Numerical approximation for a variable-order nonlinear reaction–subdiffusion equation

Fractional reaction–subdiffusion equations are widely used in recent years to simulate physical phenomena. In this paper, we consider a variable-order nonlinear reaction–subdiffusion equation. A numerical approximation method is proposed to solve the equation. Its convergence and stability are analyzed by Fourier analysis. By means of the technique for improving temporal accuracy, we also propose an improved numerical approximation. Finally, the effectiveness of the theoretical results is demonstrated by numerical examples.

The analytical solution and numerical solution of the fractional diffusion-wave equation with damping

Fractional partial differential equations have been applied to many problems in physics, finance, and engineering. Numerical methods and error estimates of these equations are currently a very active area of research. In this paper we consider a fractional diffusionwave equation with damping. We derive the analytical solution for the equation using the method of separation of variables. An implicit difference approximation is constructed. Stability and convergence are proved by the energy method. Finally, two numerical examples are presented to show the effectiveness of this approximation.

A computationally effective alternating direction method for the space and time fractional Bloch–Torrey equation in 3-D

The space and time fractional Bloch–Torrey equation (ST-FBTE) has been used to study anomalous diffusion in the human brain. Numerical methods for solving ST-FBTE in three-dimensions are computationally demanding. In this paper, we propose a computationally effective fractional alternating direction method (FADM) to overcome this problem. We consider ST-FBTE on a finite domain where the time and space derivatives are replaced by the Caputo–Djrbashian and the sequential Riesz fractional derivatives, respectively. The stability and convergence properties of the FADM are discussed. Finally, some numerical results for ST-FBTE are given to confirm our theoretical findings.

Examining the influence of participant performance factors on contractor satisfaction : a structural equation model

Participant performance is critical to the success of projects. At the same time, enhancing the satisfaction of participants not only helps in problem solving but also improves their motivation and cooperation. However, previous research related to participant satisfaction is primarily concerned with clients and customers and relatively little attention has been paid to contractors. This paper investigates how the performance of project participants affects contractor project satisfaction in terms of the client's clarity of objectives (OC) and promptness of payments (PP), designer carefulness (DC), construction risk management (RM), the effectiveness their contribution (EW) and mutual respect and trust (RT). With 125 valid responses from contractors in Malaysia, a contractor satisfaction model is developed based on structural equation modelling. The results demonstrate the necessity for dividing abstract satisfaction into two dimensions, comprising economic-related satisfaction (ES) and production-related satisfaction (PS), with DC, OC, PP and RM having significant effects on ES, while DC, OC, EW and RM influence PS. In addition, the model tests the indirect effects of these performance variables on ES and PS. In particular, OC indirectly affects ES and PS through mediation of RM and DC respectively. The results also provide opportunities for improving contractor satisfaction and supplementing the contractor selection criteria for clients.

A finite volume method for solving the two-sided time-space fractional advection-dispersion equation

We present a finite volume method to solve the time-space two-sided fractional advection-dispersion equation on a one-dimensional domain. The spatial discretisation employs fractionally-shifted Grünwald formulas to discretise the Riemann-Liouville fractional derivatives at control volume faces in terms of function values at the nodes. We demonstrate how the finite volume formulation provides a natural, convenient and accurate means of discretising this equation in conservative form, compared to using a conventional finite difference approach. Results of numerical experiments are presented to demonstrate the effectiveness of the approach.

A comparison of finite difference and finite volume methods for solving the space-fractional advection-dispersion equation with variable coefficients

Transport processes within heterogeneous media may exhibit non- classical diffusion or dispersion which is not adequately described by the classical theory of Brownian motion and Fick’s law. We consider a space-fractional advection-dispersion equation based on a fractional Fick’s law. Zhang et al. [Water Resources Research, 43(5)(2007)] considered such an equation with variable coefficients, which they dis- cretised using the finite difference method proposed by Meerschaert and Tadjeran [Journal of Computational and Applied Mathematics, 172(1):65-77 (2004)]. For this method the presence of variable coef- ficients necessitates applying the product rule before discretising the Riemann–Liouville fractional derivatives using standard and shifted Gru ̈nwald formulas, depending on the fractional order. As an alternative, we propose using a finite volume method that deals directly with the equation in conservative form. Fractionally-shifted Gru ̈nwald formulas are used to discretise the Riemann–Liouville fractional derivatives at control volume faces, eliminating the need for product rule expansions. We compare the two methods for several case studies, highlighting the convenience of the finite volume approach.

