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.

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.

Analytical solutions for the multi-term time-space Caputo-Riesz fractional advection-diffusion equations on a finite domain

Generalized fractional partial differential equations have now found wide application for describing important physical phenomena, such as subdiffusive and superdiffusive processes. However, studies of generalized multi-term time and space fractional partial differential equations are still under development. In this paper, the multi-term time-space Caputo-Riesz fractional advection diffusion equations (MT-TSCR-FADE) with Dirichlet nonhomogeneous boundary conditions are considered. The multi-term time-fractional derivatives are defined in the Caputo sense, whose orders belong to the intervals [0, 1], [1, 2] and [0, 2], respectively. These are called respectively the multi-term time-fractional diffusion terms, the multi-term time-fractional wave terms and the multi-term time-fractional mixed diffusion-wave terms. The space fractional derivatives are defined as Riesz fractional derivatives. Analytical solutions of three types of the MT-TSCR-FADE are derived with Dirichlet boundary conditions. By using Luchko's Theorem (Acta Math. Vietnam., 1999), we proposed some new techniques, such as a spectral representation of the fractional Laplacian operator and the equivalent relationship between fractional Laplacian operator and Riesz fractional derivative, that enabled the derivation of the analytical solutions for the multi-term time-space Caputo-Riesz fractional advection-diffusion equations. © 2012.

Multiple distribution static synchronous compensators for distribution feeder voltage support

The potential of multiple distribution static synchronous compensators (DSTATCOMs) to improve the voltage profile of radial distribution networks has been reported in the literature by few authors. However, the operation of multiple DSTATCOMs across a distribution feeder may introduce control interactions and/or voltage instability. This study proposes a control scheme that alleviates interactions among controllers and enhances proper reactive power sharing among DSTATCOMs. A generalised mathematical model is presented to analyse the interactions among any number of DSTATCOMs in the network. The criterion for controller design is developed by conducting eigenvalue analysis on this mathematical model. The proposed control scheme is tested in time domain on a sample radial distribution feeder installed with multiple DSTATCOMs and test results are presented.

Quantum key distribution in the classical authenticated key exchange framework

Key establishment is a crucial primitive for building secure channels in a multi-party setting. Without quantum mechanics, key establishment can only be done under the assumption that some computational problem is hard. Since digital communication can be easily eavesdropped and recorded, it is important to consider the secrecy of information anticipating future algorithmic and computational discoveries which could break the secrecy of past keys, violating the secrecy of the confidential channel. Quantum key distribution (QKD) can be used generate secret keys that are secure against any future algorithmic or computational improvements. QKD protocols still require authentication of classical communication, although existing security proofs of QKD typically assume idealized authentication. It is generally considered folklore that QKD when used with computationally secure authentication is still secure against an unbounded adversary, provided the adversary did not break the authentication during the run of the protocol. We describe a security model for quantum key distribution extending classical authenticated key exchange (AKE) security models. Using our model, we characterize the long-term security of the BB84 QKD protocol with computationally secure authentication against an eventually unbounded adversary. By basing our model on traditional AKE models, we can more readily compare the relative merits of various forms of QKD and existing classical AKE protocols. This comparison illustrates in which types of adversarial environments different quantum and classical key agreement protocols can be secure.

Characteristics of the Internet for Criminal Child Sexual Abuse by Online Groomers

This paper presents a number of characteristics of the Internet that makes it attractive to online groomers. Relevant Internet characteristics include disconnected personal communication, mediating technology, universality, network externalities, distribution channel, time moderator, low‐cost standard, electronic double, electronic double manipulation, information asymmetry, infinite virtual capacity, independence in time and space, cyberspace, and dynamic social network. Potential sex offenders join virtual communities, where they meet other persons who have the same interest. A virtual community provides an online meeting place where people with similar interests can communicate and find useful information. Communication between members may be via email, bulletin boards, online chat, web‐based conferencing or other computer‐based media.

Ethnic differences in body fat distribution among Asian pre-pubertal children: A cross-sectional multicenter study

Background Ethnic differences in body fat distribution contribute to ethnic differences in cardiovascular morbidities and diabetes. However few data are available on differences in fat distribution in Asian children from various backgrounds. Therefore, the current study aimed to explore ethnic differences in body fat distribution among Asian children from four countries. Methods A total of 758 children aged 8-10 y from China, Lebanon, Malaysia and Thailand were recruited using a non-random purposive sampling approach to enrol children encompassing a wide BMI range. Height, weight, waist circumference (WC), fat mass (FM, derived from total body water [TBW] estimation using the deuterium dilution technique) and skinfold thickness (SFT) at biceps, triceps, subscapular, supraspinale and medial calf were collected. Results After controlling for height and weight, Chinese and Thai children had a significantly higher WC than their Lebanese and Malay counterparts. Chinese and Thais tended to have higher trunk fat deposits than Lebanese and Malays reflected in trunk SFT, trunk/upper extremity ratio or supraspinale/upper extremity ratio after adjustment for age and total body fat. The subscapular/supraspinale skinfold ratio was lower in Chinese and Thais compared with Lebanese and Malays after correcting for trunk SFT. Conclusions Asian pre-pubertal children from different origins vary in body fat distribution. These results indicate the importance of population-specific WC cut-off points or other fat distribution indices to identify the population at risk of obesity-related health problems.