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.

Enhancing digital road map with lane details extracted from large-scale stereo aerial imagery using object-oriented image analysis

Precise, up-to-date and increasingly detailed road maps are crucial for various advanced road applications, such as lane-level vehicle navigation, and advanced driver assistant systems. With the very high resolution (VHR) imagery from digital airborne sources, it will greatly facilitate the data acquisition, data collection and updates if the road details can be automatically extracted from the aerial images. In this paper, we proposed an effective approach to detect road lane information from aerial images with employment of the object-oriented image analysis method. Our proposed algorithm starts with constructing the DSM and true orthophotos from the stereo images. The road lane details are detected using an object-oriented rule based image classification approach. Due to the affection of other objects with similar spectral and geometrical attributes, the extracted road lanes are filtered with the road surface obtained by a progressive two-class decision classifier. The generated road network is evaluated using the datasets provided by Queensland department of Main Roads. The evaluation shows completeness values that range between 76% and 98% and correctness values that range between 82% and 97%.

Towards an automatic road lane marks extraction based on ISODATA segmentation and shadow detection from large-scale aerial images

The automatic extraction of road features from remote sensed images has been a topic of great interest within the photogrammetric and remote sensing communities for over 3 decades. Although various techniques have been reported in the literature, it is still challenging to efficiently extract the road details with the increasing of image resolution as well as the requirement for accurate and up-to-date road data. In this paper, we will focus on the automatic detection of road lane markings, which are crucial for many applications, including lane level navigation and lane departure warning. The approach consists of four steps: i) data preprocessing, ii) image segmentation and road surface detection, iii) road lane marking extraction based on the generated road surface, and iv) testing and system evaluation. The proposed approach utilized the unsupervised ISODATA image segmentation algorithm, which segments the image into vegetation regions, and road surface based only on the Cb component of YCbCr color space. A shadow detection method based on YCbCr color space is also employed to detect and recover the shadows from the road surface casted by the vehicles and trees. Finally, the lane marking features are detected from the road surface using the histogram clustering. The experiments of applying the proposed method to the aerial imagery dataset of Gympie, Queensland demonstrate the efficiency of the approach.

Theoretical investigation of thermal tweezers for parallel manipulation of atoms and nanoparticles on surfaces

