A meshless method based on Point Interpolation Method (PIM) for the space fractional diffusion equation

This paper aims to develop a meshless approach based on the Point Interpolation Method (PIM) for numerical simulation of a space fractional diffusion equation. Two fully-discrete schemes for the one-dimensional space fractional diffusion equation are obtained by using the PIM and the strong-forms of the space diffusion equation. Numerical examples with different nodal distributions are studied to validate and investigate the accuracy and efficiency of the newly developed meshless approach.

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.

A fast semi-implicit difference method for a nonlinear two-sided space-fractional diffusion equation with variable diffusivity coefficients

In this paper, we derive a new nonlinear two-sided space-fractional diffusion equation with variable coefficients from the fractional Fick’s law. A semi-implicit difference method (SIDM) for this equation is proposed. The stability and convergence of the SIDM are discussed. For the implementation, we develop a fast accurate iterative method for the SIDM by decomposing the dense coefficient matrix into a combination of Toeplitz-like matrices. This fast iterative method significantly reduces the storage requirement of O(n2)O(n2) and computational cost of O(n3)O(n3) down to n and O(nlogn)O(nlogn), where n is the number of grid points. The method retains the same accuracy as the underlying SIDM solved with Gaussian elimination. Finally, some numerical results are shown to verify the accuracy and efficiency of the new method.

A Crank--Nicolson ADI spectral method for a two-dimensional riesz space fractional nonlinear reaction-diffusion equation

In this paper, a new alternating direction implicit Galerkin--Legendre spectral method for the two-dimensional Riesz space fractional nonlinear reaction-diffusion equation is developed. The temporal component is discretized by the Crank--Nicolson method. The detailed implementation of the method is presented. The stability and convergence analysis is strictly proven, which shows that the derived method is stable and convergent of order $2$ in time. An optimal error estimate in space is also obtained by introducing a new orthogonal projector. The present method is extended to solve the fractional FitzHugh--Nagumo model. Numerical results are provided to verify the theoretical analysis.

Stability and convergence of a new finite volume method for a two-sided space-fractional diffusion equation

In this paper, we consider a two-sided space-fractional diffusion equation with variable coefficients on a finite domain. Firstly, based on the nodal basis functions, we present a new fractional finite volume method for the two-sided space-fractional diffusion equation and derive the implicit scheme and solve it in matrix form. Secondly, we prove the stability and convergence of the implicit fractional finite volume method and conclude that the method is unconditionally stable and convergent. Finally, some numerical examples are given to show the effectiveness of the new numerical method, and the results are in excellent agreement with 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.

Outcome evaluation framework for the Queensland Alcohol Ignition Interlock Program. Report to Queensland Department of Transport and Main Roads (TMR 3814)

The Queensland Department of Transport and Main Roads (TMR) required an evaluation framework for the Queensland Alcohol Ignition Interlock Program (AIIP). The objective of this project was to develop a framework to evaluate the AIIP in terms of its effect on road safety outcomes.

Numeracy counts in the statistical reasoning equation

The relationship between mathematics and statistical reasoning frequently receives comment (Vere-Jones 1995, Moore 1997); however most of the research into the area tends to focus on maths anxiety. Gnaldi (Gnaldi 2003) showed that in a statistics course for psychologists, the statistical understanding of students at the end of the course depended on students’ basic numeracy, rather than the number or level of previous mathematics courses the student had undertaken. As part of a study into the development of statistical thinking at the interface between secondary and tertiary education, students enrolled in an introductory data analysis subject were assessed regarding their statistical reasoning ability, basic numeracy skills and attitudes towards statistics. This work reports on the relationships between these factors and in particular the importance of numeracy to statistical reasoning.

