Computationally efficient methods for solving time-variable-order time-space fractional reaction-diffusion equation

Fractional differential equations are becoming more widely accepted as a powerful tool in modelling anomalous diffusion, which is exhibited by various materials and processes. Recently, researchers have suggested that rather than using constant order fractional operators, some processes are more accurately modelled using fractional orders that vary with time and/or space. In this paper we develop computationally efficient techniques for solving time-variable-order time-space fractional reaction-diffusion equations (tsfrde) using the finite difference scheme. We adopt the Coimbra variable order time fractional operator and variable order fractional Laplacian operator in space where both orders are functions of time. Because the fractional operator is nonlocal, it is challenging to efficiently deal with its long range dependence when using classical numerical techniques to solve such equations. The novelty of our method is that the numerical solution of the time-variable-order tsfrde is written in terms of a matrix function vector product at each time step. This product is approximated efficiently by the Lanczos method, which is a powerful iterative technique for approximating the action of a matrix function by projecting onto a Krylov subspace. Furthermore an adaptive preconditioner is constructed that dramatically reduces the size of the required Krylov subspaces and hence the overall computational cost. Numerical examples, including the variable-order fractional Fisher equation, are presented to demonstrate the accuracy and efficiency of the approach.

Quality of work life and turnover intention in primary healthcare organisations : a cross-sectional study of registered nurses in Saudi Arabia

