Numerical methods for fractional partial differential equations with Riesz space fractional derivatives

In this paper, we consider the numerical solution of a fractional partial differential equation with Riesz space fractional derivatives (FPDE-RSFD) on a finite domain. Two types of FPDE-RSFD are considered: the Riesz fractional diffusion equation (RFDE) and the Riesz fractional advection–dispersion equation (RFADE). The RFDE is obtained from the standard diffusion equation by replacing the second-order space derivative with the Riesz fractional derivative of order αset membership, variant(1,2]. The RFADE is obtained from the standard advection–dispersion equation by replacing the first-order and second-order space derivatives with the Riesz fractional derivatives of order βset membership, variant(0,1) and of order αset membership, variant(1,2], respectively. Firstly, analytic solutions of both the RFDE and RFADE are derived. Secondly, three numerical methods are provided to deal with the Riesz space fractional derivatives, namely, the L1/L2-approximation method, the standard/shifted Grünwald method, and the matrix transform method (MTM). Thirdly, the RFDE and RFADE are transformed into a system of ordinary differential equations, which is then solved by the method of lines. Finally, numerical results are given, which demonstrate the effectiveness and convergence of the three numerical methods.

Novel analytical and numerical methods for solving fractional dynamical systems

During the past three decades, the subject of fractional calculus (that is, calculus of integrals and derivatives of arbitrary order) has gained considerable popularity and importance, mainly due to its demonstrated applications in numerous diverse and widespread fields in science and engineering. For example, fractional calculus has been successfully applied to problems in system biology, physics, chemistry and biochemistry, hydrology, medicine, and finance. In many cases these new fractional-order models are more adequate than the previously used integer-order models, because fractional derivatives and integrals enable the description of the memory and hereditary properties inherent in various materials and processes that are governed by anomalous diffusion. Hence, there is a growing need to find the solution behaviour of these fractional differential equations. However, the analytic solutions of most fractional differential equations generally cannot be obtained. As a consequence, approximate and numerical techniques are playing an important role in identifying the solution behaviour of such fractional equations and exploring their applications. The main objective of this thesis is to develop new effective numerical methods and supporting analysis, based on the finite difference and finite element methods, for solving time, space and time-space fractional dynamical systems involving fractional derivatives in one and two spatial dimensions. A series of five published papers and one manuscript in preparation will be presented on the solution of the space fractional diffusion equation, space fractional advectiondispersion equation, time and space fractional diffusion equation, time and space fractional Fokker-Planck equation with a linear or non-linear source term, and fractional cable equation involving two time fractional derivatives, respectively. One important contribution of this thesis is the demonstration of how to choose different approximation techniques for different fractional derivatives. Special attention has been paid to the Riesz space fractional derivative, due to its important application in the field of groundwater flow, system biology and finance. We present three numerical methods to approximate the Riesz space fractional derivative, namely the L1/ L2-approximation method, the standard/shifted Gr¨unwald method, and the matrix transform method (MTM). The first two methods are based on the finite difference method, while the MTM allows discretisation in space using either the finite difference or finite element methods. Furthermore, we prove the equivalence of the Riesz fractional derivative and the fractional Laplacian operator under homogeneous Dirichlet boundary conditions – a result that had not previously been established. This result justifies the aforementioned use of the MTM to approximate the Riesz fractional derivative. After spatial discretisation, the time-space fractional partial differential equation is transformed into a system of fractional-in-time differential equations. We then investigate numerical methods to handle time fractional derivatives, be they Caputo type or Riemann-Liouville type. This leads to new methods utilising either finite difference strategies or the Laplace transform method for advancing the solution in time. The stability and convergence of our proposed numerical methods are also investigated. Numerical experiments are carried out in support of our theoretical analysis. We also emphasise that the numerical methods we develop are applicable for many other types of fractional partial differential equations.

