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.

Governors highway safety associations and transportation planning: Exploratory factor analysis and structural equation modeling

The Intermodal Surface Transportation Efficiency Act (ISTEA) of 1991 mandated the consideration of safety in the regional transportation planning process. As part of National Cooperative Highway Research Program Project 8-44, "Incorporating Safety into the Transportation Planning Process," we conducted a telephone survey to assess safety-related activities and expertise at Governors Highway Safety Associations (GHSAs), and GHSA relationships with metropolitan planning organizations (MPOs) and state departments of transportation (DOTs). The survey results were combined with statewide crash data to enable exploratory modeling of the relationship between GHSA policies and programs and statewide safety. The modeling objective was to illuminate current hurdles to ISTEA implementation, so that appropriate institutional, analytical, and personnel improvements can be made. The study revealed that coordination of transportation safety across DOTs, MPOs, GHSAs, and departments of public safety is generally beneficial to the implementation of safety. In addition, better coordination is characterized by more positive and constructive attitudes toward incorporating safety into planning.

Adapting to the work-life interface : the influence of individual differences, work and family on well-being, mental health and work engagement

Bronfenbrenner.s Bioecological Model, expressed as the developmental equation, D f PPCT, is the theoretical framework for two studies that bring together diverse strands of psychology to study the work-life interface of working adults. Occupational and organizational psychology is focused on the demands and resources of work and family, without emphasising the individual in detail. Health and personality psychology examine the individual but without emphasis on the individual.s work and family roles. The current research used Bronfenbrenner.s theoretical framework to combine individual differences, work and family to understand how these factors influence the working adult.s psychological functioning. Competent development has been defined as high well-being (measured as life satisfaction and psychological well-being) and high work engagement (as work vigour, work dedication and absorption in work) and as the absence of mental illness (as depression, anxiety and stress) and the absence of burnout (as emotional exhaustion, cynicism and professional efficacy). Study 1 and 2 were linked, with Study 1 as a cross-sectional survey and Study 2, a prospective panel study that followed on from the data used in Study1. Participants were recruited from a university and from a large public hospital to take part in a 3-wave, online study where they completed identical surveys at 3-4 month intervals (N = 470 at Time 1 and N = 198 at Time 3). In Study 1, hierarchical multiple regressions were used to assess the effects of individual differences (Block 1, e.g. dispositional optimism, coping self-efficacy, perceived control of time, humour), work and family variables (Block 2, e.g. affective commitment, skill discretion, work hours, children, marital status, family demands) and the work-life interface (Block 3, e.g. direction and quality of spillover between roles, work-life balance) on the outcomes. There were a mosaic of predictors of the outcomes with a group of seven that were the most frequent significant predictors and which represented the individual (dispositional optimism and coping self-efficacy), the workplace (skill discretion, affective commitment and job autonomy) and the work-life interface (negative work-to-family spillover and negative family-to-work spillover). Interestingly, gender and working hours were not important predictors. The effects of job social support, generally and for work-life issues, perceived control of time and egalitarian gender roles on the outcomes were mediated by negative work-to-family spillover, particularly for emotional exhaustion. Further, the effect of negative spillover on depression, anxiety and work engagement was moderated by the individual.s personal and workplace resources. Study 2 modelled the longitudinal relationships between the group of the seven most frequent predictors and the outcomes. Using a set of non-nested models, the relative influences of concurrent functioning, stability and change over time were assessed. The modelling began with models at Time 1, which formed the basis for confirmatory factor analysis (CFA) to establish the underlying relationships between the variables and calculate the composite variables for the longitudinal models. The CFAs were well fitting with few modifications to ensure good fit. However, using burnout and work engagement together required additional analyses to resolve poor fit, with one factor (representing a continuum from burnout to work engagement) being the only acceptable solution. Five different longitudinal models were investigated as the Well-Being, Mental Distress, Well-Being-Mental Health, Work Engagement and Integrated models using differing combinations of the outcomes. The best fitting model for each was a reciprocal model that was trimmed of trivial paths. The strongest paths were the synchronous correlations and the paths within variables over time. The reciprocal paths were more variable with weak to mild effects. There was evidence of gain and loss spirals between the variables over time, with a slight net gain in resources that may provide the mechanism for the accumulation of psychological advantage over a lifetime. The longitudinal models also showed that there are leverage points at which personal, psychological and managerial interventions can be targeted to bolster the individual and provide supportive workplace conditions that also minimise negative spillover. Bronfenbrenner.s developmental equation has been a useful framework for the current research, showing the importance of the person as central to the individual.s experience of the work-life interface. By taking control of their own life, the individual can craft a life path that is most suited to their own needs. Competent developmental outcomes were most likely where the person was optimistic and had high self-efficacy, worked in a job that they were attached to and which allowed them to use their talents and without too much negative spillover between their work and family domains. In this way, individuals had greater well-being, better mental health and greater work engagement at any one time and across time.

