Computationally efficient methods for solving time-variable-order time-space fractional reaction-diffusion equation

Fractional differential equations are becoming more widely accepted as a powerful tool in modelling anomalous diffusion, which is exhibited by various materials and processes. Recently, researchers have suggested that rather than using constant order fractional operators, some processes are more accurately modelled using fractional orders that vary with time and/or space. In this paper we develop computationally efficient techniques for solving time-variable-order time-space fractional reaction-diffusion equations (tsfrde) using the finite difference scheme. We adopt the Coimbra variable order time fractional operator and variable order fractional Laplacian operator in space where both orders are functions of time. Because the fractional operator is nonlocal, it is challenging to efficiently deal with its long range dependence when using classical numerical techniques to solve such equations. The novelty of our method is that the numerical solution of the time-variable-order tsfrde is written in terms of a matrix function vector product at each time step. This product is approximated efficiently by the Lanczos method, which is a powerful iterative technique for approximating the action of a matrix function by projecting onto a Krylov subspace. Furthermore an adaptive preconditioner is constructed that dramatically reduces the size of the required Krylov subspaces and hence the overall computational cost. Numerical examples, including the variable-order fractional Fisher equation, are presented to demonstrate the accuracy and efficiency of the approach.

Intensity dependent residual amplitude modulation in electro-optic phase modulators

Residual amplitude modulation (RAM) mechanisms in electro-optic phase modulators are detrimental in applications that require high purity phase modulation of the incident laser beam. While the origins of RAMare not fully understood, measurements have revealed that it depends on the beam properties of the laser as well as the properties of the medium. Here we present experimental and theoretical results that demonstrate, for the first time, the dependence of RAM production in electro-optic phase modulators on beam intensity. The results show an order of magnitude increase in the level of RAM, around 10 dB, with a fifteenfold enhancement in the input intensity from 12 to 190 mW/mm 2. We show that this intensity dependent RAM is photorefractive in origin. © 2012 Optical Society of America.

The effect of light intensity and temperature on photocatalytic water splitting

Photocatalytic water splitting is a process which could potentially lead to commercially viable solar hydrogen production. This thesis uses an engineering perspective to investigate the technology. The effect of light intensity and temperature on photocatalytic water splitting was examined to evaluate the prospect of using solar concentration to increase the feasibility of the process. P25 TiO2 films deposited on conducting glass were used as photocatalyst electrodes and coupled with platinum electrodes which were also deposited on conducting glass. These films were used to form a photocatalysis cell and illuminated with a Xenon arc lamp to simulate solar light at intensities up to 50 suns. They were also tested at temperatures between 20°C and 100°C. The reaction demonstrated a sub-linear relationship with intensity. Photocurrent was proportional to intensity with an exponential value of 0.627. Increasing temperature resulted in an exponential relationship. This proved to follow an Arrhenius relationship with an activation energy of 10.3 kJ mol-1 and a pre-exponential factor of approximately 8.7×103. These results then formed the basis of a mathematical model which extrapolated beyond the range of the experimental tests. This model shows that the loss of efficiency from performing the reaction under high light intensity is offset by the increased reaction rate and efficiency from the associated temperature increase. This is an important finding for photocatalytic water splitting. It will direct future research in system design and materials research and may provide an avenue for the commercialisation of this technology.

Intensity pulsations in distributed feedback Yb:Phosphate waveguide lasers

The intensity pulsations of a cw 1030 nm Yb:Phosphate monolithic waveguide laser with distributed feedback are described. We show that the pulsations could result from the coupling of the two orthogonal polarization modes through the two photon process of cooperative luminescence. The predictions of the presented theoretical model agree well with the observed behaviour.

