Analytical and numerical solutions for the time and space-symmetric fractional diffusion equation

We consider a time and space-symmetric fractional diffusion equation (TSS-FDE) under homogeneous Dirichlet conditions and homogeneous Neumann conditions. The TSS-FDE is obtained from the standard diffusion equation by replacing the first-order time derivative by a Caputo fractional derivative, and the second order space derivative by a symmetric fractional derivative. First, a method of separating variables expresses the analytical solution of the TSS-FDE in terms of the Mittag--Leffler function. Second, we propose two numerical methods to approximate the Caputo time fractional derivative: the finite difference method; and the Laplace transform method. The symmetric space fractional derivative is approximated using the matrix transform method. Finally, numerical results demonstrate the effectiveness of the numerical methods and to confirm the theoretical claims.

Stability and convergence of an effective numerical method for the time-space fractional Fokker-Planck equation with a nonlinear source term

Fractional Fokker-Planck equations (FFPEs) have gained much interest recently for describing transport dynamics in complex systems that are governed by anomalous diffusion and nonexponential relaxation patterns. However, effective numerical methods and analytic techniques for the FFPE are still in their embryonic state. In this paper, we consider a class of time-space fractional Fokker-Planck equations with a nonlinear source term (TSFFPE-NST), which involve the Caputo time fractional derivative (CTFD) of order α ∈ (0, 1) and the symmetric Riesz space fractional derivative (RSFD) of order μ ∈ (1, 2). Approximating the CTFD and RSFD using the L1-algorithm and shifted Grunwald method, respectively, a computationally effective numerical method is presented to solve the TSFFPE-NST. The stability and convergence of the proposed numerical method are investigated. Finally, numerical experiments are carried out to support the theoretical claims.

Computationally efficient numerical methods for time- and space-fractional Fokker–Planck equations

Fractional Fokker–Planck equations have been used to model several physical situations that present anomalous diffusion. In this paper, a class of time- and space-fractional Fokker–Planck equations (TSFFPE), which involve the Riemann–Liouville time-fractional derivative of order 1-α (α(0, 1)) and the Riesz space-fractional derivative (RSFD) of order μ(1, 2), are considered. The solution of TSFFPE is important for describing the competition between subdiffusion and Lévy flights. However, effective numerical methods for solving TSFFPE are still in their infancy. We present three computationally efficient numerical methods to deal with the RSFD, and approximate the Riemann–Liouville time-fractional derivative using the Grünwald method. The TSFFPE is then transformed into a system of ordinary differential equations (ODE), which is solved by the fractional implicit trapezoidal method (FITM). Finally, numerical results are given to demonstrate the effectiveness of these methods. These techniques can also be applied to solve other types of fractional partial differential equations.

Analytical and numerical solutions for the time and space-symmetric fractional diffusion equation

We consider a time and space-symmetric fractional diffusion equation (TSS-FDE) under homogeneous Dirichlet conditions and homogeneous Neumann conditions. The TSS-FDE is obtained from the standard diffusion equation by replacing the first-order time derivative by the Caputo fractional derivative and the second order space derivative by the symmetric fractional derivative. Firstly, a method of separating variables is used to express the analytical solution of the tss-fde in terms of the Mittag–Leffler function. Secondly, we propose two numerical methods to approximate the Caputo time fractional derivative, namely, the finite difference method and the Laplace transform method. The symmetric space fractional derivative is approximated using the matrix transform method. Finally, numerical results are presented to demonstrate the effectiveness of the numerical methods and to confirm the theoretical claims.

Measurement of standby power for selected electrical appliances in Australia

High growth in the uptake of electrical appliances is accounting for a significant increase in electricity consumption globally. In some developed countries, standby energy alone may account for about 10% of residential electricity use. The standby power for many appliances used in Australia is still well above the national goal of 1 W or less. In this paper, field measurements taken of standby power and operating power for a range of electrical appliances are presented. It was found that the difference between minimum value and maximum value of standby power could be quite large, up to 22.13 W for home theatre systems, for example. With the exception of home audio systems, however, the annual operating energy used by most electrical appliances was generally greater than the annual standby energy. Consumer behaviour and product choice can have a significant impact on standby power and operating power, which influences both energy demand and greenhouse gas emissions.

Directional earth fault relay operation in mutually coupled multiple circuit distribution lines

Many of the power utilities around the world experienced spurious tripping of directional earth fault relays in their mesh distribution networks due to induced circulating currents. This circulating current is zero sequence and induced in the healthy circuit due to the zero sequence current flow resulting from a ground fault of a parallel circuit. This paper quantitatively discusses the effects of mutual coupling on earth fault protection of distribution systems. An actual spurious tripping event is analyzed to support the theory and to present options for improved resilience to spurious tripping.

