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.

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.

Changes in intraocular pressure and ocular pulse amplitude with accommodation

Aims: To investigate the change that occurs in intraocular pressure (IOP) and ocular pulse amplitude (OPA) with accommodation in young adult myopes and emmetropes. Methods: Fifteen progressing myopic and 17 emmetropic young adult subjects had their IOP and OPA measured using the Pascal dynamic contour tonometer. Measurements were taken initially with accommodation relaxed, and then following 2 min of near fixation (accommodative demand 3 D). Baseline measurements of axial length and corneal thickness were also collected prior to the IOP measures. Results: IOP significantly decreased with accommodation in both the myopic and emmetropic subjects (mean change 1.861.1 mm Hg, p<0.0001). There was no significant difference (p>0.05) between myopes and emmetropes in terms of baseline IOP or the magnitude of change in IOP with accommodation. OPA also decreased significantly with accommodation (mean change for all subjects 0.560.5, p<0.0001). The myopic subjects (baseline OPA 2.060.7 mm Hg) exhibited a significantly lower baseline OPA (p¼0.004) than the emmetropes (baseline OPA 3.261.3 mm Hg),and a significantly lower magnitude of change in OPA with accommodation. Conclusion: IOP decreases significantly with accommodation, and changes similarly in progressing myopic and emmetropic subjects. However, differences found between progressing myopes and emmetropes in the mean OPA levels and the decrease in OPA associated with accommodation suggested some changes in IOP dynamics associated with myopia.

An advanced implicit meshless approach for the non-linear anomalous subdiffusion equation

Recently, the numerical modelling and simulation for anomalous subdiffusion equation (ASDE), which is a type of fractional partial differential equation( FPDE) and has been found with widely applications in modern engineering and sciences, are attracting more and more attentions. The current dominant numerical method for modelling ASDE is Finite Difference Method (FDM), which is based on a pre-defined grid leading to inherited issues or shortcomings. This paper aims to develop an implicit meshless approach based on the radial basis functions (RBF) for numerical simulation of the non-linear ASDE. The discrete system of equations is obtained by using the meshless shape functions and the strong-forms. The stability and convergence of this meshless approach are then discussed and theoretically proven. Several numerical examples with different problem domains are used to validate and investigate accuracy and efficiency of the newly developed meshless formulation. The results obtained by the meshless formulations are also compared with those obtained by FDM in terms of their accuracy and efficiency. It is concluded that the present meshless formulation is very effective for the modeling and simulation of the ASDE. Therefore, the meshless technique should have good potential in development of a robust simulation tool for problems in engineering and science which are governed by the various types of fractional differential equations.