Numerical simulation of a new two-dimensional variable-order fractional percolation equation in non-homogeneous porous media

Percolation flow problems are discussed in many research fields, such as seepage hydraulics, groundwater hydraulics, groundwater dynamics and fluid dynamics in porous media. Many physical processes appear to exhibit fractional-order behavior that may vary with time, or space, or space and time. The theory of pseudodifferential operators and equations has been used to deal with this situation. In this paper we use a fractional Darcys law with variable order Riemann-Liouville fractional derivatives, this leads to a new variable-order fractional percolation equation. In this paper, a new two-dimensional variable-order fractional percolation equation is considered. A new implicit numerical method and an alternating direct method for the two-dimensional variable-order fractional model is proposed. Consistency, stability and convergence of the implicit finite difference method are established. Finally, some numerical examples are given. The numerical results demonstrate the effectiveness of the methods. This technique can be used to simulate a three-dimensional variable-order fractional percolation equation.

Numerical analysis of the time variable fractional order mobile-immobile advection-dispersion model

Many physical processes exhibit fractional order behavior that varies with time 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 the time variable fractional order mobile-immobile advection-dispersion model. Numerical methods and analyses of stability and convergence for the fractional partial differential equations are quite limited and difficult to derive. This motivates us to develop efficient numerical methods as well as stability and convergence of the implicit numerical methods for the fractional order mobile immobile advection-dispersion model. In the paper, we use the Coimbra variable time fractional derivative which is more efficient from the numerical standpoint and is preferable for modeling dynamical systems. An implicit Euler approximation for the equation is proposed and then the stability of the approximation are investigated. As for the convergence of the numerical scheme we only consider a special case, i.e. the time fractional derivative is independent of time variable t. The case where the time fractional derivative depends both the time variable t and the space variable x will be considered in the future work. Finally, numerical examples are provided to show that the implicit Euler approximation is computationally efficient.

Numerical techniques for the variable order time fractional diffusion equation

In this paper we consider the variable order time fractional diffusion equation. We adopt the Coimbra variable order (VO) time fractional operator, which defines a consistent method for VO differentiation of physical variables. The Coimbra variable order fractional operator also can be viewed as a Caputo-type definition. Although this definition is the most appropriate definition having fundamental characteristics that are desirable for physical modeling, numerical methods for fractional partial differential equations using this definition have not yet appeared in the literature. Here an approximate scheme is first proposed. The stability, convergence and solvability of this numerical scheme are discussed via the technique of Fourier analysis. Numerical examples are provided to show that the numerical method is computationally efficient. Crown Copyright © 2012 Published by Elsevier Inc. All rights reserved.

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.

Numerical solution of stress intensity factors of multiple cracks in great number with eigen COD boundary integral equations

Based on the eigen crack opening displacement (COD) boundary integral equations, a newly developed computational approach is proposed for the analysis of multiple crack problems. The eigen COD particularly refers to a crack in an infinite domain under fictitious traction acting on the crack surface. With the concept of eigen COD, the multiple cracks in great number can be solved by using the conventional displacement discontinuity boundary integral equations in an iterative fashion with a small size of system matrix. The interactions among cracks are dealt with by two parts according to the distances of cracks to the current crack. The strong effects of cracks in adjacent group are treated with the aid of the local Eshelby matrix derived from the traction BIEs in discrete form. While the relatively week effects of cracks in far-field group are treated in the iteration procedures. Numerical examples are provided for the stress intensity factors of multiple cracks, up to several thousands in number, with the proposed approach. By comparing with the analytical solutions in the literature as well as solutions of the dual boundary integral equations, the effectiveness and the efficiencies of the proposed approach are verified.

A variable-stepsize Jacobian-free exponential integrator for simulating transport in heterogeneous porous media : application to wood drying

A Jacobian-free variable-stepsize method is developed for the numerical integration of the large, stiff systems of differential equations encountered when simulating transport in heterogeneous porous media. Our method utilises the exponential Rosenbrock-Euler method, which is explicit in nature and requires a matrix-vector product involving the exponential of the Jacobian matrix at each step of the integration process. These products can be approximated using Krylov subspace methods, which permit a large integration stepsize to be utilised without having to precondition the iterations. This means that our method is truly "Jacobian-free" - the Jacobian need never be formed or factored during the simulation. We assess the performance of the new algorithm for simulating the drying of softwood. Numerical experiments conducted for both low and high temperature drying demonstrates that the new approach outperforms (in terms of accuracy and efficiency) existing simulation codes that utilise the backward Euler method via a preconditioned Newton-Krylov strategy.