A major focus of research in nanotechnology is the development of novel, high throughput techniques for fabrication of arbitrarily shaped surface nanostructures of sub 100 nm to atomic scale. A related pursuit is the development of simple and efficient means for parallel manipulation and redistribution of adsorbed atoms, molecules and nanoparticles on surfaces – adparticle manipulation. These techniques will be used for the manufacture of nanoscale surface supported functional devices in nanotechnologies such as quantum computing, molecular electronics and lab-on-achip, as well as for modifying surfaces to obtain novel optical, electronic, chemical, or mechanical properties. A favourable approach to formation of surface nanostructures is self-assembly. In self-assembly, nanostructures are grown by aggregation of individual adparticles that diffuse by thermally activated processes on the surface. The passive nature of this process means it is generally not suited to formation of arbitrarily shaped structures. The self-assembly of nanostructures at arbitrary positions has been demonstrated, though these have typically required a pre-patterning treatment of the surface using sophisticated techniques such as electron beam lithography. On the other hand, a parallel adparticle manipulation technique would be suited for directing the selfassembly process to occur at arbitrary positions, without the need for pre-patterning the surface. There is at present a lack of techniques for parallel manipulation and redistribution of adparticles to arbitrary positions on the surface. This is an issue that needs to be addressed since these techniques can play an important role in nanotechnology. In this thesis, we propose such a technique – thermal tweezers. In thermal tweezers, adparticles are redistributed by localised heating of the surface. This locally enhances surface diffusion of adparticles so that they rapidly diffuse away from the heated regions. Using this technique, the redistribution of adparticles to form a desired pattern is achieved by heating the surface at specific regions. In this project, we have focussed on the holographic implementation of this approach, where the surface is heated by holographic patterns of interfering pulsed laser beams. This implementation is suitable for the formation of arbitrarily shaped structures; the only condition is that the shape can be produced by holographic means. In the simplest case, the laser pulses are linearly polarised and intersect to form an interference pattern that is a modulation of intensity along a single direction. Strong optical absorption at the intensity maxima of the interference pattern results in approximately a sinusoidal variation of the surface temperature along one direction. The main aim of this research project is to investigate the feasibility of the holographic implementation of thermal tweezers as an adparticle manipulation technique. Firstly, we investigate theoretically the surface diffusion of adparticles in the presence of sinusoidal modulation of the surface temperature. Very strong redistribution of adparticles is predicted when there is strong interaction between the adparticle and the surface, and the amplitude of the temperature modulation is ~100 K. We have proposed a thin metallic film deposited on a glass substrate heated by interfering laser beams (optical wavelengths) as a means of generating very large amplitude of surface temperature modulation. Indeed, we predict theoretically by numerical solution of the thermal conduction equation that amplitude of the temperature modulation on the metallic film can be much greater than 100 K when heated by nanosecond pulses with an energy ~1 mJ. The formation of surface nanostructures of less than 100 nm in width is predicted at optical wavelengths in this implementation of thermal tweezers. Furthermore, we propose a simple extension to this technique where spatial phase shift of the temperature modulation effectively doubles or triples the resolution. At the same time, increased resolution is predicted by reducing the wavelength of the laser pulses. In addition, we present two distinctly different, computationally efficient numerical approaches for theoretical investigation of surface diffusion of interacting adparticles – the Monte Carlo Interaction Method (MCIM) and the random potential well method (RPWM). Using each of these approaches we have investigated thermal tweezers for redistribution of both strongly and weakly interacting adparticles. We have predicted that strong interactions between adparticles can increase the effectiveness of thermal tweezers, by demonstrating practically complete adparticle redistribution into the low temperature regions of the surface. This is promising from the point of view of thermal tweezers applied to directed self-assembly of nanostructures. Finally, we present a new and more efficient numerical approach to theoretical investigation of thermal tweezers of non-interacting adparticles. In this approach, the local diffusion coefficient is determined from solution of the Fokker-Planck equation. The diffusion equation is then solved numerically using the finite volume method (FVM) to directly obtain the probability density of adparticle position. We compare predictions of this approach to those of the Ermak algorithm solution of the Langevin equation, and relatively good agreement is shown at intermediate and high friction. In the low friction regime, we predict and investigate the phenomenon of ‘optimal’ friction and describe its occurrence due to very long jumps of adparticles as they diffuse from the hot regions of the surface. Future research directions, both theoretical and experimental are also discussed.

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.

The evaluation of a biphasic osteochondral implant coupled with an electrospun membrane in a large animal model

Conventional clinical therapies are unable to resolve osteochondral defects adequately, hence tissue engineering solutions are sought to address the challenge. A biphasic implant which was seeded with Mesenchymal Stem Cells (MSC) and coupled with an electrospun membrane was evaluated as an alternative. This dual phase construct comprised of a Polycaprolactone (PCL) cartilage scaffold and a Polycaprolactone - Tri Calcium Phosphate (PCL - TCP) osseous matrix. Autologous MSC was seeded into the entire implant via fibrin and the construct was inserted into critically sized osteochondral defects located at the medial condyle and patellar groove of pigs. The defect was resurfaced with a PCL - collagen electrospun mesh that served as a substitute for periosteal flap in preventing cell leakage. Controls either without implanted MSC or resurfacing membrane were included. After 6 months, cartilaginous repair was observed with a low occurrence of fibrocartilage at the medial condyle. Osteochondral repair was promoted and host cartilage degeneration was arrested as shown by the superior Glycosaminoglycan (GAG) maintenance. This positive morphological outcome was supported by a higher relative Young's modulus which indicated functional cartilage restoration. Bone in growth and remodeling occurred in all groups with a higher degree of mineralization in the experimental group. Tissue repair was compromised in the absence of the implanted cells or the resurfacing membrane. Moreover healing was inferior at the patellar groove as compared to the medial condyle and this was attributed to the native biomechanical features.