A new implicit numerical method for the space and time fractional Bloch-Torrey equation

In recent years, it has been found that many phenomena in engineering, physics, chemistry and other sciences can be described very successfully by models using mathematical tools from fractional calculus. Recently, noted a new space and time fractional Bloch-Torrey equation (ST-FBTE) has been proposed (see Magin et al. (2008)), and successfully applied to analyse diffusion images of human brain tissues to provide new insights for further investigations of tissue structures. In this paper, we consider the ST-FBTE on a finite domain. The time and space derivatives in the ST-FBTE are replaced by the Caputo and the sequential Riesz fractional derivatives, respectively. Firstly, we propose a new effective implicit numerical method (INM) for the STFBTE whereby we discretize the Riesz fractional derivative using a fractional centered difference. Secondly, we prove that the implicit numerical method for the ST-FBTE is unconditionally stable and convergent, and the order of convergence of the implicit numerical method is ( T2 - α + h2 x + h2 y + h2 z ). Finally, some numerical results are presented to support our theoretical analysis.

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.

The Galerkin finite element approximation of the fractional cable equation

The cable equation is one of the most fundamental equations for modeling neuronal dynamics. Cable equations with a fractional order temporal derivative have been introduced to model electrotonic properties of spiny neuronal dendrites. In this paper, the fractional cable equation involving two integro-differential operators is considered. The Galerkin finite element approximations of the fractional cable equation are proposed. The main contribution of this work is outlined as follow: • A semi-discrete finite difference approximation in time is proposed. We prove that the scheme is unconditionally stable, and the numerical solution converges to the exact solution with order O(Δt). • A semi-discrete difference scheme for improving the order of convergence for solving the fractional cable equation is proposed, and the numerical solution converges to the exact solution with order O((Δt)2). • Based on the above semi-discrete difference approximations, Galerkin finite element approximations in space for a full discretization are also investigated. • Finally, some numerical results are given to demonstrate the theoretical analysis.

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

The field of fractional differential equations provides a means for modelling transport processes within complex media which are governed by anomalous transport. Indeed, the application to anomalous transport has been a significant driving force behind the rapid growth and expansion of the literature in the field of fractional calculus. In this paper, we present a finite volume method to solve the time-space two-sided fractional advection dispersion equation on a one-dimensional domain. Such an equation allows modelling different flow regime impacts from either side. The finite volume formulation provides a natural way to handle fractional advection-dispersion equations written in conservative form. The novel spatial discretisation employs fractionally-shifted Gr¨unwald formulas to discretise the Riemann-Liouville fractional derivatives at control volume faces in terms of function values at the nodes, while the L1-algorithm is used to discretise the Caputo time fractional derivative. Results of numerical experiments are presented to demonstrate the effectiveness of the approach.

A second-order accurate numerical approximation for the Riesz space fractional advection-dispersion equation

In this paper, we consider a space Riesz fractional advection-dispersion equation. The equation is obtained from the standard advection-diffusion equation by replacing the ¯rst-order and second-order space derivatives by the Riesz fractional derivatives of order β 1 Є (0; 1) and β2 Є(1; 2], respectively. Riesz fractional advection and dispersion terms are approximated by using two fractional centered difference schemes, respectively. A new weighted Riesz fractional ¯nite difference approximation scheme is proposed. When the weighting factor Ѳ = 1/2, a second- order accurate numerical approximation scheme for the Riesz fractional advection-dispersion equation is obtained. 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.

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.

Analytical and numerical solutions of the space and time fractional bloch-torrey equation

Fractional order dynamics in physics, particularly when applied to diffusion, leads to an extension of the concept of Brown-ian motion through a generalization of the Gaussian probability function to what is termed anomalous diffusion. As MRI is applied with increasing temporal and spatial resolution, the spin dynamics are being examined more closely; such examinations extend our knowledge of biological materials through a detailed analysis of relaxation time distribution and water diffusion heterogeneity. Here the dynamic models become more complex as they attempt to correlate new data with a multiplicity of tissue compartments where processes are often anisotropic. Anomalous diffusion in the human brain using fractional order calculus has been investigated. Recently, a new diffusion model was proposed by solving the Bloch-Torrey equation using fractional order calculus with respect to time and space (see R.L. Magin et al., J. Magnetic Resonance, 190 (2008) 255-270). However effective numerical methods and supporting error analyses for the fractional Bloch-Torrey equation are still limited. In this paper, the space and time fractional Bloch-Torrey equation (ST-FBTE) is considered. The time and space derivatives in the ST-FBTE are replaced by the Caputo and the sequential Riesz fractional derivatives, respectively. Firstly, we derive an analytical solution for the ST-FBTE with initial and boundary conditions on a finite domain. Secondly, we propose an implicit numerical method (INM) for the ST-FBTE, and the stability and convergence of the INM are investigated. We prove that the implicit numerical method for the ST-FBTE is unconditionally stable and convergent. Finally, we present some numerical results that support our theoretical analysis.