Purpose: The purpose of this study was to improve the retention of primary healthcare (PHC) nurses through exploring and assessing their quality of work life (QWL) and turnover intention. Design and methods: A cross-sectional survey design was used in this study. Data were collected using a questionnaire comprising four sections (Brooks’ survey of Quality of Nursing Work Life [QNWL], Anticipated Turnover Intention, open-ended questions and demographic characteristics). A convenience sample was recruited from 143 PHC centres in Jazan, Saudi Arabia. A response rate of 87% (n = 508/585) was achieved. The SPSS v17 for Windows and NVivo 8 were used for analysis purposes. Procedures and tests used in this study to analyse the quantitative data were descriptive statistics, t-test, ANOVA, General Linear Model (GLM) univariate analysis, standard multiple regression, and hierarchical multiple regression. Qualitative data obtained from responses to the open-ended questions were analysed using the NVivo 8. Findings: Quantitative findings suggested that PHC nurses were dissatisfied with their work life. Respondents’ scores ranged between 45 and 218 (mean = 139.45), which is lower than the average total score on Brooks’ Survey (147). Major influencing factors were classified under four dimensions. First, work life/home life factors: unsuitable working hours, lack of facilities for nurses, inability to balance work with family needs and inadequacy of vacations’ policy. Second, work design factors: high workload, insufficient workforce numbers, lack of autonomy and undertaking many non-nursing tasks. Third, work context factors: management practices, lack of development opportunities, and inappropriate working environment in terms of the level of security, patient care supplies and unavailability of recreation room. Finally, work world factors: negative public image of nursing, and inadequate payment. More positively, nurses were notably satisfied with their co-workers. Conversely, 40.4% (n = 205) of the respondents indicated that they intended to leave their current employment. The relationships between QWL and demographic variables of gender, age, marital status, dependent children, dependent adults, nationality, ethnicity, nursing tenure, organisational tenure, positional tenure, and payment per month were significant (p < .05). The eta squared test for these demographics indicates a small to medium effect size of the variation in QWL scores. Using the GLM univariate analysis, education level was also significantly related to the QWL (p < .05). The relationships between turnover intention and demographic variables including gender, age, marital status, dependent children, education level, nursing tenure, organisational tenure, positional tenure, and payment per month were significant (p < .05). The eta squared test for these demographics indicates a small to moderate effect size of the variation in the turnover intention scores. Using the GLM univariate analysis, the dependent adults’ variable was also significantly related to turnover intention (p < .05). Turnover intention was significantly related to QWL. Using standard multiple regression, 26% of the variance in turnover intention was explained by the QWL F (4,491), 43.71, p < .001, with R² = .263. Further analysis using hierarchical multiple regression found that the total variance explained by the model as a whole (demographics and QWL) was 32.1%, F (17.433) = 12.04, p < .001. QWL explained an additional 19% of the variance in turnover intention, after controlling for demographic variables, R squared change =.19, F change (4, 433) = 30.190, p < .001. The work context variable makes the strongest unique contribution (-.387) to explain the turnover intention, followed by the work design dimension (-.112). The qualitative findings reaffirmed the quantitative findings in terms of QWL and turnover intention. However, the home life/work life and work world dimensions were of great important to both QWL and turnover intention. The qualitative findings revealed a number of new factors that were not included in the survey questionnaire. These included being away from family, lack of family support, social and cultural aspects, accommodation facilities, transportation, building and infrastructure of PHC, nature of work, job instability, privacy at work, patients and community, and distance between home and workplace. Conclusion: Creating and maintaining a healthy work life for PHC nurses is very important to improve their work satisfaction, reduce turnover, enhance productivity and improve nursing care outcomes. Improving these factors could lead to a higher QWL and increase retention rates and therefore reinforcing the stabilisation of the nursing workforce. Significance of the research: Many countries are examining strategies to attract and retain the health care workforce, particularly nurses. This study identified factors that influence the QWL of PHC nurses as well as their turnover intention. It also determined the significant relationship between QWL and turnover intention. In addition, the present study tested Brooks’ survey of QNWL on PHC nurses for the first time. The qualitative findings of this study revealed a number of new variables regarding QWL and turnover intention of PHC nurses. These variables could be used to improve current survey instruments or to develop new research surveys. The study findings could be also used to develop and appropriately implement plans to improve QWL. This may help to enhance the home and work environments of PHC nurses, improve individual and organisational performance, and increase nurses’ commitment. This study contributes to the existing body of research knowledge by presenting new data and findings from a different country and healthcare system. It is the first of its kind in Saudi Arabia, especially in the field of PHC. It has examined the relationship between QWL and turnover intention of PHC nurses for the first time using nursing instruments. The study also offers a fresh explanation (new framework) of the relationship between QWL and turnover intention among PHC nurses, which could be used or tested by researchers in other settings. Implications for further research: Review of the extant literature reveals little in-depth research on the PHC workforce, especially in terms of QWL and organisational turnover in developing countries. Further research is required to develop a QWL tool for PHC nurses, taking into consideration the findings of the current study along with the local culture. Moreover, the revised theoretical framework of the current study could be tested in further research in other regions, countries or healthcare systems in order to identify its ability to predict the level of PHC nurses’ QWL and their intention to leave. There is a need to conduct longitudinal research on PHC organisations to gain an in-depth understanding of the determents of and changes in QWL and turnover intention of PHC nurses at various points of time. An intervention study is required to improve QWL and retention among PHC nurses using the findings of the current study. This would help to assess the impact of such strategies on reducing turnover of PHC nurses. Focusing on the location of the current study, it would be valuable to conduct another study in five years’ time to examine the percentage of actual turnover among PHC nurses compared with the reported turnover intention in the current study. Further in-depth research would also be useful to assess the impact of the local culture on the perception of expatriate nurses towards their QWL and their turnover intention. A comparative study is required between PHC centres and hospitals as well as the public and private health sector agencies in terms of QWL and turnover intention of nursing personnel. Findings may differ from sector to sector according to variations in health systems, working environments and the case mix of patients.

The non-binding vote on executive pay : a review of the CLERP 9 reform

Directors’ and executives’ remuneration, including levels of pay, accountability and transparency, is controversial. Section 250R of the CLERP (Audit Reform & Disclosure) Act 2004 that was not greatly anticipated, requires the holding of a non-binding resolution on board remuneration at companies’ annual general meetings. The reform has been criticised on the basis that, inter alia, it blurs the respective roles of shareholders and directors. This article identifies possible motivations for the imposition of the non-binding resolution in Australia. These are evaluated with reference to sources of corporate governance policy and the current state of Australia’s relevant corporate governance structures. We speculate that the non-binding vote will not amount to a substantive addition to the corporate governance regime.

