Fundamental solution and discrete random walk model for time-space fractional diffusion equation

In this paper, we consider a time-space fractional diffusion equation of distributed order (TSFDEDO). The TSFDEDO is obtained from the standard advection-dispersion equation by replacing the first-order time derivative by the Caputo fractional derivative of order α∈(0,1], the first-order and second-order space derivatives by the Riesz fractional derivatives of orders β 1∈(0,1) and β 2∈(1,2], respectively. We derive the fundamental solution for the TSFDEDO with an initial condition (TSFDEDO-IC). The fundamental solution can be interpreted as a spatial probability density function evolving in time. We also investigate a discrete random walk model based on an explicit finite difference approximation for the TSFDEDO-IC.

Visual localisation in outdoor industrial building environments

This paper presents a vision-based method of vehicle localisation that has been developed and tested on a large forklift type robotic vehicle which operates in a mainly outdoor industrial setting. The localiser uses a sparse 3D edgemap of the environment and a particle filter to estimate the pose of the vehicle. The vehicle operates in dynamic and non-uniform outdoor lighting conditions, an issue that is addressed by using knowledge of the scene to intelligently adjust the camera exposure and hence improve the quality of the information in the image. Results from the industrial vehicle are shown and compared to another laser-based localiser which acts as a ground truth. An improved likelihood metric, using peredge calculation, is presented and has shown to be 40% more accurate in estimating rotation. Visual localization results from the vehicle driving an arbitrary 1.5km path during a bright sunny period show an average position error of 0.44m and rotation error of 0.62deg.

Coding speech signals at very low rates (below 1 kb/s) with high intelligibility

This thesis presents an original approach to parametric speech coding at rates below 1 kbitsjsec, primarily for speech storage applications. Essential processes considered in this research encompass efficient characterization of evolutionary configuration of vocal tract to follow phonemic features with high fidelity, representation of speech excitation using minimal parameters with minor degradation in naturalness of synthesized speech, and finally, quantization of resulting parameters at the nominated rates. For encoding speech spectral features, a new method relying on Temporal Decomposition (TD) is developed which efficiently compresses spectral information through interpolation between most steady points over time trajectories of spectral parameters using a new basis function. The compression ratio provided by the method is independent of the updating rate of the feature vectors, hence allows high resolution in tracking significant temporal variations of speech formants with no effect on the spectral data rate. Accordingly, regardless of the quantization technique employed, the method yields a high compression ratio without sacrificing speech intelligibility. Several new techniques for improving performance of the interpolation of spectral parameters through phonetically-based analysis are proposed and implemented in this research, comprising event approximated TD, near-optimal shaping event approximating functions, efficient speech parametrization for TD on the basis of an extensive investigation originally reported in this thesis, and a hierarchical error minimization algorithm for decomposition of feature parameters which significantly reduces the complexity of the interpolation process. Speech excitation in this work is characterized based on a novel Multi-Band Excitation paradigm which accurately determines the harmonic structure in the LPC (linear predictive coding) residual spectra, within individual bands, using the concept 11 of Instantaneous Frequency (IF) estimation in frequency domain. The model yields aneffective two-band approximation to excitation and computes pitch and voicing with high accuracy as well. New methods for interpolative coding of pitch and gain contours are also developed in this thesis. For pitch, relying on the correlation between phonetic evolution and pitch variations during voiced speech segments, TD is employed to interpolate the pitch contour between critical points introduced by event centroids. This compresses pitch contour in the ratio of about 1/10 with negligible error. To approximate gain contour, a set of uniformly-distributed Gaussian event-like functions is used which reduces the amount of gain information to about 1/6 with acceptable accuracy. The thesis also addresses a new quantization method applied to spectral features on the basis of statistical properties and spectral sensitivity of spectral parameters extracted from TD-based analysis. The experimental results show that good quality speech, comparable to that of conventional coders at rates over 2 kbits/sec, can be achieved at rates 650-990 bits/sec.