Crash involvement patterns of unlicensed drivers and riders in Queensland. Report to Queensland Department of Transport and Main Roads

Unlicensed driving remains a serious problem for road safety, despite ongoing improvements in traffic law enforcement practices and technology. While it does not play a direct causative role in road crashes, unlicensed driving undermines the integrity of the driver licensing system and is associated with a range of high-risk behaviours. This report examines official road crash data from Queensland for the years 2003-08 to compare the crash involvement patterns of unlicensed drivers with those of licensed drivers and explore the scope and nature of unlicensed driving. This study replicates and extends upon two previous studies examining the involvement of unlicensed drivers in crashes in Queensland (Watson, 2004a; 2004b; Watson & Steinhardt, 2006).

Estimating the prevalence of unregistered driving in Queensland. Report to Queensland Department of Transport and Main Roads

A state-wide observational study was undertaken by the Centre for Accident Research and Road Safety – Queensland (CARRS-Q) in order to investigate the prevalence of unregistered vehicles on Queensland roads. This study was conducted on behalf of the Queensland department of Transport and Main Roads (TMR). This study builds upon research commissioned by Queensland Transport and conducted in 2000, 2003 and 2005. Vehicles were observed throughout Queensland from April 13th, 2010 to May 8th, 2010 in a mix of small rural towns, regional centres and metropolitan locations. The locations sampled for this study were restricted to destinations (e.g. shopping centres, hospitals, airports and park-and-ride facilities) rather than residential areas, and a variety of different destinations were sampled. Sampling vehicles in these types of locations provides a higher probability that the vehicles captured in the survey are being driven on a regular basis (Younglove, et al. 2004).

Links between unlicensed and unregistered driving: Literature review. Report to Queensland Department of Transport and Main Roads

Vehicle registration represents an important component of the management of the road transport system in Queensland, with most vehicles required to be registered before they can be driven or parked on a public road (Department of Transport and Main Roads, 2010b). In addition to the collection of taxes for road construction and maintenance, the current registration system also: • Sets the safety standards required for vehicles to be allowed on public roads; • Allows driver behaviour to be managed by identifying vehicles, and the responsible owners of vehicles, for enforcement purposes; and • Facilitates the collection of insurance premiums for the Queensland Compulsory Third Party (CTP) insurance scheme.

A comparison of methods to estimate the extent of unlicensed driving: Literature review. Final report to Queensland Transport

Unlicensed driving is a serious problem in many countries, despite ongoing improvements in traffic law enforcement practices and technology. The term ‘unlicensed driver' is generally used to refer to people who operate a motor vehicle or motorcycle without a valid driver's licence, including those who: • have let their licence expire; • have been disqualified or suspended from driving; • hold an inappropriate class of licence for the vehicle they drive; • drive outside the restrictions of a special licence; • don’t currently hold a licence; or • have never held a licence.

Final report : Estimating the extent of unlicensed driving: Roadside licence check survey - Wave2. Report to Queensland Department of Transport and Main Roads

This report documents a second, revised roadside licence check survey undertaken by the Queensland Police Service (QPS) in conjunction with roadside random breath testing (RBT) traffic operations. The methodology utilised was devised by the Centre for Accident Research and Road Safety – Queensland (CARRS-Q), Transport and Main Roads (TMR) and QPS, after consideration of a number of options.

An examination of the factors contributing to unlicensed driving. Report to Queensland Department of Transport and Main Roads

Unlicensed driving is a serious problem in many countries, despite ongoing improvements in traffic law enforcement practices and technology. Unlike alcohol impairment and speeding, unlicensed driving does not play a direct causative role in road crashes. However it represents a major problem for road safety in two respects. Firstly, it undermines the effectiveness of driver licensing systems by preventing the allocation of demerit points and reducing the impact of licence loss (Watson, 2004b). Secondly, there is a growing body of evidence linking unlicensed driving to a cluster of high-risk behaviours including drink driving, speeding, failure to wear seat belts and motorcycle use (Griffin & DeLaZerda, 2000; Harrison, 1997; Watson, 1997, 2004b). Consistent with this, utilising the quasi-induced exposure method, Watson (2004a) estimated that in Queensland, unlicensed drivers were almost three times more likely to be involved in a reported crash than licensed drivers.