The relationship between drivers’ perceptions toward police speed enforcement and self-reported speeding behaviour

Exceeding the speed limit and driving too fast for the conditions are regularly cited as significant contributing factors in traffic crashes, particularly fatal and serious injury crashes. Despite an extensive body of research highlighting the relationship between increased vehicle speeds and crash risk and severity, speeding remains a pervasive behaviour on Australian roads. The development of effective countermeasures designed to reduce the prevalence of speeding behaviour requires that this behaviour is well understood. The primary aim of this program of research was to develop a better understanding of the influence of drivers’ perceptions and attitudes toward police speed enforcement on speeding behaviour. Study 1 employed focus group discussions with 39 licensed drivers to explore the influence of perceptions relating to specific characteristics of speed enforcement policies and practices on drivers’ attitudes towards speed enforcement. Three primary factors were identified as being most influential: site selection; visibility; and automaticity (i.e., whether the enforcement approach is automated/camera-based or manually operated). Perceptions regarding these enforcement characteristics were found to influence attitudes regarding the perceived legitimacy and transparency of speed enforcement. Moreover, misperceptions regarding speed enforcement policies and practices appeared to also have a substantial impact on attitudes toward speed enforcement, typically in a negative direction. These findings have important implications for road safety given that prior research has suggested that the effectiveness of speed enforcement approaches may be reduced if efforts are perceived by drivers as being illegitimate, such that they do little to encourage voluntary compliance. Study 1 also examined the impact of speed enforcement approaches varying in the degree of visibility and automaticity on self-reported willingness to comply with speed limits. These discussions suggested that all of the examined speed enforcement approaches (see Section 1.5 for more details) generally showed potential to reduce vehicle speeds and encourage compliance with posted speed limits. Nonetheless, participant responses suggested a greater willingness to comply with approaches operated in a highly visible manner, irrespective of the corresponding level of automaticity of the approach. While less visible approaches were typically associated with poorer rates of driver acceptance (e.g., perceived as “sneaky” and “unfair”), participants reported that such approaches would likely encourage long-term and network-wide impacts on their own speeding behaviour, as a function of the increased unpredictability of operations and increased direct (specific deterrence) and vicarious (general deterrence) experiences with punishment. Participants in Study 1 suggested that automated approaches, particularly when operated in a highly visible manner, do little to encourage compliance with speed limits except in the immediate vicinity of the enforcement location. While speed cameras have been criticised on such grounds in the past, such approaches can still have substantial road safety benefits if implemented in high-risk settings. Moreover, site-learning effects associated with automated approaches can also be argued to be a beneficial by-product of enforcement, such that behavioural modifications are achieved even in the absence of actual enforcement. Conversely, manually operated approaches were reported to be associated with more network-wide impacts on behaviour. In addition, the reported acceptance of such methods was high, due to the increased swiftness of punishment, ability for additional illegal driving behaviours to be policed and the salutary influence associated with increased face-to-face contact with authority. Study 2 involved a quantitative survey conducted with 718 licensed Queensland drivers from metropolitan and regional areas. The survey sought to further examine the influence of the visibility and automaticity of operations on self-reported likelihood and duration of compliance. Overall, the results from Study 2 corroborated those of Study 1. All examined approaches were again found to encourage compliance with speed limits, such that all approaches could be considered to be “effective”. Nonetheless, significantly greater self-reported likelihood and duration of compliance was associated with visibly operated approaches, irrespective of the corresponding automaticity of the approach. In addition, the impact of automaticity was influenced by visibility; such that significantly greater self-reported likelihood of compliance was associated with manually operated approaches, but only when they are operated in a less visible fashion. Conversely, manually operated approaches were associated with significantly greater durations of self-reported compliance, but only when they are operated in a highly visible manner. Taken together, the findings from Studies 1 and 2 suggest that