Modified alternating direction methods for solving a two-dimensional non-continuous seepage flow with fractional derivatives

In this paper, a two-dimensional non-continuous seepage flow with fractional derivatives (2D-NCSF-FD) in uniform media is considered, which has modified the well known Darcy law. Using the relationship between Riemann-Liouville and Grunwald-Letnikov fractional derivatives, two modified alternating direction methods: a modified alternating direction implicit Euler method and a modified Peaceman-Rachford method, are proposed for solving the 2D-NCSF-FD in uniform media. The stability and consistency, thus convergence of the two methods in a bounded domain are discussed. Finally, numerical results are given.

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.

Seeing pedestrians at night : visual clutter does not mask biological motion

Although placing reflective markers on pedestrians’ major joints can make pedestrians more conspicuous to drivers at night, it has been suggested that this “biological motion” effect may be reduced when visual clutter is present. We tested whether extraneous points of light affected the ability of 12 younger and 12 older drivers to see pedestrians as they drove on a closed road at night. Pedestrians wore black clothing alone or with retroreflective markings in four different configurations. One pedestrian walked in place and was surrounded by clutter on half of the trials. Another was always surrounded by visual clutter but either walked in place or stood still. Clothing configuration, pedestrian motion, and driver age influenced conspicuity but clutter did not. The results confirm that even in the presence of visual clutter pedestrians wearing biological motion configurations are recognized more often and at greater distances than when they wear a reflective vest.

A brush with the real world : the future of inertial motion capture in live performance

3D Motion capture is a medium that plots motion, typically human motion, converting it into a form that can be represented digitally. It is a fast evolving field and recent inertial technology may provide new artistic possibilities for its use in live performance. Although not often used in this context, motion capture has a combination of attributes that can provide unique forms of collaboration with performance arts. The inertial motion capture suit used for this study has orientation sensors placed at strategic points on the body to map body motion. Its portability, real-time performance, ease of use, and its immunity from line-of-sight problems inherent in optical systems suggest it would work well as a live performance technology. Many animation techniques can be used in real-time. This research examines a broad cross-section of these techniques using four practice-led cases to assess the suitability of inertial motion capture to live performance. Although each case explores different visual possibilities, all make use of the performativity of the medium, using either an improvisational format or interactivity among stage, audience and screen that would be difficult to emulate any other way. A real-time environment is not capable of reproducing the depth and sophistication of animation people have come to expect through media. These environments take many hours to render. In time the combination of what can be produced in real-time and the tools available in a 3D environment will no doubt create their own tree of aesthetic directions in live performance. The case study looks at the potential of interactivity that this technology offers.

Numerical simulation of the Riesz fractional diffusion equation with a nonlinear source term

In this paper, A Riesz fractional diffusion equation with a nonlinear source term (RFDE-NST) is considered. This equation is commonly used to model the growth and spreading of biological species. According to the equivalent of the Riemann-Liouville(R-L) and Gr¨unwald-Letnikov(GL) fractional derivative definitions, an implicit difference approximation (IFDA) for the RFDE-NST is derived. We prove the IFDA is unconditionally stable and convergent. In order to evaluate the efficiency of the IFDA, a comparison with a fractional method of lines (FMOL) is used. Finally, two numerical examples are presented to show that the numerical results are in good agreement with our theoretical analysis.

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.