Rumination, post-traumatic growth, and distress : structural equation modelling with cancer survivors

Objective Theoretical models of post-traumatic growth (PTG) have been derived in the general trauma literature to describe the post-trauma experience that facilitates the perception of positive life changes. To develop a statistical model identifying factors that are associated with PTG, structural equation modelling (SEM) was used in the current study to assess the relationships between perception of diagnosis severity, rumination, social support, distress, and PTG. Method A statistical model of PTG was tested in a sample of participants diagnosed with a variety of cancers (N=313). Results An initial principal components analysis of the measure used to assess rumination revealed three components: intrusive rumination, deliberate rumination of benefits, and life purpose rumination. SEM results indicated that the model fit the data well and that 30% of the variance in PTG was explained by the variables. Trauma severity was directly related to distress, but not to PTG. Deliberately ruminating on benefits and social support were directly related to PTG. Life purpose rumination and intrusive rumination were associated with distress. Conclusions The model showed that in addition to having unique correlating factors, distress was not related to PTG, thereby providing support for the notion that these are discrete constructs in the post-diagnosis experience. The statistical model provides support that post-diagnosis experience is simultaneously shaped by positive and negative life changes and that one or the other outcome may be prevalent or may occur concurrently. As such, an implication for practice is the need for supportive care that is holistic in nature.

An advanced implicit meshless approach for fractional partial differential equation in computational mechanics

Recently, the numerical modelling and simulation for fractional partial differential equations (FPDE), which have been found with widely applications in modern engineering and sciences, are attracting increased attentions. The current dominant numerical method for modelling of FPDE is the explicit Finite Difference Method (FDM), which is based on a pre-defined grid leading to inherited issues or shortcomings. This paper aims to develop an implicit meshless approach based on the radial basis functions (RBF) for numerical simulation of time fractional diffusion equations. The discrete system of equations is obtained by using the RBF meshless shape functions and the strong-forms. The stability and convergence of this meshless approach are then discussed and theoretically proven. Several numerical examples with different problem domains are used to validate and investigate accuracy and efficiency of the newly developed meshless formulation. The results obtained by the meshless formations are also compared with those obtained by FDM in terms of their accuracy and efficiency. It is concluded that the present meshless formulation is very effective for the modelling and simulation for FPDE.

Numerical simulation for the variable-order Galilei invariant advection diffusion equation with a nonlinear source term

In this paper, we consider the variable-order Galilei advection diffusion equation with a nonlinear source term. A numerical scheme with first order temporal accuracy and second order spatial accuracy is developed to simulate the equation. The stability and convergence of the numerical scheme are analyzed. Besides, another numerical scheme for improving temporal accuracy is also developed. Finally, some numerical examples are given and the results demonstrate the effectiveness of theoretical analysis. Keywords: The variable-order Galilei invariant advection diffusion equation with a nonlinear source term; The variable-order Riemann–Liouville fractional partial derivative; Stability; Convergence; Numerical scheme improving temporal accuracy