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.

Efficient preconditioning of the method of lines for solving nonlinear two-sided space-fractional diffusion equations

A standard method for the numerical solution of partial differential equations (PDEs) is the method of lines. In this approach the PDE is discretised in space using �finite di�fferences or similar techniques, and the resulting semidiscrete problem in time is integrated using an initial value problem solver. A significant challenge when applying the method of lines to fractional PDEs is that the non-local nature of the fractional derivatives results in a discretised system where each equation involves contributions from many (possibly every) spatial node(s). This has important consequences for the effi�ciency of the numerical solver. First, 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. Second, since the Jacobian matrix of the system is dense (partially or fully), methods that avoid the need to form and factorise this matrix are preferred. In this paper, we consider a nonlinear two-sided space-fractional di�ffusion equation in one spatial dimension. A key contribution of this paper is to demonstrate how an eff�ective preconditioner is crucial for improving the effi�ciency of the method of lines for solving this equation. In particular, we 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.

Data breach notification law in the EU and Australia : where to now?

Mandatory data breach notification laws have been a significant legislative reform in response to unauthorized disclosures of personal information by public and private sector organizations. These laws originated in the state-based legislatures of the United States during the last decade and have subsequently garnered worldwide legislative interest. We contend that there are conceptual and practical concerns regarding mandatory data breach notification laws which limit the scope of their applicability, particularly in relation to existing information privacy law regimes. We outline these concerns here, in the light of recent European Union and Australian legal developments in this area.

Improving trust and securing data accessibility for e-health decision making by using data encryption techniques

In the medical and healthcare arena, patients‟ data is not just their own personal history but also a valuable large dataset for finding solutions for diseases. While electronic medical records are becoming popular and are used in healthcare work places like hospitals, as well as insurance companies, and by major stakeholders such as physicians and their patients, the accessibility of such information should be dealt with in a way that preserves privacy and security. Thus, finding the best way to keep the data secure has become an important issue in the area of database security. Sensitive medical data should be encrypted in databases. There are many encryption/ decryption techniques and algorithms with regard to preserving privacy and security. Currently their performance is an important factor while the medical data is being managed in databases. Another important factor is that the stakeholders should decide more cost-effective ways to reduce the total cost of ownership. As an alternative, DAS (Data as Service) is a popular outsourcing model to satisfy the cost-effectiveness but it takes a consideration that the encryption/ decryption modules needs to be handled by trustworthy stakeholders. This research project is focusing on the query response times in a DAS model (AES-DAS) and analyses the comparison between the outsourcing model and the in-house model which incorporates Microsoft built-in encryption scheme in a SQL Server. This research project includes building a prototype of medical database schemas. There are 2 types of simulations to carry out the project. The first stage includes 6 databases in order to carry out simulations to measure the performance between plain-text, Microsoft built-in encryption and AES-DAS (Data as Service). Particularly, the AES-DAS incorporates implementations of symmetric key encryption such as AES (Advanced Encryption Standard) and a Bucket indexing processor using Bloom filter. The results are categorised such as character type, numeric type, range queries, range queries using Bucket Index and aggregate queries. The second stage takes the scalability test from 5K to 2560K records. The main result of these simulations is that particularly as an outsourcing model, AES-DAS using the Bucket index shows around 3.32 times faster than a normal AES-DAS under the 70 partitions and 10K record-sized databases. Retrieving Numeric typed data takes shorter time than Character typed data in AES-DAS. The aggregation query response time in AES-DAS is not as consistent as that in MS built-in encryption scheme. The scalability test shows that the DBMS reaches in a certain threshold; the query response time becomes rapidly slower. However, there is more to investigate in order to bring about other outcomes and to construct a secured EMR (Electronic Medical Record) more efficiently from these simulations.

Benefits of publicly available data

The National Morbidity, Mortality, and Air Pollution Study (NMMAPS) was designed to examine the health effects of air pollution in the United States. The primary question was whether particulate matter was responsible for the associations between air pollution and daily mortality. Secondary questions concerned measurement error in air pollution and mortality displacement.1 Since then, NMMAPS has been used to answer many important questions in environmental epidemiology...

Analysing environmental acoustic data through collaboration and automation

Monitoring environmental health is becoming increasingly important as human activity and climate change place greater pressure on global biodiversity. Acoustic sensors provide the ability to collect data passively, objectively and continuously across large areas for extended periods. While these factors make acoustic sensors attractive as autonomous data collectors, there are significant issues associated with large-scale data manipulation and analysis. We present our current research into techniques for analysing large volumes of acoustic data efficiently. We provide an overview of a novel online acoustic environmental workbench and discuss a number of approaches to scaling analysis of acoustic data; online collaboration, manual, automatic and human-in-the loop analysis.