enforcement efforts, irrespective of their visibility or automaticity, generally encourage compliance with speed limits. However, the duration of these effects on behaviour upon removal of the enforcement efforts remains questionable and represents an area where current speed enforcement practices could possibly be improved. Overall, it appears that identifying the optimal mix of enforcement operations, implementing them at a sufficient intensity and increasing the unpredictability of enforcement efforts (e.g., greater use of less visible approaches, random scheduling) are critical elements of success. Hierarchical multiple regression analyses were also performed in Study 2 to investigate the punishment-related and attitudinal constructs that influence self-reported frequency of speeding behaviour. The research was based on the theoretical framework of expanded deterrence theory, augmented with three particular attitudinal constructs. Specifically, previous research examining the influence of attitudes on speeding behaviour has typically focussed on attitudes toward speeding behaviour in general only. This research sought to more comprehensively explore the influence of attitudes by also individually measuring and analysing attitudes toward speed enforcement and attitudes toward the appropriateness of speed limits on speeding behaviour. Consistent with previous research, a number of classical and expanded deterrence theory variables were found to significantly predict self-reported frequency of speeding behaviour. Significantly greater speeding behaviour was typically reported by those participants who perceived punishment associated with speeding to be less certain, who reported more frequent use of punishment avoidance strategies and who reported greater direct experiences with punishment. A number of interesting differences in the significant predictors among males and females, as well as younger and older drivers, were reported. Specifically, classical deterrence theory variables appeared most influential on the speeding behaviour of males and younger drivers, while expanded deterrence theory constructs appeared more influential for females. These findings have important implications for the development and implementation of speeding countermeasures. Of the attitudinal factors, significantly greater self-reported frequency of speeding behaviour was reported among participants who held more favourable attitudes toward speeding and who perceived speed limits to be set inappropriately low. Disappointingly, attitudes toward speed enforcement were found to have little influence on reported speeding behaviour, over and above the other deterrence theory and attitudinal constructs. Indeed, the relationship between attitudes toward speed enforcement and self-reported speeding behaviour was completely accounted for by attitudes toward speeding. Nonetheless, the complexity of attitudes toward speed enforcement are not yet fully understood and future research should more comprehensively explore the measurement of this construct. Finally, given the wealth of evidence (both in general and emerging from this program of research) highlighting the association between punishment avoidance and speeding behaviour, Study 2 also sought to investigate the factors that influence the self-reported propensity to use punishment avoidance strategies. A standard multiple regression analysis was conducted for exploratory purposes only. The results revealed that punishment-related and attitudinal factors significantly predicted approximately one fifth of the variance in the dependent variable. The perceived ability to avoid punishment, vicarious punishment experience, vicarious punishment avoidance and attitudes toward speeding were all significant predictors. Future research should examine these relationships more thoroughly and identify additional influential factors. In summary, the current program of research has a number of implications for road safety and speed enforcement policy and practice decision-making. The research highlights a number of potential avenues for the improvement of public education regarding enforcement efforts and provides a number of insights into punishment avoidance behaviours. In addition, the research adds strength to the argument that enforcement approaches should not only demonstrate effectiveness in achieving key road safety objectives, such as reduced vehicle speeds and associated crashes, but also strive to be transparent and legitimate, such that voluntary compliance is encouraged. A number of potential strategies are discussed (e.g., point-to-point speed cameras, intelligent speed adaptation. The correct mix and intensity of enforcement approaches appears critical for achieving optimum effectiveness from enforcement efforts, as well as enhancements in the unpredictability of operations and swiftness of punishment. Achievement of these goals should increase both the general and specific deterrent effects associated with enforcement through an increased perceived risk of detection and a more balanced exposure to punishment and punishment avoidance experiences.