Numerical methods for the variable-order fractional advection-diffusion equation with a nonlinear source term

In this paper, we consider a variable-order fractional advection-diffusion equation with a nonlinear source term on a finite domain. Explicit and implicit Euler approximations for the equation are proposed. Stability and convergence of the methods are discussed. Moreover, we also present a fractional method of lines, a matrix transfer technique, and an extrapolation method for the equation. Some numerical examples are given, and the results demonstrate the effectiveness of theoretical analysis.

Numerical schemes and multivariate extrapolation of a two-dimensional anomalous sub-diffusion equation

Anomalous dynamics in complex systems have gained much interest in recent years. In this paper, a two-dimensional anomalous subdiffusion equation (2D-ASDE) is considered. Two numerical methods for solving the 2D-ASDE are presented. Their stability, convergence and solvability are discussed. A new multivariate extrapolation is introduced to improve the accuracy. Finally, numerical examples are given to demonstrate the effectiveness of the schemes and confirm the theoretical analysis.

Stability and convergence of a new explicit finite-difference approximation for the variable-order nonlinear fractional diffusion equation

In this paper, we consider the variable-order nonlinear fractional diffusion equation View the MathML source where xRα(x,t) is a generalized Riesz fractional derivative of variable order View the MathML source and the nonlinear reaction term f(u,x,t) satisfies the Lipschitz condition |f(u1,x,t)-f(u2,x,t)|less-than-or-equals, slantL|u1-u2|. A new explicit finite-difference approximation is introduced. The convergence and stability of this approximation are proved. Finally, some numerical examples are provided to show that this method is computationally efficient. The proposed method and techniques are applicable to other variable-order nonlinear fractional differential equations.

Robot path planning in dynamic environments using a simulated annealing based approach

Mobile robots are widely used in many industrial fields. Research on path planning for mobile robots is one of the most important aspects in mobile robots research. Path planning for a mobile robot is to find a collision-free route, through the robot’s environment with obstacles, from a specified start location to a desired goal destination while satisfying certain optimization criteria. Most of the existing path planning methods, such as the visibility graph, the cell decomposition, and the potential field are designed with the focus on static environments, in which there are only stationary obstacles. However, in practical systems such as Marine Science Research, Robots in Mining Industry, and RoboCup games, robots usually face dynamic environments, in which both moving and stationary obstacles exist. Because of the complexity of the dynamic environments, research on path planning in the environments with dynamic obstacles is limited. Limited numbers of papers have been published in this area in comparison with hundreds of reports on path planning in stationary environments in the open literature. Recently, a genetic algorithm based approach has been introduced to plan the optimal path for a mobile robot in a dynamic environment with moving obstacles. However, with the increase of the number of the obstacles in the environment, and the changes of the moving speed and direction of the robot and obstacles, the size of the problem to be solved increases sharply. Consequently, the performance of the genetic algorithm based approach deteriorates significantly. This motivates the research of this work. This research develops and implements a simulated annealing algorithm based approach to find the optimal path for a mobile robot in a dynamic environment with moving obstacles. The simulated annealing algorithm is an optimization algorithm similar to the genetic algorithm in principle. However, our investigation and simulations have indicated that the simulated annealing algorithm based approach is simpler and easier to implement. Its performance is also shown to be superior to that of the genetic algorithm based approach in both online and offline processing times as well as in obtaining the optimal solution for path planning of the robot in the dynamic environment. The first step of many path planning methods is to search an initial feasible path for the robot. A commonly used method for searching the initial path is to randomly pick up some vertices of the obstacles in the search space. This is time consuming in both static and dynamic path planning, and has an important impact on the efficiency of the dynamic path planning. This research proposes a heuristic method to search the feasible initial path efficiently. Then, the heuristic method is incorporated into the proposed simulated annealing algorithm based approach for dynamic robot path planning. Simulation experiments have shown that with the incorporation of the heuristic method, the developed simulated annealing algorithm based approach requires much shorter processing time to get the optimal solutions in the dynamic path planning problem. Furthermore, the quality of the solution, as characterized by the length of the planned path, is also improved with the incorporated heuristic method in the simulated annealing based approach for both online and offline path planning.

