Numerical algorithms for time-fractional subdiffusion equation with second-order accuracy

This article aims to fill in the gap of the second-order accurate schemes for the time-fractional subdiffusion equation with unconditional stability. Two fully discrete schemes are first proposed for the time-fractional subdiffusion equation with space discretized by finite element and time discretized by the fractional linear multistep methods. These two methods are unconditionally stable with maximum global convergence order of $O(\tau+h^{r+1})$ in the $L^2$ norm, where $\tau$ and $h$ are the step sizes in time and space, respectively, and $r$ is the degree of the piecewise polynomial space. The average convergence rates for the two methods in time are also investigated, which shows that the average convergence rates of the two methods are $O(\tau^{1.5}+h^{r+1})$. Furthermore, two improved algorithms are constrcted, they are also unconditionally stable and convergent of order $O(\tau^2+h^{r+1})$. Numerical examples are provided to verify the theoretical analysis. The comparisons between the present algorithms and the existing ones are included, which show that our numerical algorithms exhibit better performances than the known ones.

High order unconditionally stable difference schemes for the Riesz space-fractional telegraph equation

In this paper, a class of unconditionally stable difference schemes based on the Pad´e approximation is presented for the Riesz space-fractional telegraph equation. Firstly, we introduce a new variable to transform the original dfferential equation to an equivalent differential equation system. Then, we apply a second order fractional central difference scheme to discretise the Riesz space-fractional operator. Finally, we use (1, 1), (2, 2) and (3, 3) Pad´e approximations to give a fully discrete difference scheme for the resulting linear system of ordinary differential equations. Matrix analysis is used to show the unconditional stability of the proposed algorithms. Two examples with known exact solutions are chosen to assess the proposed difference schemes. Numerical results demonstrate that these schemes provide accurate and efficient methods for solving a space-fractional hyperbolic equation.

Numerical analysis for the time distributed-order and Riesz space fractional diffusions on bounded domains

Subdiffusion equations with distributed-order fractional derivatives describe some important physical phenomena. In this paper, we consider the time distributed-order and Riesz space fractional diffusions on bounded domains with Dirichlet boundary conditions. Here, the time derivative is defined as the distributed-order fractional derivative in the Caputo sense, and the space derivative is defined as the Riesz fractional derivative. First, we discretize the integral term in the time distributed-order and Riesz space fractional diffusions using numerical approximation. Then the given equation can be written as a multi-term time–space fractional diffusion. Secondly, we propose an implicit difference method for the multi-term time–space fractional diffusion. Thirdly, using mathematical induction, we prove the implicit difference method is unconditionally stable and convergent. Also, the solvability for our method is discussed. Finally, two numerical examples are given to show that the numerical results are in good agreement with our theoretical analysis.

A Novel High Order Space-Time Spectral Method for the Time Fractional Fokker--Planck Equation

The fractional Fokker-Planck equation is an important physical model for simulating anomalous diffusions with external forces. Because of the non-local property of the fractional derivative an interesting problem is to explore high accuracy numerical methods for fractional differential equations. In this paper, a space-time spectral method is presented for the numerical solution of the time fractional Fokker-Planck initial-boundary value problem. The proposed method employs the Jacobi polynomials for the temporal discretization and Fourier-like basis functions for the spatial discretization. Due to the diagonalizable trait of the Fourier-like basis functions, this leads to a reduced representation of the inner product in the Galerkin analysis. We prove that the time fractional Fokker-Planck equation attains the same approximation order as the time fractional diffusion equation developed in [23] by using the present method. That indicates an exponential decay may be achieved if the exact solution is sufficiently smooth. Finally, some numerical results are given to demonstrate the high order accuracy and efficiency of the new numerical scheme. The results show that the errors of the numerical solutions obtained by the space-time spectral method decay exponentially.

Generalized element load method for first- and second-order element solutions with element load effect

The finite element method in principle adaptively divides the continuous domain with complex geometry into discrete simple subdomain by using an approximate element function, and the continuous element loads are also converted into the nodal load by means of the traditional lumping and consistent load methods, which can standardise a plethora of element loads into a typical numerical procedure, but element load effect is restricted to the nodal solution. It in turn means the accurate continuous element solutions with the element load effects are merely restricted to element nodes discretely, and further limited to either displacement or force field depending on which type of approximate function is derived. On the other hand, the analytical stability functions can give the accurate continuous element solutions due to element loads. Unfortunately, the expressions of stability functions are very diverse and distinct when subjected to different element loads that deter the numerical routine for practical applications. To this end, this paper presents a displacement-based finite element function (generalised element load method) with a plethora of element load effects in the similar fashion that never be achieved by the stability function, as well as it can generate the continuous first- and second-order elastic displacement and force solutions along an element without loss of accuracy considerably as the analytical approach that never be achieved by neither the lumping nor consistent load methods. Hence, the salient and unique features of this paper (generalised element load method) embody its robustness, versatility and accuracy in continuous element solutions when subjected to the great diversity of transverse element loads.

