficacy scores remained significantly associated (p<0.05) with a lower risk of recurrence, while male gender and a history of DVT remained significant risk factors for recurrence. Conclusion Results indicate that leg elevation, compression hosiery, high levels of self-efficacy and strong social support will help prevent recurrence.