Numerical techniques for the variable order time fractional diffusion equation

In this paper we consider the variable order time fractional diffusion equation. We adopt the Coimbra variable order (VO) time fractional operator, which defines a consistent method for VO differentiation of physical variables. The Coimbra variable order fractional operator also can be viewed as a Caputo-type definition. Although this definition is the most appropriate definition having fundamental characteristics that are desirable for physical modeling, numerical methods for fractional partial differential equations using this definition have not yet appeared in the literature. Here an approximate scheme is first proposed. The stability, convergence and solvability of this numerical scheme are discussed via the technique of Fourier analysis. Numerical examples are provided to show that the numerical method is computationally efficient. Crown Copyright © 2012 Published by Elsevier Inc. All rights reserved.

Numerical methods for solving a two-dimensional variable-order anomalous subdiffusion equation

Anomalous subdiffusion equations have in recent years received much attention. In this paper, we consider a two-dimensional variable-order anomalous subdiffusion equation. Two numerical methods (the implicit and explicit methods) are developed to solve the equation. Their stability, convergence and solvability are investigated by the Fourier method. Moreover, the effectiveness of our theoretical analysis is demonstrated by some numerical examples. © 2011 American Mathematical Society.

The heart of quality teaching : a values-based pedagogy for pre-service teacher education

In an ever changing world the adults of the future will be faced with many challenges. To cope with these challenges it seems apparent that values education will need to become paramount within a child.s education. A considerable number of research studies have indicated that values education is a critical component within education (Lovat & Toomey, 2007b). Building on this research Lovat (2006) claimed that values education was the missing link in quality teaching The concept of quality teaching had risen to the fore within educational research literature in the late 20th century with the claim that it is the teacher who makes the difference in schooling (Hattie, 2004). Thus, if teachers make such a difference to student learning, achievement and well-being, then it must hold true that pre-service teacher education programmes are vital in ensuring the development of quality teachers for our schools. The gap that this current research programme addressed was to link the fields of values education, quality teaching and pre-service teacher education. This research programme aimed to determine the impact of a values-based pedagogy on the development of quality teaching dimensions within pre-service teacher education. The values-based pedagogy that was investigated in this research programme was Philosophy in the Classroom. The research programme adopted a nested case study design based on the constructivist-interpretative paradigm in examining a unit within a pre-service teacher education programme at a Queensland university. The methodology utilised was qualitative where the main source of data was via interviews. In total, 43 pre-service teachers participated in three studies in order to determine if their involvement in a unit where the focus was on introducing pre-service teachers to an explicit values-based pedagogy impacted on their knowledge, skills and confidence in terms of quality teaching dimensions. The research programme was divided into three separate studies in order to address the two research questions: 1. In what ways do pre-service teachers perceive they are being prepared to become quality teachers? 2. Is there a connection between an explicit values-based pedagogy in pre-service teacher education and the development of pre-service teachers. understanding of quality teaching? Study One provided insight into 21 pre-service teachers. understandings of quality teaching. These 21 participants had not engaged in an explicit values-based pedagogy. Study Two involved the interviewing of 22 pre-service teachers at two separate points in time . prior to exposure to a unit that employed a values-explicit pedagogy and post this subject.s lecture content delivery. Study Three reported on and analysed individual case studies of five pre-service teachers who had participated in Study Two Time 1 and Time 2, as well as a third time following their field experience where they had practice in teaching the values explicit pedagogy. The results of the research demonstrate that an explicit values-based pedagogy introduced into a teacher education programme has a positive impact on the development of pre-service teachers. understanding of quality teaching skills and knowledge. The teaching and practice of a values-based pedagogy positively impacted on pre-service teachers with increases of knowledge, skills and confidence demonstrated on the quality teaching dimensions of intellectual quality, a supportive classroom environment, recognition of difference, connectedness and values. These findings were reinforced through the comparison of pre-service teachers who had participated in the explicit values-based pedagogical approach, with a sample of pre-service teachers who had not engaged in this same values-based pedagogical approach. A solid values-based pedagogy and practice can and does enhance pre-service teachers. understanding of quality teaching. These findings surrounding the use of a values-based pedagogy in pre-service teacher education to enhance quality teaching knowledge and skills has contributed theoretically to the field of educational research, as well having practical implications for teacher education institutions and teacher educators.

Monocular amblyopia and higher order aberrations

This study compared the corneal and total higher order aberrations between the fellow eyes in monocular amblyopia. Nineteen amblyopic subjects (8 refractive and 11 strabismic) (mean age 30 ± 11 years) were recruited. A range of biometric and optical measurements were collected from the amblyopic and non-amblyopic eye including; axial length, corneal topography and total higher order aberrations. For a sub-group of eleven non-presbyopic subjects (6 refractive and 5 strabismic amblyopes, mean age 29 ± 10 years) total higher order aberrations were also measured during accommodation (2.5 D stimuli). Amblyopic eyes were significantly shorter and more hyperopic compared to non-amblyopic eyes and the interocular difference in axial length correlated with both the magnitude of anisometropia and amblyopia (both p < 0.01). Significant differences in higher order aberrations were observed between fellow eyes, which varied with the type of amblyopia. Refractive amblyopes displayed higher levels of 4th order corneal aberrations C(4, 0)(spherical aberration), C(4, 2)(secondary astigmatism 90°) and C(4, −2)(secondary astigmatism along 45°) in the amblyopic eye compared to the non-amblyopic eye. Strabismic amblyopes exhibited significantly higher levels of C(3, 3)(trefoil) in the amblyopic eye for both corneal and total higher order aberrations. During accommodation, the amblyopic eye displayed a significantly greater lag of accommodation compared to the non-amblyopic eye, while the changes in higher order aberrations were similar in magnitude between fellow eyes. Asymmetric visual experience during development appears to be associated with asymmetries in higher order aberrations, in some cases proportional to the magnitude of anisometropia and dependent upon the amblyogenic factor.

Enabling USDL by tools

Fundamental tooling is required in order to apply USDL in practical settings. This chapter discusses three fundamental types of tools for USDL. First, USDL editors have been developed for expert and casual users, respectively. Second, several USDL repositories have been built to allow editors accessing and storing USDL descriptions. Third, our generic USDL marketplace allows providers to describe their services once and potentially trade them anywhere. In addition, the iosyncrasies of service trading as opposed to the simpler case of product trading. The chapter also presents several deployment scenarios of such tools to foster individual value chains and support new business models across organizational boundaries. We close the chapter with an application of USDL in the context of service engineering.