Assessing of circuit breaker restrike risks using computer simulation and wavelet analysis

A breaker restrike is an abnormal arcing phenomenon, leading to a possible breaker failure. Eventually, this failure leads to interruption of the transmission and distribution of the electricity supply system until the breaker is replaced. Before 2008, there was little evidence in the literature of monitoring techniques based on restrike measurement and interpretation produced during switching of capacitor banks and shunt reactor banks in power systems. In 2008 a non-intrusive radiometric restrike measurement method and a restrike hardware detection algorithm were developed by M.S. Ramli and B. Kasztenny. However, the limitations of the radiometric measurement method are a band limited frequency response as well as limitations in amplitude determination. Current restrike detection methods and algorithms require the use of wide bandwidth current transformers and high voltage dividers. A restrike switch model using Alternative Transient Program (ATP) and Wavelet Transforms which support diagnostics are proposed. Restrike phenomena become a new diagnostic process using measurements, ATP and Wavelet Transforms for online interrupter monitoring. This research project investigates the restrike switch model Parameter „A. dielectric voltage gradient related to a normal and slowed case of the contact opening velocity and the escalation voltages, which can be used as a diagnostic tool for a vacuum circuit-breaker (CB) at service voltages between 11 kV and 63 kV. During current interruption of an inductive load at current quenching or chopping, a transient voltage is developed across the contact gap. The dielectric strength of the gap should rise to a point to withstand this transient voltage. If it does not, the gap will flash over, resulting in a restrike. A straight line is fitted through the voltage points at flashover of the contact gap. This is the point at which the gap voltage has reached a value that exceeds the dielectric strength of the gap. This research shows that a change in opening contact velocity of the vacuum CB produces a corresponding change in the slope of the gap escalation voltage envelope. To investigate the diagnostic process, an ATP restrike switch model was modified with contact opening velocity computation for restrike waveform signature analyses along with experimental investigations. This also enhanced a mathematical CB model with the empirical dielectric model for SF6 (sulphur hexa-fluoride) CBs at service voltages above 63 kV and a generalised dielectric curve model for 12 kV CBs. A CB restrike can be predicted if there is a similar type of restrike waveform signatures for measured and simulated waveforms. The restrike switch model applications are used for: computer simulations as virtual experiments, including predicting breaker restrikes; estimating the interrupter remaining life of SF6 puffer CBs; checking system stresses; assessing point-on-wave (POW) operations; and for a restrike detection algorithm development using Wavelet Transforms. A simulated high frequency nozzle current magnitude was applied to an Equation (derived from the literature) which can calculate the life extension of the interrupter of a SF6 high voltage CB. The restrike waveform signatures for a medium and high voltage CB identify its possible failure mechanism such as delayed opening, degraded dielectric strength and improper contact travel. The simulated and measured restrike waveform signatures are analysed using Matlab software for automatic detection. Experimental investigation of a 12 kV vacuum CB diagnostic was carried out for the parameter determination and a passive antenna calibration was also successfully developed with applications for field implementation. The degradation features were also evaluated with a predictive interpretation technique from the experiments, and the subsequent simulation indicates that the drop in voltage related to the slow opening velocity mechanism measurement to give a degree of contact degradation. A predictive interpretation technique is a computer modeling for assessing switching device performance, which allows one to vary a single parameter at a time; this is often difficult to do experimentally because of the variable contact opening velocity. The significance of this thesis outcome is that it is a non-intrusive method developed using measurements, ATP and Wavelet Transforms to predict and interpret a breaker restrike risk. The measurements on high voltage circuit-breakers can identify degradation that can interrupt the distribution and transmission of an electricity supply system. It is hoped that the techniques for the monitoring of restrike phenomena developed by this research will form part of a diagnostic process that will be valuable for detecting breaker stresses relating to the interrupter lifetime. Suggestions for future research, including a field implementation proposal to validate the restrike switch model for ATP system studies and the hot dielectric strength curve model for SF6 CBs, are given in Appendix A.

Evaluating the success of large-scale, integrated information systems through the lens of IS-impact and IS-support