The effect of red blood cell orientation on the electrical impedance of pulsatile blood with implications for impedance cardiography

Impedance cardiography is an application of bioimpedance analysis primarily used in a research setting to determine cardiac output. It is a non invasive technique that measures the change in the impedance of the thorax which is attributed to the ejection of a volume of blood from the heart. The cardiac output is calculated from the measured impedance using the parallel conductor theory and a constant value for the resistivity of blood. However, the resistivity of blood has been shown to be velocity dependent due to changes in the orientation of red blood cells induced by changing shear forces during flow. The overall goal of this thesis was to study the effect that flow deviations have on the electrical impedance of blood, both experimentally and theoretically, and to apply the results to a clinical setting. The resistivity of stationary blood is isotropic as the red blood cells are randomly orientated due to Brownian motion. In the case of blood flowing through rigid tubes, the resistivity is anisotropic due to the biconcave discoidal shape and orientation of the cells. The generation of shear forces across the width of the tube during flow causes the cells to align with the minimal cross sectional area facing the direction of flow. This is in order to minimise the shear stress experienced by the cells. This in turn results in a larger cross sectional area of plasma and a reduction in the resistivity of the blood as the flow increases. Understanding the contribution of this effect on the thoracic impedance change is a vital step in achieving clinical acceptance of impedance cardiography. Published literature investigates the resistivity variations for constant blood flow. In this case, the shear forces are constant and the impedance remains constant during flow at a magnitude which is less than that for stationary blood. The research presented in this thesis, however, investigates the variations in resistivity of blood during pulsataile flow through rigid tubes and the relationship between impedance, velocity and acceleration. Using rigid tubes isolates the impedance change to variations associated with changes in cell orientation only. The implications of red blood cell orientation changes for clinical impedance cardiography were also explored. This was achieved through measurement and analysis of the experimental impedance of pulsatile blood flowing through rigid tubes in a mock circulatory system. A novel theoretical model including cell orientation dynamics was developed for the impedance of pulsatile blood through rigid tubes. The impedance of flowing blood was theoretically calculated using analytical methods for flow through straight tubes and the numerical Lattice Boltzmann method for flow through complex geometries such as aortic valve stenosis. The result of the analytical theoretical model was compared to the experimental impedance measurements through rigid tubes. The impedance calculated for flow through a stenosis using the Lattice Boltzmann method provides results for comparison with impedance cardiography measurements collected as part of a pilot clinical trial to assess the suitability of using bioimpedance techniques to assess the presence of aortic stenosis. The experimental and theoretical impedance of blood was shown to inversely follow the blood velocity during pulsatile flow with a correlation of -0.72 and -0.74 respectively. The results for both the experimental and theoretical investigations demonstrate that the acceleration of the blood is an important factor in determining the impedance, in addition to the velocity. During acceleration, the relationship between impedance and velocity is linear (r2 = 0.98, experimental and r2 = 0.94, theoretical). The relationship between the impedance and velocity during the deceleration phase is characterised by a time decay constant, ô , ranging from 10 to 50 s. The high level of agreement between the experimental and theoretically modelled impedance demonstrates the accuracy of the model developed here. An increase in the haematocrit of the blood resulted in an increase in the magnitude of the impedance change due to changes in the orientation of red blood cells. The time decay constant was shown to decrease linearly with the haematocrit for both experimental and theoretical results, although the slope of this decrease was larger in the experimental case. The radius of the tube influences the experimental and theoretical impedance given the same velocity of flow. However, when the velocity was divided by the radius of the tube (labelled the reduced average velocity) the impedance response was the same for two experimental tubes with equivalent reduced average velocity but with different radii. The temperature of the blood was also shown to affect the impedance with the impedance decreasing as the temperature increased. These results are the first published for the impedance of pulsatile blood. The experimental impedance change measured orthogonal to the direction of flow is in the opposite direction to that measured in the direction of flow. These results indicate that the impedance of blood flowing through rigid cylindrical tubes is axisymmetric along the radius. This has not previously been verified experimentally. Time frequency analysis of the experimental results demonstrated that the measured impedance contains the same frequency components occuring at the same time point in the cycle as the velocity signal contains. This suggests that the impedance contains many of the fluctuations of the velocity signal. Application of a theoretical steady flow model to pulsatile flow presented here has verified that the steady flow model is not adequate in calculating the impedance of pulsatile blood flow. The success of the new theoretical model over the steady flow model demonstrates that the velocity profile is important in determining the impedance of pulsatile blood. The clinical application of the impedance of blood flow through a stenosis was theoretically modelled using the Lattice Boltzman method (LBM) for fluid flow through complex geometeries. The impedance of blood exiting a narrow orifice was calculated for varying degrees of stenosis. Clincial impedance cardiography measurements were also recorded for both aortic valvular stenosis patients (n = 4) and control subjects (n = 4) with structurally normal hearts. This pilot trial was used to corroborate the results of the LBM. Results from both investigations showed that the decay time constant for impedance has potential in the assessment of aortic valve stenosis. In the theoretically modelled case (LBM results), the decay time constant increased with an increase in the degree of stenosis. The clinical results also showed a statistically significant difference in time decay constant between control and test subjects (P = 0.03). The time decay constant calculated for test subjects (ô = 180 - 250 s) is consistently larger than that determined for control subjects (ô = 50 - 130 s). This difference is thought to be due to difference in the orientation response of the cells as blood flows through the stenosis. Such a non-invasive technique using the time decay constant for screening of aortic stenosis provides additional information to that currently given by impedance cardiography techniques and improves the value of the device to practitioners. However, the results still need to be verified in a larger study. While impedance cardiography has not been widely adopted clinically, it is research such as this that will enable future acceptance of the method.

