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.

Travel behavior of workers in Dhaka and their attitude towards road pricing

Dhaka, the capital of Bangladesh, is facing severe traffic congestion. Owing to the flaws in past land use and transport planning decisions, uncontrolled population growth and urbanization, Dhaka’s traffic condition is worsening. Road space is widely regarded in the literature as a utility, so a common view of transport economists is that its usage ought to be charged. Road pricing policy has proven to be effective in managing travel demand, in order to reduce traffic congestion from road networks in a number of cities including London, Stockholm and Singapore. Road pricing as an economic mechanism to manage travel demand can be more effective and user-friendly when revenue is hypothecated into supply alternatives such as improvements to the transit system. This research investigates the feasibility of adopting road pricing in Dhaka with respect to a significant Bus Rapid Transit (BRT) project. Because both are very new concepts for the population of Dhaka, public acceptability would be a principal issue driving their success or failure. This paper explores the travel behaviour of workers in Dhaka and public perception toward Road Pricing with regards to work trips- based on worker’s travel behaviour. A revealed preference and stated preference survey has been conducted on sample of workers in Dhaka. They were asked limited demographic questions, their current travel behaviour and at the end they had been given several hypothetical choices of integrated BRT and road pricing to choose from. Key finding from the survey is the objective of integrated road pricing; subsidies Bus rapid Transit by road pricing to get reduced BRT fare; cannot be achieved in Dhaka. This is because most of the respondent stated that they would choose the cheapest option Walk-BRT-Walk, even though this would be more time consuming and uncomfortable as they have to walk from home to BRT station and also from BRT station to home. Proper economic analysis has to be carried out to find out the appropriate fare of BRT and road charge with some incentive for the low income people.

The relationship between unsafe working conditions and workers’ behavior and their impacts on injury severity in the U.S.

Unsafe acts of workers (e.g. misjudgment, inappropriate operation) become the major root causes of construction accidents when they are combined with unsafe working conditions (e.g. working surface conditions, weather) on a construction site. The overarching goal of the research presented in this paper is to explore ways to prevent unsafe acts of workers and reduce the likelihood of construction accidents occurring. The study specifically aims to (1) understand the relationships between human behavior related and working condition related risk factors, (2) identify the significant behavior and condition factors and their impacts on accident types (e.g. struck by/against, caught in/between, falling, shock, inhalation/ingestion/absorption, respiratory failure) and injury severity (e.g. fatality, hospitalized, non-hospitalized), and (3) analyze the fundamental accident-injury relationship on how each accident type contributes to the injury severity. The study reviewed 9,358 accidents which occurred in the U.S. construction industry between 2002 and 2011. The large number of accident samples supported reliable statistical analyses. The analysis identified a total of 17 significant correlations between behavior and condition factors and distinguished key risk factors that highly impacted on the determination of accident types and injury severity. The research outcomes will assist safety managers to control specific unsafe acts of workers by eliminating the associated unsafe working conditions and vice versa. They also can prioritize risk factors and pay more attention to controlling them in order to achieve a safer working environment.

Biology or behavior : which is the strongest contributor to weight gain?

Combating unhealthy weight gain is a major public health and clinical management issue. The extent of research into the etiology and pathophysiology of obesity has produced a wealth of evidence regarding the contributing factors. While aspects of the environment are ‘obesogenic’, weight gain is not inevitable for every individual. What then explains potentially unhealthy weight gain in individuals living within an environment where others remain lean? In this paper we explore the biological compensation that acts in response to a reduced energy intake by reducing energy needs, in order to defend against weight loss. We then examine the evidence that there is only a weak biological compensation to surplus energy supply, and that this allows behavior to drive weight gain. The extent to which biology impacts behavior is also considered.

Experimental studies of the shear behavior and strength of LiteSteel beams with stiffened web openings

Abstract: LiteSteel beam (LSB) is a new cold-formed steel hollow flange channel section produced using a patented manufacturing process. It is commonly used as flexural members in residential, industrial and commercial buildings. Current practice in flooring systems is to include openings in the web element of floor joists or bearers so that building services can be located within them. Test results have shown that the shear capacity of LSBs can be reduced considerably by the inclusion of web openings. A cost effective method of eliminating the detrimental effects of a large web opening is to attach suitable stiffeners around the web openings of LSBs. A detailed experimental study consisting of 17 shear tests was therefore undertaken to investigate the shear behaviour and strength of LSBs with stiffened circular web openings. Both plate and stud stiffeners with varying sizes and thicknesses were attached to the web elements of LSBs using a number of screw-fastening arrangements in order to develop a suitable stiffening arrangement for LSBs. Simply supported test specimens of LSBs with an aspect ratio of 1.5 were loaded at mid-span until failure. This paper presents the details of this experimental study of LSBs with stiffened web openings, and the results of their shear capacities and associated behavioural characteristics. Suitable screw-fastened plate stiffener arrangements have been recommended in order to restore the original shear capacity of LSBs.