Protecting Australia's children: A cross-jurisdictional review of domestic violence protection order legislation

Increasingly, domestic violence is being treated as a child protection issue, and children affected by domestic violence are recognised as experiencing a form of child abuse. Domestic violence protection order legislation – as a key legal response to domestic violence – may offer an important legal option for the protection of children affected by domestic violence. In this article, we consider the research that establishes domestic violence as a form of child abuse, and review the provisions of State and Territory domestic violence protection order legislation to assess whether they demonstrate an adequate focus on the protection of children.

Application of higher-order spectra for automated grading of diabetic maculopathy

Diabetic macular edema (DME) is one of the most common causes of visual loss among diabetes mellitus patients. Early detection and successive treatment may improve the visual acuity. DME is mainly graded into non-clinically significant macular edema (NCSME) and clinically significant macular edema according to the location of hard exudates in the macula region. DME can be identified by manual examination of fundus images. It is laborious and resource intensive. Hence, in this work, automated grading of DME is proposed using higher-order spectra (HOS) of Radon transform projections of the fundus images. We have used third-order cumulants and bispectrum magnitude, in this work, as features, and compared their performance. They can capture subtle changes in the fundus image. Spectral regression discriminant analysis (SRDA) reduces feature dimension, and minimum redundancy maximum relevance method is used to rank the significant SRDA components. Ranked features are fed to various supervised classifiers, viz. Naive Bayes, AdaBoost and support vector machine, to discriminate No DME, NCSME and clinically significant macular edema classes. The performance of our system is evaluated using the publicly available MESSIDOR dataset (300 images) and also verified with a local dataset (300 images). Our results show that HOS cumulants and bispectrum magnitude obtained an average accuracy of 95.56 and 94.39 % for MESSIDOR dataset and 95.93 and 93.33 % for local dataset, respectively.

Numerical treatment of a two-dimensional variable-order fractional nonlinear reaction-diffusion model

A two-dimensional variable-order fractional nonlinear reaction-diffusion model is considered. A second-order spatial accurate semi-implicit alternating direction method for a two-dimensional variable-order fractional nonlinear reaction-diffusion model is proposed. Stability and convergence of the semi-implicit alternating direct method are established. Finally, some numerical examples are given to support our theoretical analysis. These numerical techniques can be used to simulate a two-dimensional variable order fractional FitzHugh-Nagumo model in a rectangular domain. This type of model can be used to describe how electrical currents flow through the heart, controlling its contractions, and are used to ascertain the effects of certain drugs designed to treat arrhythmia.

Nonlinear analysis for the pre- and post-yield behaviour of a hybrid structure by a higher-order element with the refined plastic hinge approach

Efficient and accurate geometric and material nonlinear analysis of the structures under ultimate loads is a backbone to the success of integrated analysis and design, performance-based design approach and progressive collapse analysis. This paper presents the advanced computational technique of a higher-order element formulation with the refined plastic hinge approach which can evaluate the concrete and steel-concrete structure prone to the nonlinear material effects (i.e. gradual yielding, full plasticity, strain-hardening effect when subjected to the interaction between axial and bending actions, and load redistribution) as well as the nonlinear geometric effects (i.e. second-order P-d effect and P-D effect, its associate strength and stiffness degradation). Further, this paper also presents the cross-section analysis useful to formulate the refined plastic hinge approach.

Does size matter? Knowledge-based development of second-order city-regions in Finland

Achieving knowledge-based urban development (KBUD) profoundly depends on not only encouraging the development of economic activities, but also strengthening the societal, environmental and governance bases of city-regions. In recent years, a number of global city-regions have been investigated from the angle of this multidimensional perspective, which has provided a new comprehension in the development processes of primate city-regions. However, there is a knowledge gap in understanding how KBUD works in the second-order city-region (SOCR) context. This warrants more attention as SOCRs potentially help secure balanced development and territorial cohesion. This paper aims to empirically investigate KBUD performances of SOCRs in order to generate new insights. An assessment framework is utilised in the Finnish context, where the findings provide a nationally benchmarked snapshot of the degree of achievements of SOCRs based on numerous KBUD performance areas. The results shed light on the unique Finnish urban and regional development process, and provide lessons for other SOCRs.