Multiple types of child maltreatment and adolescent mental health in Viet Nam

Objective To examine the prevalence of multiple types of maltreatment (MTM), potentially confounding factors and associations with depression, anxiety and self-esteem among adolescents in Viet Nam. Methods In 2006 we conducted a cross-sectional survey of 2591 students (aged 12–18 years; 52.1% female) from randomly-selected classes in eight secondary schools in urban (Hanoi) and rural (Hai Duong) areas of northern Viet Nam (response rate, 94.7%). Sequential multiple regression analyses were performed to estimate the relative influence of individual, family and social characteristics and of eight types of maltreatment, including physical, emotional and sexual abuse and physical or emotional neglect, on adolescent mental health. Findings Females reported more neglect and emotional abuse, whereas males reported more physical abuse, but no statistically significant difference was found between genders in the prevalence of sexual abuse. Adolescents were classified as having nil (32.6%), one (25.9%), two (20.7%), three (14.5%) or all four (6.3%) maltreatment types. Linear bivariate associations between MTM and depression, anxiety and low self-esteem were observed. After controlling for demographic and family factors, MTM showed significant independent effects. The proportions of the variance explained by the models ranged from 21% to 28%. Conclusion The combined influence of adverse individual and family background factors and of child maltreatment upon mental health in adolescents in Viet Nam is consistent with research in non-Asian countries. Emotional abuse was strongly associated with each health indicator. In Asian communities where child abuse is often construed as severe physical violence, it is important to emphasize the equally pernicious effects of emotional maltreatment.

A method for evaluating lean assembly process at design stage

Lean product design has the potential to reduce the overall product development time and cost and can improve the quality of a product. However, it has been found that no or little work has been carried out to provide an integrated framework of "lean design" and to quantitatively evaluate the effectiveness of lean practices/principles in product development process. This research proposed an integrated framework for lean design process and developed a dynamic decision making tool based on Methods Time Measurement (MTM) approach for assessing the impact of lean design on the assembly process. The proposed integrated lean framework demonstrates the lean processes to be followed in the product design and assembly process in order to achieve overall leanness. The decision tool consists of a central database, the lean design guidelines, and MTM analysis. Microsoft Access and C# are utilized to develop the user interface to use the MTM analysis as decision making tool. MTM based dynamic tool is capable of estimating the assembly time, costs of parts and labour of various alternatives of a design and hence is able to achieve optimum design. A case study is conducted to test and validate the functionality of the MTM Analysis as well as to verify the lean guidelines proposed for product development.

The Imperial War Museum's social interpretation project

This report represents the output from research undertaken by University of Salford and MTM London as part of the joint Digital R&D Fund for Arts and Culture, operated by Nesta, Arts Council England and the AHRC. University of Salford and MTM London received funding from the programme to act as researchers on the Social Interpretation (SI) project, which was led by the Imperial War Museum (IWM) and their technical partners, The Centre for Digital Humanities, University College London, Knowledge Integration, and Gooii. The project was carried out between October 2011 and October 2012.

A systematic approach for selecting lean strategies and assessing leanness in manufacturing organizations