Efficient solution of two-sided nonlinear space-fractional diffusion equations using fast Poisson preconditioners

We develop a fast Poisson preconditioner for the efficient numerical solution of a class of two-sided nonlinear space fractional diffusion equations in one and two dimensions using the method of lines. Using the shifted Gr¨unwald finite difference formulas to approximate the two-sided(i.e. the left and right Riemann-Liouville) fractional derivatives, the resulting semi-discrete nonlinear systems have dense Jacobian matrices owing to the non-local property of fractional derivatives. We employ a modern initial value problem solver utilising backward differentiation formulas and Jacobian-free Newton-Krylov methods to solve these systems. For efficient performance of the Jacobianfree Newton-Krylov method it is essential to apply an effective preconditioner to accelerate the convergence of the linear iterative solver. The key contribution of our work is to generalise the fast Poisson preconditioner, widely used for integer-order diffusion equations, so that it applies to the two-sided space fractional diffusion equation. A number of numerical experiments are presented to demonstrate the effectiveness of the preconditioner and the overall solution strategy.

A banded preconditioner for the two-sided, nonlinear space-fractional diffusion equation

The method of lines is a standard method for advancing the solution of partial differential equations (PDEs) in time. In one sense, the method applies equally well to space-fractional PDEs as it does to integer-order PDEs. However, there is a significant challenge when solving space-fractional PDEs in this way, owing to the non-local nature of the fractional derivatives. Each equation in the resulting semi-discrete system involves contributions from every spatial node in the domain. This has important consequences for the efficiency of the numerical solver, especially when the system is large. First, the Jacobian matrix of the system is dense, and hence methods that avoid the need to form and factorise this matrix are preferred. Second, since the cost of evaluating the discrete equations is high, it is essential to minimise the number of evaluations required to advance the solution in time. In this paper, we show how an effective preconditioner is essential for improving the efficiency of the method of lines for solving a quite general two-sided, nonlinear space-fractional diffusion equation. A key contribution is to show, how to construct suitable banded approximations to the system Jacobian for preconditioning purposes that permit high orders and large stepsizes to be used in the temporal integration, without requiring dense matrices to be formed. The results of numerical experiments are presented that demonstrate the effectiveness of this approach.

A novel numerical approximation for the space fractional advection-dispersion equation

In this paper, we consider a space fractional advection–dispersion equation. The equation is obtained from the standard advection–diffusion equation by replacing the first- and second-order space derivatives by the Riesz fractional derivatives of order β1 ∈ (0, 1) and β2 ∈ (1, 2], respectively. The fractional advection and dispersion terms are approximated by using two fractional centred difference schemes. A new weighted Riesz fractional finite-difference approximation scheme is proposed. When the weighting factor θ equals 12, a second-order accuracy scheme is obtained. The stability, consistency and convergence of the numerical approximation scheme are discussed. A numerical example is given to show that the numerical results are in good agreement with our theoretical analysis.

The use of a Riesz fractional differential-based approach for texture enhancement in image processing

Abstract: Texture enhancement is an important component of image processing, with extensive application in science and engineering. The quality of medical images, quantified using the texture of the images, plays a significant role in the routine diagnosis performed by medical practitioners. Previously, image texture enhancement was performed using classical integral order differential mask operators. Recently, first order fractional differential operators were implemented to enhance images. Experiments conclude that the use of the fractional differential not only maintains the low frequency contour features in the smooth areas of the image, but also nonlinearly enhances edges and textures corresponding to high-frequency image components. However, whilst these methods perform well in particular cases, they are not routinely useful across all applications. To this end, we applied the second order Riesz fractional differential operator to improve upon existing approaches of texture enhancement. Compared with the classical integral order differential mask operators and other fractional differential operators, our new algorithms provide higher signal to noise values, which leads to superior image quality.

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.

Mice selectively bred for High and Low fear behavior show differences in the number of pMAPK (p44/42 ERK) expressing neurons in lateral amygdala following Pavlovian fear conditioning

Individual variability in the acquisition, consolidation and extinction of conditioned fear potentially contributes to the development of fear pathology including posttraumatic stress disorder (PTSD). Pavlovian fear conditioning is a key tool for the study of fundamental aspects of fear learning. Here, we used a selected mouse line of High and Low Pavlovian conditioned fear created from an advanced intercrossed line (AIL) in order to begin to identify the cellular basis of phenotypic divergence in Pavlovian fear conditioning. We investigated whether phosphorylated MAPK (p44/42 ERK/MAPK), a protein kinase required in the amygdala for the acquisition and consolidation of Pavlovian fear memory, is differentially expressed following Pavlovian fear learning in the High and Low fear lines. We found that following Pavlovian auditory fear conditioning, High and Low line mice differ in the number of pMAPK-expressing neurons in the dorsal sub nucleus of the lateral amygdala (LAd). In contrast, this difference was not detected in the ventral medial (LAvm) or ventral lateral (LAvl) amygdala sub nuclei or in control animals. We propose that this apparent increase in plasticity at a known locus of fear memory acquisition and consolidation relates to intrinsic differences between the two fear phenotypes. These data provide important insights into the micronetwork mechanisms encoding phenotypic differences in fear. Understanding the circuit level cellular and molecular mechanisms that underlie individual variability in fear learning is critical for the development of effective treatment of fear-related illnesses such as PTSD.