This study proceeds from a central interest in the importance of systematically evaluating operational large-scale integrated information systems (IS) in organisations. The study is conducted within the IS-Impact Research Track at Queensland University of Technology (QUT). The goal of the IS-Impact Track is, "to develop the most widely employed model for benchmarking information systems in organizations for the joint benefit of both research and practice" (Gable et al, 2009). The track espouses programmatic research having the principles of incrementalism, tenacity, holism and generalisability through replication and extension research strategies. Track efforts have yielded the bicameral IS-Impact measurement model; the ‘impact’ half includes Organisational-Impact and Individual-Impact dimensions; the ‘quality’ half includes System-Quality and Information-Quality dimensions. Akin to Gregor’s (2006) analytic theory, the ISImpact model is conceptualised as a formative, multidimensional index and is defined as "a measure at a point in time, of the stream of net benefits from the IS, to date and anticipated, as perceived by all key-user-groups" (Gable et al., 2008, p: 381). The study adopts the IS-Impact model (Gable, et al., 2008) as its core theory base. Prior work within the IS-Impact track has been consciously constrained to Financial IS for their homogeneity. This study adopts a context-extension strategy (Berthon et al., 2002) with the aim "to further validate and extend the IS-Impact measurement model in a new context - i.e. a different IS - Human Resources (HR)". The overarching research question is: "How can the impacts of large-scale integrated HR applications be effectively and efficiently benchmarked?" This managerial question (Cooper & Emory, 1995) decomposes into two more specific research questions – In the new HR context: (RQ1): "Is the IS-Impact model complete?" (RQ2): "Is the ISImpact model valid as a 1st-order formative, 2nd-order formative multidimensional construct?" The study adhered to the two-phase approach of Gable et al. (2008) to hypothesise and validate a measurement model. The initial ‘exploratory phase’ employed a zero base qualitative approach to re-instantiating the IS-Impact model in the HR context. The subsequent ‘confirmatory phase’ sought to validate the resultant hypothesised measurement model against newly gathered quantitative data. The unit of analysis for the study is the application, ‘ALESCO’, an integrated large-scale HR application implemented at Queensland University of Technology (QUT), a large Australian university (with approximately 40,000 students and 5000 staff). Target respondents of both study phases were ALESCO key-user-groups: strategic users, management users, operational users and technical users, who directly use ALESCO or its outputs. An open-ended, qualitative survey was employed in the exploratory phase, with the objective of exploring the completeness and applicability of the IS-Impact model’s dimensions and measures in the new context, and to conceptualise any resultant model changes to be operationalised in the confirmatory phase. Responses from 134 ALESCO users to the main survey question, "What do you consider have been the impacts of the ALESCO (HR) system in your division/department since its implementation?" were decomposed into 425 ‘impact citations.’ Citation mapping using a deductive (top-down) content analysis approach instantiated all dimensions and measures of the IS-Impact model, evidencing its content validity in the new context. Seeking to probe additional (perhaps negative) impacts; the survey included the additional open question "In your opinion, what can be done better to improve the ALESCO (HR) system?" Responses to this question decomposed into a further 107 citations which in the main did not map to IS-Impact, but rather coalesced around the concept of IS-Support. Deductively drawing from relevant literature, and working inductively from the unmapped citations, the new ‘IS-Support’ construct, including the four formative dimensions (i) training, (ii) documentation, (iii) assistance, and (iv) authorisation (each having reflective measures), was defined as: "a measure at a point in time, of the support, the [HR] information system key-user groups receive to increase their capabilities in utilising the system." Thus, a further goal of the study became validation of the IS-Support construct, suggesting the research question (RQ3): "Is IS-Support valid as a 1st-order reflective, 2nd-order formative multidimensional construct?" With the aim of validating IS-Impact within its nomological net (identification through structural relations), as in prior work, Satisfaction was hypothesised as its immediate consequence. The IS-Support construct having derived from a question intended to probe IS-Impacts, too was hypothesised as antecedent to Satisfaction, thereby suggesting the research question (RQ4): "What is the relative contribution of IS-Impact and IS-Support to Satisfaction?" With the goal of testing the above research questions, IS-Impact, IS-Support and Satisfaction were operationalised in a quantitative survey instrument. Partial least squares (PLS) structural equation modelling employing 221 valid responses largely evidenced the validity of the commencing IS-Impact model in the HR context. ISSupport too was validated as operationalised (including 11 reflective measures of its 4 formative dimensions). IS-Support alone explained 36% of Satisfaction; IS-Impact alone 70%; in combination both explaining 71% with virtually all influence of ISSupport subsumed by IS-Impact. Key study contributions to research include: (1) validation of IS-Impact in the HR context, (2) validation of a newly conceptualised IS-Support construct as important antecedent of Satisfaction, and (3) validation of the redundancy of IS-Support when gauging IS-Impact. The study also makes valuable contributions to practice, the research track and the sponsoring organisation.

The persistence of phase-separation in LiFePO4 with two-dimensional Li+ transport : the Cahn-Hilliard-reaction equation and the role of defects

We examine the solution of the two-dimensional Cahn-Hilliard-reaction (CHR) equation in the xy plane as a model of Li+ intercalation into LiFePO4 material. We validate our numerical solution against the solution of the depth-averaged equation, which has been used to model intercalation in the limit of highly orthotropic diffusivity and gradient penalty tensors. We then examine the phase-change behaviour in the full CHR system as these parameters become more isotropic, and find that as the Li+ diffusivity is increased in the x direction, phase separation persists at high currents, even in small crystals with averaged coherency strain included. The resulting voltage curves decrease monotonically, which has previously been considered a hallmark of crystals that fill homogeneously.

Leadership [2nd Asia-Pacific ed.]

This second edition of Leadership continues to offer a balanced approach to the study of leadership, drawing on Australasian practices and international theory. It looks at the characteristics of leaders in a wide variety of Australasian settings - organisations in the private, public, and not-for-profit sectors, as well as in politics and in our community. Traditional content such as charismatic, transformational, contingency, and situational theories of leadership are covered in detail, along with the power, influence, motivation, coaching, communication, and team building aspects of leadership. The text also introduces contemporary issues, such as entrepreneurship, knowledge management, leadership in international contexts, and the importance of ethics and social responsibility.