Job satisfaction and intention to leave among critical care nurses in Saudi Arabia

Aim The purpose of this study was to examine the relationship between registered nurses’ (RN) job satisfaction and their intention to leave critical care nursing in Saudi Arabia. Background Many studies have identified critical care areas as stressful work environments for nurses and have identified factors contributing to job satisfaction and staff retention. However, very little research has examined these relationships in the Saudi context. Design and Methods This study utilised an exploratory, cross-sectional survey design to examine the relationship between RN job satisfaction and intention to leave at King Abdul-Aziz University Hospital, Saudi Arabia. Respondents completed a self-administered survey including demographic items and validated measures of job satisfaction and intention to leave. A convenience sample of 182 RNs working in critical care areas during the data collection period were included. Results Regression analysis predicting RN intention to leave found that demographic variables including age, parental status and length of ICU experience, and three of the job satisfaction subscales including perceived workload, professional support and pay and prospects for promotion, were significantly associated with the outcome variable. Conclusion This study adds to the existing literature on the relationship between job satisfaction and intention to leave critical care areas among RNs working in Saudi Arabia. These findings point to the need for management and policy interventions targeting nurses’ workloads, professional support and pay and promotion in order to improve nurse retention.

Numerical simulation of a new two-dimensional variable-order fractional percolation equation in non-homogeneous porous media

Percolation flow problems are discussed in many research fields, such as seepage hydraulics, groundwater hydraulics, groundwater dynamics and fluid dynamics in porous media. Many physical processes appear to exhibit fractional-order behavior that may vary with time, or space, or space and time. The theory of pseudodifferential operators and equations has been used to deal with this situation. In this paper we use a fractional Darcys law with variable order Riemann-Liouville fractional derivatives, this leads to a new variable-order fractional percolation equation. In this paper, a new two-dimensional variable-order fractional percolation equation is considered. A new implicit numerical method and an alternating direct method for the two-dimensional variable-order fractional model is proposed. Consistency, stability and convergence of the implicit finite difference method are established. Finally, some numerical examples are given. The numerical results demonstrate the effectiveness of the methods. This technique can be used to simulate a three-dimensional variable-order fractional percolation equation.

Numerical analysis of the time variable fractional order mobile-immobile advection-dispersion model

Many physical processes exhibit fractional order behavior that varies with time or space. The continuum of order in the fractional calculus allows the order of the fractional operator to be considered as a variable. In this paper, we consider the time variable fractional order mobile-immobile advection-dispersion model. Numerical methods and analyses of stability and convergence for the fractional partial differential equations are quite limited and difficult to derive. This motivates us to develop efficient numerical methods as well as stability and convergence of the implicit numerical methods for the fractional order mobile immobile advection-dispersion model. In the paper, we use the Coimbra variable time fractional derivative which is more efficient from the numerical standpoint and is preferable for modeling dynamical systems. An implicit Euler approximation for the equation is proposed and then the stability of the approximation are investigated. As for the convergence of the numerical scheme we only consider a special case, i.e. the time fractional derivative is independent of time variable t. The case where the time fractional derivative depends both the time variable t and the space variable x will be considered in the future work. Finally, numerical examples are provided to show that the implicit Euler approximation is computationally efficient.

Numerical techniques for the variable order time fractional diffusion equation

In this paper we consider the variable order time fractional diffusion equation. We adopt the Coimbra variable order (VO) time fractional operator, which defines a consistent method for VO differentiation of physical variables. The Coimbra variable order fractional operator also can be viewed as a Caputo-type definition. Although this definition is the most appropriate definition having fundamental characteristics that are desirable for physical modeling, numerical methods for fractional partial differential equations using this definition have not yet appeared in the literature. Here an approximate scheme is first proposed. The stability, convergence and solvability of this numerical scheme are discussed via the technique of Fourier analysis. Numerical examples are provided to show that the numerical method is computationally efficient. Crown Copyright © 2012 Published by Elsevier Inc. All rights reserved.

Numerical methods for solving a two-dimensional variable-order anomalous subdiffusion equation

Anomalous subdiffusion equations have in recent years received much attention. In this paper, we consider a two-dimensional variable-order anomalous subdiffusion equation. Two numerical methods (the implicit and explicit methods) are developed to solve the equation. Their stability, convergence and solvability are investigated by the Fourier method. Moreover, the effectiveness of our theoretical analysis is demonstrated by some numerical examples. © 2011 American Mathematical Society.