Molecular orientational order of nitroxide radicals in liquid crystalline media

The orientational distribution of a set of stable nitroxide radicals in aligned liquid crystals 5CB (nematic) and 8CB (smectic A) was studied in detail by numerical simulation of EPR spectra. The order parameters up to the 10th rank were measured. The directions of the principal orientation axes of the radicals were determined. It was shown that the ordering of the probe molecules is controlled by their interaction with the matrix molecules more than the inherent geometry of the probes themselves. The rigid fused phenanthrene-based (A5) and 2-azaphenalene (A4) nitroxides as well as the rigid core elongated C11 and 5α-cholestane (CLS) nitroxides were found to be most sensitive to the orientation of the liquid crystal matrixes.

Numerical analysis of a new space-time variable fractional order advection-dispersion equation

Many physical processes appear to exhibit fractional order behavior that may vary with time and/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 a new space–time variable fractional order advection–dispersion equation on a finite domain. The equation is obtained from the standard advection–dispersion equation by replacing the first-order time derivative by Coimbra’s variable fractional derivative of order α(x)∈(0,1]α(x)∈(0,1], and the first-order and second-order space derivatives by the Riemann–Liouville derivatives of order γ(x,t)∈(0,1]γ(x,t)∈(0,1] and β(x,t)∈(1,2]β(x,t)∈(1,2], respectively. We propose an implicit Euler approximation for the equation and investigate the stability and convergence of the approximation. Finally, numerical examples are provided to show that the implicit Euler approximation is computationally efficient.

On the order of the fractional Laplacian in determining the spatio-temporal evolution of a space-fractional model of cardiac electrophysiology

Space-fractional operators have been used with success in a variety of practical applications to describe transport processes in media characterised by spatial connectivity properties and high structural heterogeneity altering the classical laws of diffusion. This study provides a systematic investigation of the spatio-temporal effects of a space-fractional model in cardiac electrophysiology. We consider a simplified model of electrical pulse propagation through cardiac tissue, namely the monodomain formulation of the Beeler-Reuter cell model on insulated tissue fibres, and obtain a space-fractional modification of the model by using the spectral definition of the one-dimensional continuous fractional Laplacian. The spectral decomposition of the fractional operator allows us to develop an efficient numerical method for the space-fractional problem. Particular attention is paid to the role played by the fractional operator in determining the solution behaviour and to the identification of crucial differences between the non-fractional and the fractional cases. We find a positive linear dependence of the depolarization peak height and a power law decay of notch and dome peak amplitudes for decreasing orders of the fractional operator. Furthermore, we establish a quadratic relationship in conduction velocity, and quantify the increasingly wider action potential foot and more pronounced dispersion of action potential duration, as the fractional order is decreased. A discussion of the physiological interpretation of the presented findings is made.