Export decisions within Taiwanese electrical and electronic SMEs : the role of management characteristics and attitudes

In this globalized environment, Taiwanese firms have been very successful in achieving growth via international market expansion. In particular, the Taiwanese electronics industry has shown a dynamism lacking in comparable industries around the world. However, in recent years there has been a move by many of the larger Taiwanese manufacturing firms to outsource their manufacturing to low-cost producers such as China in order to remain competitive. Conversely, most Taiwanese small- to medium-sized enterprises (SMEs) have retained their production facilities in Taiwan. These SMEs seek to expand their sales beyond the domestic market by employing an export strategy, making a significant socioeconomic contribution to the domestic and regional economies. This paper highlights the key dimensions such as enhancing factors (benefits/advantages), inhibiting factors (barriers/costs), and managerial factors (characteristics/commitment) that play an important role in the internationalization of SMEs located within the Taiwanese electronics industry. A logistic regression model is used to predict the probability of a firm being an exporter.

An advanced implicit meshless approach for fractional partial differential equation in computational mechanics

Recently, the numerical modelling and simulation for fractional partial differential equations (FPDE), which have been found with widely applications in modern engineering and sciences, are attracting increased attentions. The current dominant numerical method for modelling of FPDE is the explicit Finite Difference Method (FDM), which is based on a pre-defined grid leading to inherited issues or shortcomings. This paper aims to develop an implicit meshless approach based on the radial basis functions (RBF) for numerical simulation of time fractional diffusion equations. The discrete system of equations is obtained by using the RBF meshless shape functions and the strong-forms. The stability and convergence of this meshless approach are then discussed and theoretically proven. Several numerical examples with different problem domains are used to validate and investigate accuracy and efficiency of the newly developed meshless formulation. The results obtained by the meshless formations are also compared with those obtained by FDM in terms of their accuracy and efficiency. It is concluded that the present meshless formulation is very effective for the modelling and simulation for FPDE.

Matching cognitive styles to computer-based instruction : an approach for enhanced learning in electrical engineering

An approach aimed at enhancing learning by matching individual students' preferred cognitive styles to computer-based instructional (CBI) material is presented. This approach was used in teaching some components of a third-year unit in an electrical engineering course at the Queensland University of Technology. Cognitive style characteristics of perceiving and processing information were considered. The bimodal nature of cognitive styles (analytic/imager, analytic/verbalizer, wholist/imager and wholist/verbalizer) was examined in order to assess the full ramification of cognitive styles on learning. In a quasi-experimental format, students' cognitive styles were analysed by cognitive style analysis (CSA) software. On the basis of the CSA results the system defaulted students to either matched or mismatched CBI material. The consistently better performance by the matched group suggests potential for further investigations where the limitations cited in this paper are eliminated. Analysing the differences between cognitive styles on individual test tasks also suggests that certain test tasks may better suit certain cognitive styles.

Biceps brachii muscle oxygenation in electrical muscle stimulation

The purpose of this study was to compare between electrical muscle stimulation (EMS) and maximal voluntary (VOL) isometric contractions of the elbow flexors for changes in biceps brachii muscle oxygenation (tissue oxygenation index, TOI) and haemodynamics (total haemoglobin volume, tHb = oxygenated-Hb + deoxygenated-Hb) determined by near-infrared spectroscopy (NIRS). The biceps brachii muscle of 10 healthy men (23–39 years) was electrically stimulated at high frequency (75 Hz) via surface electrodes to evoke 50 intermittent (4-s contraction, 15-s relaxation) isometric contractions at maximum tolerated current level (EMS session). The contralateral arm performed 50 intermittent (4-s contraction, 15-s relaxation) maximal voluntary isometric contractions (VOL session) in a counterbalanced order separated by 2–3 weeks. Results indicated that although the torque produced during EMS was approximately 50% of VOL (P<0Æ05), there was no significant difference in the changes in TOI amplitude or TOI slope between EMS and VOL over the 50 contractions. However, the TOI amplitude divided by peak torque was approximately 50% lower for EMS than VOL (P<0Æ05), which indicates EMS was less efficient than VOL. This seems likely because of the difference in the muscles involved in the force production between conditions. Mean decrease in tHb amplitude during the contraction phases was significantly (P<0Æ05) greater for EMS than VOL from the 10th contraction onwards, suggesting that the muscle blood volume was lower in EMS than VOL. It is concluded that local oxygen demand of the biceps brachii sampled by NIRS is similar between VOL and EMS.