Implicit difference approximation for the time fractional diffusion equation

In this paper, we consider a time fractional diffusion equation on a finite domain. The equation is obtained from the standard diffusion equation by replacing the first-order time derivative by a fractional derivative (of order $0<\alpha<1$ ). We propose a computationally effective implicit difference approximation to solve the time fractional diffusion equation. Stability and convergence of the method are discussed. We prove that the implicit difference approximation (IDA) is unconditionally stable, and the IDA is convergent with $O(\tau+h^2)$, where $\tau$ and $h$ are time and space steps, respectively. Some numerical examples are presented to show the application of the present technique.

Constructive development of the solutions of linear equations in introductory ordinary differential equations

The solution of linear ordinary differential equations (ODEs) is commonly taught in first year undergraduate mathematics classrooms, but the understanding of the concept of a solution is not always grasped by students until much later. Recognising what it is to be a solution of a linear ODE and how to postulate such solutions, without resorting to tables of solutions, is an important skill for students to carry with them to advanced studies in mathematics. In this study we describe a teaching and learning strategy that replaces the traditional algorithmic, transmission presentation style for solving ODEs with a constructive, discovery based approach where students employ their existing skills as a framework for constructing the solutions of first and second order linear ODEs. We elaborate on how the strategy was implemented and discuss the resulting impact on a first year undergraduate class. Finally we propose further improvements to the strategy as well as suggesting other topics which could be taught in a similar manner.

The influence of instructions and body-scaling as constraints on decision-making processes in team sports

Team games conceptualized as dynamical systems engender a view of emergent decision-making behaviour under constraints, although specific effects of instructional and body-scaling constraints have yet to be verified empirically. For this purpose, we studied the effects of task and individual constraints on decision-making processes in basketball. Eleven experienced female players performed 350 trials in 1 vs. 1 sub-phases of basketball in which an attacker tried to perturb the stable state of a dyad formed with a defender (i.e. break the symmetry). In Experiment 1, specific instructions (neutral, risk taking or conservative) were manipulated to observe effects on emergent behaviour of the dyadic system. When attacking players were given conservative instructions, time to cross court mid-line and variability of the attacker's trajectory were significantly greater. In Experiment 2, body-scaling of participants was manipulated by creating dyads with different height relations. When attackers were considerably taller than defenders, there were fewer occurrences of symmetry-breaking. When attackers were considerably shorter than defenders, time to cross court mid-line was significantly shorter than when dyads were composed of athletes of similar height or when attackers were considerably taller than defenders. The data exemplify how interacting task and individual constraints can influence emergent decision-making processes in team ball games.

Ecological approaches to cognition and action in sport and exercise : ask not only what you do, but where you do it

In recent decades, concepts and ideas from James J. Gibson’s theory of direct perception in ecological psychology have been applied to the study of how perception and action regulate sport performance. This article examines the influence of different streams of thought in ecological psychology for studying cognition and action in the diverse behavioural contexts of sport and exercise. In discussing the origins of ecological psychology it can be concluded that psychologists such as Lewin, and to some extent Heider, provided the initial impetus for the development of key ideas. We argue that the papers in this special issue clarify that the different schools of thinking in ecological psychology have much to contribute to theoretical and practical developments in sport and exercise psychology. For example, Gibson emphasized and formalized how the individual is coupled with the environment; Brunswik raised the issue of the ontology of probability in human behaviour and the problem of representative design for experimental task constraints; Barker looked carefully into extra-individual behavioural contexts and Bronfenbrenner presented insights pertinent to the relations between behaviour contexts, and macro influences on behaviour. In this overview, we highlight essential issues from the main schools of thought of relevance to the contexts of sport and exercise, and we consider some potential theoretical linkages with dynamical systems theory.

Constraints on the complete optimization of human motion

In sport and exercise biomechanics, forward dynamics analyses or simulations have frequently been used in attempts to establish optimal techniques for performance of a wide range of motor activities. However, the accuracy and validity of these simulations is largely dependent on the complexity of the mathematical model used to represent the neuromusculoskeletal system. It could be argued that complex mathematical models are superior to simple mathematical models as they enable basic mechanical insights to be made and individual-specific optimal movement solutions to be identified. Contrary to some claims in the literature, however, we suggest that it is currently not possible to identify the complete optimal solution for a given motor activity. For a complete optimization of human motion, dynamical systems theory implies that mathematical models must incorporate a much wider range of organismic, environmental and task constraints. These ideas encapsulate why sports medicine specialists need to adopt more individualized clinical assessment procedures in interpreting why performers' movement patterns may differ.

Information for regulating action in sport : metastability and emergence of tactical solutions under ecological constraints

The aims of this chapter are twofold. First, we show how experiments related to nonlinear dynamical systems theory can bring about insights on the interconnectedness of different information sources for action. These include the amount of information as emphasised in conventional models of cognition and action in sport and the nature of perceptual information typically emphasised in the ecological approach. The second aim was to show how, through examining the interconnectedness of these information sources, one can study the emergence of novel tactical solutions in sport; and design experiments where tactical/decisional creativity can be observed. Within this approach it is proposed that perceptual and affective information can be manipulated during practice so that the athlete's cognitive and action systems can be transposed to a meta-stable dynamical performance region where the creation of novel action information may reside.

Optimisation of performance in top level athletes : an action-based coping approach : a commentary

INTRODUCTION In their target article, Yuri Hanin and Muza Hanina outlined a novel multidisciplinary approach to performance optimisation for sport psychologists called the Identification-Control-Correction (ICC) programme. According to the authors, this empirically-verified, psycho-pedagogical strategy is designed to improve the quality of coaching and consistency of performance in highly skilled athletes and involves a number of steps including: (i) identifying and increasing self-awareness of ‘optimal’ and ‘non-optimal’ movement patterns for individual athletes; (ii) learning to deliberately control the process of task execution; and iii), correcting habitual and random errors and managing radical changes of movement patterns. Although no specific examples were provided, the ICC programme has apparently been successful in enhancing the performance of Olympic-level athletes. In this commentary, we address what we consider to be some important issues arising from the target article. We specifically focus attention on the contentious topic of optimization in neurobiological movement systems, the role of constraints in shaping emergent movement patterns and the functional role of movement variability in producing stable performance outcomes. In our view, the target article and, indeed, the proposed ICC programme, would benefit from a dynamical systems theoretical backdrop rather than the cognitive scientific approach that appears to be advocated. Although Hanin and Hanina made reference to, and attempted to integrate, constructs typically associated with dynamical systems theoretical accounts of motor control and learning (e.g., Bernstein’s problem, movement variability, etc.), these ideas required more detailed elaboration, which we provide in this commentary.