Building prosperous knowledge cities : policies, plans and metrics

This unique book reveals the procedural aspects of knowledge-based urban planning, development and assessment. Concentrating on major knowledge city building processes, and providing state-of-the-art experiences and perspectives, this important compendium explores innovative models, approaches and lessons learnt from a number of key case studies across the world. Many cities worldwide, in order to brand themselves as knowledge cities, have undergone major transformations in the 21st century. This book provides a thorough understanding of these transformations and the key issues in building prosperous knowledge cities by focusing particularly on the policy-making, planning process and performance assessment aspects. The contributors reveal theoretical and conceptual foundations of knowledge cities and their development approach of knowledge-based urban development. They present best-practice examples from a number of key case studies across the globe. This important book provides readers with a thorough understanding of the key issues in planning and developing prosperous knowledge cities of the knowledge economy era, which will prove invaluable to national, state/regional and city governments’ planning and development departments. Academics, postgraduate and undergraduate students of regional and urban studies will also find this path-breaking book an intriguing read.

Monte Carlo evaluation of collapsed-cone convolution calculations in head and neck radiotherapy treatment plans

The presence of air and bone interfaces makes the dose distribution for head and neck cancer treatments difficult to accurately predict. This study compared planning system dose calculations using the collapsed-cone convolution algorithm with EGSnrcMonte Carlo simulation results obtained using the Monte Carlo DICOMToolKit software, for one oropharynx, two paranasal sinus and three nodal treatment plans. The difference between median doses obtained from the treatment planning and Monte Carlo calculations was found to be greatest in two bilateral treatments: 4.8%for a retropharyngeal node irradiation and 6.7% for an ethmoid paranasal sinus treatment. These deviations in median dose were smaller for two unilateral treatments: 0.8% for an infraclavicular node irradiation and 2.8% for a cervical node treatment. Examination of isodose distributions indicated that the largest deviations between Monte Carlo simulation and collapsed-cone convolution calculations were seen in the bilateral treatments, where the increase in calculated dose beyond air cavities was most significant.

A variable-stepsize Jacobian-free exponential integrator for simulating transport in heterogeneous porous media : application to wood drying

A Jacobian-free variable-stepsize method is developed for the numerical integration of the large, stiff systems of differential equations encountered when simulating transport in heterogeneous porous media. Our method utilises the exponential Rosenbrock-Euler method, which is explicit in nature and requires a matrix-vector product involving the exponential of the Jacobian matrix at each step of the integration process. These products can be approximated using Krylov subspace methods, which permit a large integration stepsize to be utilised without having to precondition the iterations. This means that our method is truly "Jacobian-free" - the Jacobian need never be formed or factored during the simulation. We assess the performance of the new algorithm for simulating the drying of softwood. Numerical experiments conducted for both low and high temperature drying demonstrates that the new approach outperforms (in terms of accuracy and efficiency) existing simulation codes that utilise the backward Euler method via a preconditioned Newton-Krylov strategy.

Effects of speeding and headway related variable message signs on driver behaviour and attitudes

This research project examined objective measures of driver behaviour and road users' perceptions on the usefulness and effectiveness of three specific VMS (Variable Message Signs) interventions to improve speeding and headway behaviours. The interventions addressed speeding behaviour alone (intervention 1), headway behaviour alone (intervention 2) and a combination of speeding and headway behaviour (intervention 3). Six VMS were installed along a segment of the Bruce Highway, with a series of three signs for each of the northbound and southbound traffic. Speeds and headway distances were measured with loop detectors installed before and after each VMS. Messages addressing speeding and headway were devised for display on the VMS. A driver could receive a message if they were detected as exceeding the posted speed limit (of 90km/hr) or if the distance to the vehicle in front was less than 2.0s. In addition to the on-road objective measurement of speeding and headway behaviours, the research project elicited self-reported responses to the speeding and headway messages from a sample of drivers via a community-based survey. The survey sought to examine the drivers' beliefs about the effectiveness of the signs and messages, and their views about the role, use, and effectiveness of VMS more generally.

A variable stepsize implementation for stochastic differential equations

Stochastic differential equations (SDEs) arise from physical systems where the parameters describing the system can only be estimated or are subject to noise. Much work has been done recently on developing higher order Runge-Kutta methods for solving SDEs numerically. Fixed stepsize implementations of numerical methods have limitations when, for example, the SDE being solved is stiff as this forces the stepsize to be very small. This paper presents a completely general variable stepsize implementation of an embedded Runge Kutta pair for solving SDEs numerically; in this implementation, there is no restriction on the value used for the stepsize, and it is demonstrated that the integration remains on the correct Brownian path.