Tear film surface quality with soft contact lenses using dynamic-area high-speed videokeratoscopy

Objectives. To evaluate the performance of the dynamic-area high-speed videokeratoscopy technique in the assessment of tear film surface quality with and without the presence of soft contact lenses on eye. Methods. Retrospective data from a tear film study using basic high-speed videokeratoscopy, captured at 25 frames per second, (Kopf et al., 2008, J Optom) were used. Eleven subjects had tear film analysis conducted in the morning, midday and evening on the first and seventh day of one week of no lens wear. Five of the eleven subjects then completed an extra week of hydrogel lens wear followed by a week of silicone hydrogel lens wear. Analysis was performed on a 6 second period of the inter-blink recording. The dynamic-area high-speed videokeratoscopy technique uses the maximum available area of Placido ring pattern reflected from the tear interface and eliminates regions of disturbance due to shadows from the eyelashes. A value of tear film surface quality was derived using image rocessing techniques, based on the quality of the reflected ring pattern orientation. Results. The group mean tear film surface quality and the standard deviations for each of the conditions (bare eye, hydrogel lens, and silicone hydrogel lens) showed a much lower coefficient of variation than previous methods (average reduction of about 92%). Bare eye measurements from the right and left eyes of eleven individuals showed high correlation values (Pearson’s correlation r = 0.73, p < 0.05). Repeated measures ANOVA across the 6 second period of measurement in the normal inter-blink period for the bare eye condition showed no statistically significant changes. However, across the 6 second inter-blink period with both contact lenses, statistically significant changes were observed (p < 0.001) for both types of contact lens material. Overall, wearing hydrogel and silicone hydrogel lenses caused the tear film surface quality to worsen compared with the bare eye condition (repeated measures ANOVA, p < 0.0001 for both hydrogel and silicone hydrogel). Conclusions. The results suggest that the dynamic-area method of high-speed videokeratoscopy was able to distinguish and quantify the subtle, but systematic worsening of tear film surface quality in the inter-blink interval in contact lens wear. It was also able to clearly show a difference between bare eye and contact lens wearing conditions.

Assessment of tear film surface quality using dynamic-area high-speed videokeratoscopy

A new method for noninvasive assessment of tear film surface quality (TFSQ) is proposed. The method is based on high-speed videokeratoscopy in which the corneal area for the analysis is dynamically estimated in a manner that removes videokeratoscopy interference from the shadows of eyelashes but not that related to the poor quality of the precorneal tear film that is of interest. The separation between the two types of seemingly similar videokeratoscopy interference is achieved by region-based classification in which the overall noise is first separated from the useful signal (unaltered videokeratoscopy pattern), followed by a dedicated interference classification algorithm that distinguishes between the two considered interferences. The proposed technique provides a much wider corneal area for the analysis of TFSQ than the previously reported techniques. A preliminary study with the proposed technique, carried out for a range of anterior eye conditions, showed an effective behavior in terms of noise to signal separation, interference classification, as well as consistent TFSQ results. Subsequently, the method proved to be able to not only discriminate between the bare eye and the lens on eye conditions but also to have the potential to discriminate between the two types of contact lenses.

Cosmoscapes and the promotion of uncosmopolitan values

Globality generates increasingly diffuse networks of human and non-human innovators, carriers and icons of exotic, polyethnic cosmopolitan difference; and this diffusion is increasingly hard to ignore or police (Latour 1993). In fact, such global networks of material-symbolic exchange can frequently have the unintended consequence of promoting status systems and cultural relationships founded on uncosmopolitan values such as cultural appropriation and status-based social exclusion. Moreover, this materialsymbolic engagement with cosmopolitan difference could also be rather mundane, engaged in routinely without any great reflexive consciousness or capacity to destabilise current relations of cultural power, or interpreted unproblematically as just one component of a person’s social environment. Indeed, Beck’s (2006) argument is that cosmopolitanism, in an age of global risk, is being forced upon us unwillingly, so there should be no surprise if it is a bitter pill for some to swallow. Within these emergent cosmopolitan networks, which we call ‘cosmoscapes’, there is no certainty about the development of ethical or behavioural stances consistent with claims foundational to the current literature on cosmopolitanism. Reviewing historical and contemporary studies of globality and its dynamic generative capacity, this paper considers such literatures in the context of studies of cultural consumption and social status. When one positions these diverse bodies of literature against one another, it becomes clear that the possibility of widespread cosmopolitan cultural formations is largely unpromising.

Thermal tweezers for manipulation of adatoms and nanoparticles on surfaces heated by interfering laser pulses

We conduct the detailed numerical investigation of a nanomanipulation and nanofabrication technique—thermal tweezers with dynamic evolution of surface temperature, caused by absorption of interfering laser pulses in a thin metalfilm or any other absorbing surface. This technique uses random Brownian forces in the presence of strong temperature modulation (surfacethermophoresis) for effective manipulation of particles/adatoms with nanoscale resolution. Substantial redistribution of particles on the surface is shown to occur with the typical size of the obtained pattern elements of ∼100 nm, which is significantly smaller than the wavelength of the incident pulses used (532 nm). It is also demonstrated that thermal tweezers based on surfacethermophoresis of particles/adatoms are much more effective in achieving permanent high maximum-to-minimum concentration ratios than bulk thermophoresis, which is explained by the interaction of diffusing particles with the periodic lattice potential on the surface. Typically required pulse regimes including pulse lengths and energies are also determined. The approach is applicable for reproducing any holographically achievable surfacepatterns, and can thus be used for engineering properties of surfaces including nanopatterning and design of surface metamaterials.

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.