Evaluation of a workplace alcohol and other drug policy

BACKGROUND The current impetus for developing alcohol and/or other drugs (AODs) workplace policies in Australia is to reduce workplace AOD impairment, improve safety, and prevent AOD-related injury in the workplace. For these policies to be effective, they need to be informed by scientific evidence. Evidence to inform the development and implementation of effective workplace AOD policies is currently lacking. There does not currently appear to be conclusive evidence for the effectiveness of workplace AOD policies in reducing impairment and preventing AOD-related injury. There is also no apparent evidence regarding which factors facilitate or impede the success of an AOD policy, or whether, for example, unsuccessful policy outcomes were due to poor policy or merely poor implementation of the policy. It was the aim of this research to undertake a process, impact, and outcome evaluation of a workplace AOD policy, and to contribute to the body of knowledge on the development and implementation of effective workplace AOD policies. METHODS The research setting was a state-based power-generating industry in Australia between May 2008 and May 2010. Participants for the process evaluation study were individuals who were integral to either the development or the implementation of the workplace AOD policy, or both of these processes (key informants), and comprised the majority of individuals who were involved in the process of developing and/or implementing the workplace AOD policy. The sample represented the two main groups of interest—management and union delegates/employee representatives—from all three of the participating organisations. For the impact and outcome evaluation studies, the population included all employees from the three participating organisations, and participants were all employees who consented to participate in the study and who completed both the pre-and post-policy implementation questionnaires. Qualitative methods in the form of interviews with key stakeholders were used to evaluate the process of developing and implementing the workplace AOD policy. In order to evaluate the impact of the policy with regard to the risk factors for workplace AOD impairment, and the outcome of the policy in terms of reducing workplace AOD impairment, quantitative methods in the form of a non-randomised single group pre- and post-test design were used. Changes from Time 1 (pre) to Time 2 (post) in the risk factors for workplace AOD impairment, and changes in the behaviour of interest—(self-reported) workplace AOD impairment—were measured. An integration of the findings from the process, impact, and outcome evaluation studies was undertaken using a combination of qualitative and quantitative methods. RESULTS For the process evaluation study Study respondents indicated that their policy was developed in the context of comparable industries across Australia developing workplace AOD policies, and that this was mainly out of concern for the deleterious health and safety impacts of workplace AOD impairment. Results from the process evaluation study also indicated that in developing and implementing the workplace AOD policy, there were mainly ‗winners', in terms of health and safety in the workplace. While there were some components of the development and implementation of the policy that were better done than others, and the process was expensive and took a long time, there were, overall, few unanticipated consequences to implementing the policy and it was reported to be thorough and of a high standard. Findings also indicated that overall the policy was developed and implemented according to best-practice in that: consultation during the policy development phase (with all the main stakeholders) was extensive; the policy was comprehensive; there was universal application of the policy to all employees; changes in the workplace (with regard to the policy) were gradual; and, the policy was publicised appropriately. Furthermore, study participants' responses indicated that the role of an independent external expert, who was trusted by all stakeholders, was integral to the success of the policy. For the impact and outcome evaluation studies Notwithstanding the limitations of pre- and post-test study designs with regard to attributing cause to the intervention, the findings from the impact evaluation study indicated that following policy implementation, statistically significant positive changes with regard to workplace AOD impairment were recorded for the following variables (risk factors for workplace AOD impairment): Knowledge; Attitudes; Perceived Behavioural Control; Perceptions of the Certainty of being punished for coming to work impaired by AODs; Perceptions of the Swiftness of punishment for coming to work impaired by AODs; and Direct and Indirect Experience with Punishment Avoidance for workplace AOD impairment. There were, however, no statistically significant positive changes following policy implementation for Behavioural Intentions, Subjective Norms, and Perceptions of the Severity of punishment for workplace AOD impairment. With regard to the outcome evaluation, there was a statistically significant reduction in self-reported workplace AOD impairment following the implementation of the policy. As with the impact evaluation, these findings need to be interpreted in light of the limitations of the study design in being able to attribute cause to the intervention alone. The findings from the outcome evaluation study also showed that while a positive change in self-reported workplace AOD impairment following implementation of the policy did not appear to be related to gender, age group, or employment type, it did appear to be related to levels of employee general alcohol use, cannabis use, site type, and employment role. Integration of the process, impact, and outcome evaluation studies There appeared to be qualitative support for the relationship between the process of developing and implementing the policy, and the impact of the policy in changing the risk factors for workplace AOD impairment. That is, overall the workplace AOD policy was developed and implemented well and, following its implementation, there were positive changes in the majority of measured risk factors for workplace AOD impairment. Quantitative findings lend further support for a relationship between the process and impact of the policy, in that there was a statistically significant association between employee perceived fidelity of the policy (related to the process of the policy) and positive changes in some risk factors for workplace AOD impairment (representing the impact of the policy). Findings also indicated support for the relationship between the impact of the policy in changing the risk factors for workplace AOD impairment and the outcome of the policy in reducing workplace AOD impairment: positive changes in the risk factors for workplace AOD impairment (impact) were related to positive changes in self reported workplace AOD impairment (representing the main goal and outcome of the policy). CONCLUSIONS The findings from the research indicate support for the conclusion that the policy was appropriately implemented and that it achieved its objectives and main goal. The Doctoral research findings also addressed a number of gaps in the literature on workplace AOD impairment, namely: the likely effectiveness of AOD policies for reducing AOD impairment in the workplace, which factors in the development and implementation of a workplace AOD policy are likely to facilitate or impede the effectiveness of the policy to reduce workplace AOD impairment, and which employee groups are less likely to respond well to policies of this type. The findings from this research not only represent an example of translational, applied research—through the evaluation of the study industry's policy—but also add to the body of knowledge on workplace AOD policies and provide policy-makers with evidence which may be useful in the development and implementation of effective workplace AOD policies. Importantly, the findings espouse the importance of scientific evidence in the development, implementation, and evaluation of workplace AOD policies.

A new type of high-order elements based on the mesh-free interpolations

We propose a new type of high-order elements that incorporates the mesh-free Galerkin formulations into the framework of finite element method. Traditional polynomial interpolation is replaced by mesh-free interpolations in the present high-order elements, and the strain smoothing technique is used for integration of the governing equations based on smoothing cells. The properties of high-order elements, which are influenced by the basis function of mesh-free interpolations and boundary nodes, are discussed through numerical examples. It can be found that the basis function has significant influence on the computational accuracy and upper-lower bounds of energy norm, when the strain smoothing technique retains the softening phenomenon. This new type of high-order elements shows good performance when quadratic basis functions are used in the mesh-free interpolations and present elements prove advantageous in adaptive mesh and nodes refinement schemes. Furthermore, it shows less sensitive to the quality of element because it uses the mesh-free interpolations and obeys the Weakened Weak (W2) formulation as introduced in [3, 5].