Effects of Mobile Phone Distraction at the Onset of Amber Light: Analysis of Driving Simulator Data

Distraction whilst driving on an approach to a signalized intersection is particularly dangerous, as potential vehicular conflicts and resulting angle collisions tend to be severe. This study examines the decisions of distracted drivers during the onset of amber lights. Driving simulator data were obtained from a sample of 58 drivers under baseline and handheld mobile phone conditions at the University of IOWA - National Advanced Driving Simulator. Explanatory variables include age, gender, cell phone use, distance to stop-line, and speed. An iterative combination of decision tree and logistic regression analyses are employed to identify main effects, non-linearities, and interactions effects. Results show that novice (16-17 years) and younger (18-25 years) drivers’ had heightened amber light running risk while distracted by cell phone, and speed and distance thresholds yielded significant interaction effects. Driver experience captured by age has a multiplicative effect with distraction, making the combined effect of being inexperienced and distracted particularly risky. Solutions are needed to combat the use of mobile phones whilst driving.

Business Process Data Compliance

Most approaches to business process compliance are restricted to the analysis of the structure of processes. It has been argued that full regulatory compliance requires information on not only the structure of processes but also on what the tasks in a process do. To this end Governatori and Sadiq[2007] proposed to extend business processes with semantic annotations. We propose a methodology to automatically extract one kind of such annotations; in particular the annotations related to the data schema and templates linked to the various tasks in a business process.

NMR relaxation behavior and quadrupole coupling constants of 39K and 23Na ions in glycerol. Comparisons with 39K tissue data

The quadrupole coupling constants (qcc) for39K and23Na ions in glycerol have been calculated from linewidths measured as a function of temperature (which in turn results in changes in solution viscosity). The qcc of39K in glycerol is found to be 1.7 MHz, and that of23Na is 1.6 MHz. The relaxation behavior of39K and23Na ions in glycerol shows magnetic field and temperature dependence consistent with the equations for transverse relaxation more commonly used to describe the reorientation of nuclei in a molecular framework with intramolecular field gradients. It is shown, however, that τc is not simply proportional to the ratio of viscosity/temperature (ηT). The 39K qcc in glycerol and the value of 1.3 MHz estimated for this nucleus in aqueous solution are much greater than values of 0.075 to 0.12 MHz calculated from T2 measurements of39K in freshly excised rat tissues. This indicates that, in biological samples, processes such as exchange of potassium between intracellular compartments or diffusion of ions through locally ordered regions play a significant role in determining the effective quadrupole coupling constant and correlation time governing39K relaxation. T1 and T2 measurements of rat muscle at two magnetic fields also indicate that a more complex correlation function may be required to describe the relaxation of39K in tissue. Similar results and conclusions are found for23Na.

Leveraging social network data for analytical CRM strategies : the introduction of social BI

The skyrocketing trend for social media on the Internet greatly alters analytical Customer Relationship Management (CRM). Against this backdrop, the purpose of this paper is to advance the conceptual design of Business Intelligence (BI) systems with data identified from social networks. We develop an integrated social network data model, based on an in-depth analysis of Facebook. The data model can inform the design of data warehouses in order to offer new opportunities for CRM analyses, leading to a more consistent and richer picture of customers? characteristics, needs, wants, and demands. Four major contributions are offered. First, Social CRM and Social BI are introduced as emerging fields of research. Second, we develop a conceptual data model to identify and systematize the data available on online social networks. Third, based on the identified data, we design a multidimensional data model as an early contribution to the conceptual design of Social BI systems and demonstrate its application by developing management reports in a retail scenario. Fourth, intellectual challenges for advancing Social CRM and Social BI are discussed.

Alternating direction implicit numerical method for the space and time fractional Bloch-Torrey equation in 3-D

Recently, some authors have considered a new diffusion model–space and time fractional Bloch-Torrey equation (ST-FBTE). Magin et al. (2008) have derived analytical solutions with fractional order dynamics in space (i.e., _ = 1, β an arbitrary real number, 1 < β ≤ 2) and time (i.e., 0 < α < 1, and β = 2), respectively. Yu et al. (2011) have derived an analytical solution and an effective implicit numerical method for solving ST-FBTEs, and also discussed the stability and convergence of the implicit numerical method. However, due to the computational overheads necessary to perform the simulations for nuclear magnetic resonance (NMR) in three dimensions, they present a study based on a two-dimensional example to confirm their theoretical analysis. Alternating direction implicit (ADI) schemes have been proposed for the numerical simulations of classic differential equations. The ADI schemes will reduce a multidimensional problem to a series of independent one-dimensional problems and are thus computationally efficient. In this paper, we consider the numerical solution of a 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. A fractional alternating direction implicit scheme (FADIS) for the ST-FBTE in 3-D is proposed. Stability and convergence properties of the FADIS are discussed. Finally, some numerical results for ST-FBTE are given.

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.

Numerical methods for solving the multi-term time-fractional wave-diffusion equations

In this paper, the multi-term time-fractional wave diffusion equations are considered. The multiterm time fractional derivatives are defined in the Caputo sense, whose orders belong to the intervals [0,1], [1,2), [0,2), [0,3), [2,3) and [2,4), respectively. Some computationally effective numerical methods are proposed for simulating the multi-term time-fractional wave-diffusion equations. The numerical results demonstrate the effectiveness of theoretical analysis. These methods and techniques can also be extended to other kinds of the multi-term fractional time-space models with fractional Laplacian.