Lean strategies have been developed to eliminate or reduce manufacturing waste and thus improve operational efficiency in manufacturing processes. However, implementing lean strategies requires a large amount of resources and, in practice, manufacturers encounter difficulties in selecting appropriate lean strategies within their resource constraints. There is currently no systematic methodology available for selecting appropriate lean strategies within a manufacturer's resource constraints. In the lean transformation process, it is also critical to measure the current and desired leanness levels in order to clearly evaluate lean implementation efforts. Despite the fact that many lean strategies are utilized to reduce or eliminate manufacturing waste, little effort has been directed towards properly assessing the leanness of manufacturing organizations. In practice, a single or specific group of metrics (either qualitative or quantitative) will only partially measure the overall leanness. Existing leanness assessment methodologies do not offer a comprehensive evaluation method, integrating both quantitative and qualitative lean measures into a single quantitative value for measuring the overall leanness of an organization. This research aims to develop mathematical models and a systematic methodology for selecting appropriate lean strategies and evaluating the leanness levels in manufacturing organizations. Mathematical models were formulated and a methodology was developed for selecting appropriate lean strategies within manufacturers' limited amount of available resources to reduce their identified wastes. A leanness assessment model was developed by using the fuzzy concept to assess the leanness level and to recommend an optimum leanness value for a manufacturing organization. In the proposed leanness assessment model, both quantitative and qualitative input factors have been taken into account. Based on program developed in MATLAB and C#, a decision support tool (DST) was developed for decision makers to select lean strategies and evaluate the leanness value based on the proposed models and methodology hence sustain the lean implementation efforts. A case study was conducted to demonstrate the effectiveness of these proposed models and methodology. Case study results suggested that out of 10 wastes identified, the case organization (ABC Limited) is able to improve a maximum of six wastes from the selected workstation within their resource limitations. The selected wastes are: unnecessary motion, setup time, unnecessary transportation, inappropriate processing, work in process and raw material inventory and suggested lean strategies are: 5S, Just-In-Time, Kanban System, the Visual Management System (VMS), Cellular Manufacturing, Standard Work Process using method-time measurement (MTM), and Single Minute Exchange of Die (SMED). From the suggested lean strategies, the impact of 5S was demonstrated by measuring the leanness level of two different situations in ABC. After that, MTM was suggested as a standard work process for further improvement of the current leanness value. The initial status of the organization showed a leanness value of 0.12. By applying 5S, the leanness level significantly improved to reach 0.19 and the simulation of MTM as a standard work method shows the leanness value could be improved to 0.31. The optimum leanness value of ABC was calculated to be 0.64. These leanness values provided a quantitative indication of the impacts of improvement initiatives in terms of the overall leanness level to the case organization. Sensitivity analsysis and a t-test were also performed to validate the model proposed. This research advances the current knowledge base by developing mathematical models and methodologies to overcome lean strategy selection and leanness assessment problems. By selecting appropriate lean strategies, a manufacturer can better prioritize implementation efforts and resources to maximize the benefits of implementing lean strategies in their organization. The leanness index is used to evaluate an organization's current (before lean implementation) leanness state against the state after lean implementation and to establish benchmarking (the optimum leanness state). Hence, this research provides a continuous improvement tool for a lean manufacturing organization.

Further development of discrete computational techniques for calculation of restricted diffusion propagators in porous media

Magnetic resonance is a well-established tool for structural characterisation of porous media. Features of pore-space morphology can be inferred from NMR diffusion-diffraction plots or the time-dependence of the apparent diffusion coefficient. Diffusion NMR signal attenuation can be computed from the restricted diffusion propagator, which describes the distribution of diffusing particles for a given starting position and diffusion time. We present two techniques for efficient evaluation of restricted diffusion propagators for use in NMR porous-media characterisation. The first is the Lattice Path Count (LPC). Its physical essence is that the restricted diffusion propagator connecting points A and B in time t is proportional to the number of distinct length-t paths from A to B. By using a discrete lattice, the number of such paths can be counted exactly. The second technique is the Markov transition matrix (MTM). The matrix represents the probabilities of jumps between every pair of lattice nodes within a single timestep. The propagator for an arbitrary diffusion time can be calculated as the appropriate matrix power. For periodic geometries, the transition matrix needs to be defined only for a single unit cell. This makes MTM ideally suited for periodic systems. Both LPC and MTM are closely related to existing computational techniques: LPC, to combinatorial techniques; and MTM, to the Fokker-Planck master equation. The relationship between LPC, MTM and other computational techniques is briefly discussed in the paper. Both LPC and MTM perform favourably compared to Monte Carlo sampling, yielding highly accurate and almost noiseless restricted diffusion propagators. Initial tests indicate that their computational performance is comparable to that of finite element methods. Both LPC and MTM can be applied to complicated pore-space geometries with no analytic solution. We discuss the new methods in the context of diffusion propagator calculation in porous materials and model biological tissues.

Effect of organic gate dielectric material properties on interfacial charging and discharging of pentacene MIM device

The effect of material properties of an environmentally friendly, optically transparent dielectric material, polyterpenol, on the carrier transients within the pentacene-based double-layer MTM device was investigated. Polyterpenol films were RF plasma polymerised under varied process conditions, with resultant films differing in surface chemistry and morphology. Independent of type of polyterpenol, time-resolved EFISHG study of IZO/polyterpenol/pentacene/Au structures showed similar transient behaviour with carriers injected into pentacene from Au electrode only, confirming polyterpenol to be a suitable blocking layer for visualisation of single-species carrier transportation during charging and discharging under different bias conditions. Polyterpenol fabricated under higher input power show better promise due to higher chemical and thermal stability, improved uniformity, and absence of defects.