Induction of multiple reinstatements of ethanol- and sucrose-seeking behavior in Long–Evans rats by the α-2 adrenoreceptor antagonist yohimbine

Rationale Developing models to efficiently explore the mechanisms by which stress can mediate reinstatement of drug-seeking behavior is crucial to the development of new pharmacotherapies for alcohol use disorders. Objectives We examined the effects of multiple reinstatement sessions using the pharmacological stressor, yohimbine, in ethanol- and sucrose-seeking rats in order to develop a more efficient model of stress-induced reinstatement. Methods Long–Evans rats were trained to self-administer 10% ethanol with a sucrose-fading procedure, 20% ethanol without a sucrose-fading procedure, or 5% sucrose in 30-min operant self-administration sessions, followed by extinction training. After reaching extinction criteria, the animals were tested once per week with yohimbine vehicle and yohimbine (2 mg/kg), respectively, 30 min prior to the reinstatement sessions or blood collection. Levels of reinstatement and plasma corticosterone (CORT) were determined each week for four consecutive weeks. Results Yohimbine induced reinstatement of ethanol- and sucrose-seeking in each of the 4 weeks. Interestingly, the magnitude of the reinstatement decreased for the 10% ethanol group after the first reinstatement session but remained stable for the 20% ethanol group trained without sucrose. Plasma CORT levels in response to injection of both vehicle and yohimbine were significantly higher in the ethanol-trained animals compared to sucrose controls. Conclusions The stable reinstatement in the 20% ethanol group supports the use of this training procedure in studies using within-subject designs with multiple yohimbine reinstatement test sessions. Additionally, these results indicate that the hormonal response to stressors can be altered following extinction from self-administration of relatively modest amounts of ethanol.

Insight into the fractional calculus via a spreadsheet

Many students of calculus are not aware that the calculus they have learned is a special case (integer order) of fractional calculus. Fractional calculus is the study of arbitrary order derivatives and integrals and their applications. The article begins by stating a naive question from a student in a paper by Larson (1974) and establishes, for polynomials and exponential functions, that they can be deformed into their derivative using the μ-th order fractional derivatives for 0<μ<1. Through the power of Excel we illustrate the continuous deformations dynamically through conditional formatting. Some applications are discussed and a connection made to mathematics education.

The use of a Riesz fractional differential-based approach for texture enhancement in image processing

Texture enhancement is an important component of image processing that finds extensive application in science and engineering. The quality of medical images, quantified using the imaging texture, plays a significant role in the routine diagnosis performed by medical practitioners. Most image texture enhancement is performed using classical integral order differential mask operators. Recently, first order fractional differential operators were used to enhance images. Experimentation with these methods led to the conclusion that fractional differential operators not only maintain the low frequency contour features in the smooth areas of the image, but they also nonlinearly enhance edges and textures corresponding to high frequency image components. However, whilst these methods perform well in particular cases, they are not routinely useful across all applications. To this end, we apply the second order Riesz fractional differential operator to improve upon existing approaches of texture enhancement. Compared with the classical integral order differential mask operators and other first order fractional differential operators, we find that our new algorithms provide higher signal to noise values and superior image quality.

A semi-alternating direction method for a 2-D fractional FitzHugh–Nagumo monodomain model on an approximate irregular domain

A FitzHugh-Nagumo monodomain model has been used to describe the propagation of the electrical potential in heterogeneous cardiac tissue. In this paper, we consider a two-dimensional fractional FitzHugh-Nagumo monodomain model on an irregular domain. The model consists of a coupled Riesz space fractional nonlinear reaction-diffusion model and an ordinary differential equation, describing the ionic fluxes as a function of the membrane potential. Secondly, we use a decoupling technique and focus on solving the Riesz space fractional nonlinear reaction-diffusion model. A novel spatially second-order accurate semi-implicit alternating direction method (SIADM) for this model on an approximate irregular domain is proposed. Thirdly, stability and convergence of the SIADM are proved. Finally, some numerical examples are given to support our theoretical analysis and these numerical techniques are employed to simulate a two-dimensional fractional Fitzhugh-Nagumo model on both an approximate circular and an approximate irregular domain.