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.

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.

Wavelet based approaches and high resolution methods for complex process models

Many industrial processes and systems can be modelled mathematically by a set of Partial Differential Equations (PDEs). Finding a solution to such a PDF model is essential for system design, simulation, and process control purpose. However, major difficulties appear when solving PDEs with singularity. Traditional numerical methods, such as finite difference, finite element, and polynomial based orthogonal collocation, not only have limitations to fully capture the process dynamics but also demand enormous computation power due to the large number of elements or mesh points for accommodation of sharp variations. To tackle this challenging problem, wavelet based approaches and high resolution methods have been recently developed with successful applications to a fixedbed adsorption column model. Our investigation has shown that recent advances in wavelet based approaches and high resolution methods have the potential to be adopted for solving more complicated dynamic system models. This chapter will highlight the successful applications of these new methods in solving complex models of simulated-moving-bed (SMB) chromatographic processes. A SMB process is a distributed parameter system and can be mathematically described by a set of partial/ordinary differential equations and algebraic equations. These equations are highly coupled; experience wave propagations with steep front, and require significant numerical effort to solve. To demonstrate the numerical computing power of the wavelet based approaches and high resolution methods, a single column chromatographic process modelled by a Transport-Dispersive-Equilibrium linear model is investigated first. Numerical solutions from the upwind-1 finite difference, wavelet-collocation, and high resolution methods are evaluated by quantitative comparisons with the analytical solution for a range of Peclet numbers. After that, the advantages of the wavelet based approaches and high resolution methods are further demonstrated through applications to a dynamic SMB model for an enantiomers separation process. This research has revealed that for a PDE system with a low Peclet number, all existing numerical methods work well, but the upwind finite difference method consumes the most time for the same degree of accuracy of the numerical solution. The high resolution method provides an accurate numerical solution for a PDE system with a medium Peclet number. The wavelet collocation method is capable of catching up steep changes in the solution, and thus can be used for solving PDE models with high singularity. For the complex SMB system models under consideration, both the wavelet based approaches and high resolution methods are good candidates in terms of computation demand and prediction accuracy on the steep front. The high resolution methods have shown better stability in achieving steady state in the specific case studied in this Chapter.

Thermal studies of D.C. traction motors

This paper describes a thorough thermal study on a fleet of DC traction motors which were found to suffer from overheating after 3 years of full operation. Overheating of these traction motors is attributed partly because of the higher than expected number of starts and stops between train terminals. Another probable cause of overheating is the design of the traction motor and/or its control strategy. According to the motor manufacturer, a current shunt is permanently connected across the motor field winding. Hence, some of the armature current is bypassed into the current shunt. The motor then runs above its rated speed in the field weakening mode. In this study, a finite difference model has been developed to simulate the temperature profile at different parts inside the traction motor. In order to validate the simulation result, an empty vehicle loaded with drums of water was also used to simulate the full pay-load of a light rail vehicle experimentally. The authors report that the simulation results agree reasonably well with experimental data, and it is likely that the armature of the traction motor will run cooler if its field shunt is disconnected at low speeds

Shaping and re-shaping social capital in buyer-supplier relationships

Social capital plays an important role in explaining how value is created from firms' network relationships, but little is understood about how social capital is shaped over time and how it is re-shaped when firms consolidate their network ties. In response, this study explores the evolution of social capital in buyer–supplier relationships through a case study of a company undertaking radical product innovation, and examines the corresponding changes in the firm's network of buyer–supplier relationships. The analysis shows that social capital is built in a decidedly non-linear and non-uniform manner. The study also reveals considerable interaction among the dimensions of social capital throughout the evolution of the firm's network, and emphasizes the importance of the cognitive dimension—a feature receiving little attention thus far. The evidence shows, too, that efforts to strengthen social capital need to increase when network ties are sacrificed to prevent unintended consequences for firms' longer-term value creation.

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.