Numerical analysis of the time variable fractional order mobile-immobile advection-dispersion model

Many physical processes exhibit fractional order behavior that varies with time or space. The continuum of order in the fractional calculus allows the order of the fractional operator to be considered as a variable. In this paper, we consider the time variable fractional order mobile-immobile advection-dispersion model. Numerical methods and analyses of stability and convergence for the fractional partial differential equations are quite limited and difficult to derive. This motivates us to develop efficient numerical methods as well as stability and convergence of the implicit numerical methods for the fractional order mobile immobile advection-dispersion model. In the paper, we use the Coimbra variable time fractional derivative which is more efficient from the numerical standpoint and is preferable for modeling dynamical systems. An implicit Euler approximation for the equation is proposed and then the stability of the approximation are investigated. As for the convergence of the numerical scheme we only consider a special case, i.e. the time fractional derivative is independent of time variable t. The case where the time fractional derivative depends both the time variable t and the space variable x will be considered in the future work. Finally, numerical examples are provided to show that the implicit Euler approximation is computationally efficient.

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.

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.

Not a dirty word : arts entrepreneurship and higher education

While the majority of creative, performing, and literary artists are self-employed, relatively few tertiary arts schools attempt to develop capabilities for venture creation and management (and entrepreneurship more broadly) and still fewer do so effectively. This article asks why this is the case. It addresses underlying conceptual and philosophical issues encountered by arts educators, arguing that in all three senses of the term: new venture creation; career self-management; and being enterprising, entrepreneurship is essential to career success in the arts. However, the practice of entrepreneurship in the arts is significantly different from the practice of entrepreneurship in business, in terms of the artist’s drivers and aims, as well as the nature of entrepreneurial opportunities, contexts and processes. These differences mean that entrepreneurship curricula cannot simply be imported from Business schools. This article also examines the arts-idiosyncratic challenge of negotiating distinctive and potentially conflicting entrepreneurial aims, using career identity theory. It concludes by suggesting strategies by which adaptive entrepreneurial artist identities can be developed through higher education programs.

Whe' yu' from?

"Whe' yu' from?" The question was put to me as I wandered, camera in hand, in the old square of Spanish Town, Jamaica's former capital. The local man, lounging in the shade of one of the colonial Georgian buildings that enclose the square, was mildly curious about what he took to be a typical white tourish photgraphing the sights of the decayed historic town. At that time, my home was in Kingston where i lived with my wife and baby son. I was then working in the Jamaican Government Town Planning Department in a job that took me all over the island. Turning to my questioner, I replied, "Kingston". There was a brief pause, and then the man spoke again: "No Man! Whe' yu' really from?" I still have difficulties when asked this question. Where am I from? What does this question mean? Does it refer to where I was born, where I spent my previous life or where I live now? Does it have a broader meaning, an enquiry about my origins in terms of background and previous experience? The following chapters are my attempt to answer these questions for my own satisfaction and, I hope, for the amusement of others who may be interested in the life of an ordinary English boy whose dream to travel and see the world was realized in ways he could not possibly have imagined. Finding an appropriate title for this book was difficult. Thursday's Child, North and South and War and Peace all came to mind but, unfortunately for me, those titles had been appropriated by other writers. Thursdays's Child is quite a popular book title, presumably because people who were born on that day and, in the words of the nursery rhyme, had 'far to go', are especially likely to have travellers' tales to tell or life stories of the rags-to-riches variety. Born on a Thursday, I have travelled a lot and I suppose that I have gone far in life. Coming from a working class family, I 'got on' by 'getting a good education' and a 'good job'. I decided against adding to the list of Thursday's Children. North and South would have reflected my life in Britain, spent in both the North and South of England, and my later years, divided between the Northern and Southern Hemispheres of the globe, as well as in countries commonly referred to as the 'advanced' North and the 'underdeveloped' South. North and South has already been appropriated by Mrs Gaskell, something that did not deter one popular American writer from using the title for a book of his. My memories of World War Two and the years afterwards made War and Peace a possible candidate, but readers expectnig an epic tale of Tolstoyan proportions may have been disappointed. To my knowledge, no other book has the title "Whe' Yu' From?". I am grateful to the Jamaican man whose question lingered in my memory and provided the title of this memoir, written decades later. This book is a word picture. It is, in a sense, a self-portrait, and like all portraits, it captures something of the character, it attempts to tell the truth, but it is not the whole truth. This is because it is not my intention to write my entire life story; rather I wish to tell about some of the things in my experience of life that have seemed important or interesting to me. Unlike a painted portrait, the picture I have created is intended to suggest the passage of time. While, for most of us in Western society, time is linear and unidirectional, like the flight of an arrov or the trajectory of a bullet, memory rearranges things, calling up images of the past in no particular order, making connections that may link events in various patterns, circular, web-like, superimposed. The stream of consciousness is very unlike that of streams we encounter in the physical world. Connections are made in all directions; thoughts hop back and forth in time and space, from topic to topic. My book is a composition drawn from periods, events and thoughts as I remember them. Like life itself, it is made up of patches, some good, some bad, but in my experience, always fascinating. In recording my memories, I have been as accurate as possible. Little of what I have written is about spectacular sights and strange customs. Much of it focuses on my more modest explorations includng observations of everyday things that have attracted my attention. Reading through the chapters, I am struck by my childhood freedom to roam and engage in 'dangerous' activities like climbing trees and playing beside streams, things that many children today are no longer allowed to enjoy. Also noticeable is the survival of traditions and superstitions from the distant past. Obvious too, is my preoccupation with place names, both official ones that appear on maps and sign boards and those used by locals and children, names rarely seen in print. If there is any uniting theme to be found in what I have written, it must be my education in the fields, woods and streets of my English homeland, in the various other countries in which I have lived and travelled, as well as more formally from books and in classrooms. Much of my book is concerned with people and places. Many of the people I mention are among those who have been, and often have remained, important and close to me. Others I remember from only the briefest of encounters, but they remain in my memory because of some specific incident or circumstance that fixed a lasting image in my mind. Some of my closest friends and relatives, however, appear nowhere in these pages or they receive only the slightest mention. This is not because they played an unimportant roles in my life. It is because this book is not the whole story. Among those whe receive little or no mention are some who are especially close to me, with whom I have shared happy and sad times and who have shown me and my family much kindness, giving support when this was needed. Some I have known since childhood and have popped up at various times in my life, often in different parts of the world. Although years may pass without me seeing them, in an important sense they are always with me. These people know who they are. I hope that they know how much I love and appreciate them. When writing my memoir, I consulted a few of the people mentioned in this book, but in the main, I have relied on my own memory, asided by daiary and notebook entries and old correspondence. In the preparation of this manuscript, I benefited greatly from the expert advice and encouragement of Neil Marr of BeWrite Books. My wife Anne, inspiration for this book, also contributed in the valuable role of critic. She has my undying gratitude.