A derivation of high-frequency asymptotic values of 3D added mass and damping based on properties of the Cummins' equation

This brief paper provides a novel derivation of the known asymptotic values of three-dimensional (3D) added mass and damping of marine structures in waves. The derivation is based on the properties of the convolution terms in the Cummins's Equation as derived by Ogilvie. The new derivation is simple and no approximations or series expansions are made. The results follow directly from the relative degree and low-frequency asymptotic properties of the rational representation of the convolution terms in the frequency domain. As an application, the extrapolation of damping values at high frequencies for the computation of retardation functions is also discussed.

A meshfree-based local Galerkin method with condensation of degree of freedom for elastic dynamic analysis

Condensation technique of degree of freedom is first proposed to improve the computational efficiency of meshfree method with Galerkin weak form for elastic dynamic analysis. In the present method, scattered nodes without connectivity are divided into several subsets by cells with arbitrary shape. Local discrete equation is established over each cell by using moving Kriging interpolation, in which the nodes that located in the cell are used for approximation. Then local discrete equations can be simplified by condensation of degree of freedom, which transfers equations of inner nodes to equations of boundary nodes based on cells. The global dynamic system equations are obtained by assembling all local discrete equations and are solved by using the standard implicit Newmark’s time integration scheme. In the scheme of present method, the calculation of each cell is carried out by meshfree method, and local search is implemented in interpolation. Numerical examples show that the present method has high computational efficiency and good accuracy in solving elastic dynamic problems.

Predicting emotional exhaustion among haemodialysis nurses : a structural equation model using Kanter’s structural empowerment theory

Aim To test an explanatory model of the relationships between the nursing work environment, job satisfaction, job stress and emotional exhaustion for haemodialysis nurses, drawing on Kanter's theory of organizational empowerment. Background Understanding the organizational predictors of burnout (emotional exhaustion) in haemodialysis nurses is critical for staff retention and improving nurse and patient outcomes. Previous research has demonstrated high levels of emotional exhaustion among haemodialysis nurses, yet the relationships between nurses' work environment, job satisfaction, stress and emotional exhaustion in this population are poorly understood. Design A cross-sectional online survey. Methods 417 nurses working in haemodialysis units completed an online survey between October 2011–April 2012 using validated measures of the work environment, job satisfaction, job stress and emotional exhaustion. Results Overall, the structural equation model demonstrated adequate fit and we found partial support for the hypothesized relationships. Nurses' work environment had a direct positive effect on job satisfaction, explaining 88% of the variance. Greater job satisfaction, in turn, predicted lower job stress, explaining 82% of the variance. Job satisfaction also had an indirect effect on emotional exhaustion by mitigating job stress. However, job satisfaction did not have a direct effect on emotional exhaustion. Conclusion The work environment of haemodialysis nurses is pivotal to the development of job satisfaction. Nurses' job satisfaction also predicts their level of job stress and emotional exhaustion. Our findings suggest staff retention can be improved by creating empowering work environments that promote job satisfaction among haemodialysis nurses.

A finite volume scheme with preconditioned Lanczos method for two-dimensional space-fractional reaction–diffusion equations

Fractional differential equations have been increasingly used as a powerful tool to model the non-locality and spatial heterogeneity inherent in many real-world problems. However, a constant challenge faced by researchers in this area is the high computational expense of obtaining numerical solutions of these fractional models, owing to the non-local nature of fractional derivatives. In this paper, we introduce a finite volume scheme with preconditioned Lanczos method as an attractive and high-efficiency approach for solving two-dimensional space-fractional reaction–diffusion equations. The computational heart of this approach is the efficient computation of a matrix-function-vector product f(A)bf(A)b, where A A is the matrix representation of the Laplacian obtained from the finite volume method and is non-symmetric. A key aspect of our proposed approach is that the popular Lanczos method for symmetric matrices is applied to this non-symmetric problem, after a suitable transformation. Furthermore, the convergence of the Lanczos method is greatly improved by incorporating a preconditioner. Our approach is show-cased by solving the fractional Fisher equation including a validation of the solution and an analysis of the behaviour of the model.

Scaling laws for localised states in a nonlocal amplitude equation

It is well known that, although a uniform magnetic field inhibits the onset of small amplitude thermal convection in a layer of fluid heated from below, isolated convection cells may persist if the fluid motion within them is sufficiently vigorous to expel magnetic flux. Such fully nonlinear(‘‘convecton’’) solutions for magnetoconvection have been investigated by several authors. Here we explore a model amplitude equation describing this separation of a fluid layer into a vigorously convecting part and a magnetically-dominated part at rest. Our analysis elucidates the origin of the scaling laws observed numerically to form the boundaries in parameter space of the region of existence of these localised states, and importantly, for the lowest thermal forcing required to sustain them.