A numerical method for the fractional Fitzhugh–Nagumo monodomain model

A fractional FitzHugh–Nagumo monodomain model with zero Dirichlet boundary conditions is presented, generalising the standard monodomain model that describes the propagation of the electrical potential in heterogeneous cardiac tissue. The model consists of a coupled fractional Riesz space nonlinear reaction-diffusion model and a system of ordinary differential equations, describing the ionic fluxes as a function of the membrane potential. We solve this model by decoupling the space-fractional partial differential equation and the system of ordinary differential equations at each time step. Thus, this means treating the fractional Riesz space nonlinear reaction-diffusion model as if the nonlinear source term is only locally Lipschitz. The fractional Riesz space nonlinear reaction-diffusion model is solved using an implicit numerical method with the shifted Grunwald–Letnikov approximation, and the stability and convergence are discussed in detail in the context of the local Lipschitz property. Some numerical examples are given to show the consistency of our computational approach.

Tubule detection in testis images using boundary weighting and circular shortest path

In studies of germ cell transplantation, measureing tubule diameters and counting cells from different populations using antibodies as markers are very important. Manual measurement of tubule sizes and cell counts is a tedious and sanity grinding work. In this paper, we propose a new boundary weighting based tubule detection method. We first enhance the linear features of the input image and detect the approximate centers of tubules. Next, a boundary weighting transform is applied to the polar transformed image of each tubule region and a circular shortest path is used for the boundary detection. Then, ellipse fitting is carried out for tubule selection and measurement. The algorithm has been tested on a dataset consisting of 20 images, each having about 20 tubules. Experiments show that the detection results of our algorithm are very close to the results obtained manually. © 2013 IEEE.

A Boundary-Oriented Chinese Segmentation Method Using NGram Mutual Information

This paper describes our participation in the Chinese word segmentation task of CIPS-SIGHAN 2010. We implemented an n-gram mutual information (NGMI) based segmentation algorithm with the mixed-up features from unsupervised, supervised and dictionarybased segmentation methods. This algorithm is also combined with a simple strategy for out-of-vocabulary (OOV) word recognition. The evaluation for both open and closed training shows encouraging results of our system. The results for OOV word recognition in closed training evaluation were however found unsatisfactory.

A reduction of diffusion in PVA Fricke hydrogels

A modification to the PVA-FX hydrogel whereby the chelating agent, xylenol orange, was partially bonded to the gelling agent, poly-vinyl alcohol, resulted in an 8% reduction in the post irradiation Fe3+ diffusion, adding approximately 1 hour to the useful timespan between irradiation and readout. This xylenol orange functionalised poly-vinyl alcohol hydrogel had an OD dose sensitivity of 0.014 Gy−1 and a diffusion rate of 0.133 mm2 h−1. As this partial bond yields only incremental improvement, it is proposed that more efficient methods of bonding xylenol orange to poly-vinyl alcohol be investigated to further reduce the diffusion in Fricke gels.

Green urbanism and diffusion issues

This paper addresses the research question, ‘What are the diffusion determinants for green urbanism innovations in Australia?’ This is a significant topic given the global movement towards green urbanism. The study reported here is based on desktop research that provides new insights through (1) synthesis of the latest research findings on green urbanism innovations and (2) interpretation of diffusion issues through our innovation system model. Although innovation determinants have been studied extensively overseas and in Australia, there is presently a gap in the literature when it comes to these determinants for green urbanism in Australia. The current paper fills this gap. Using a conceptual framework drawn from the innovation systems literature, this paper synthesises and interprets the literature to map the current state of green urbanism innovations in Australia and to analyse the drivers for, and obstacles to, their optimal diffusion. The results point to the importance of collaboration between project-based actors in the implementation of green urbanism. Education, training and regulation across the product system is also required to improve the cultural and technical context for implementation. The results are limited by their exploratory nature and future research is planned to quantify barriers to green urbanism.

Further development of discrete computational techniques for calculation of restricted diffusion propagators in porous media

Magnetic resonance is a well-established tool for structural characterisation of porous media. Features of pore-space morphology can be inferred from NMR diffusion-diffraction plots or the time-dependence of the apparent diffusion coefficient. Diffusion NMR signal attenuation can be computed from the restricted diffusion propagator, which describes the distribution of diffusing particles for a given starting position and diffusion time. We present two techniques for efficient evaluation of restricted diffusion propagators for use in NMR porous-media characterisation. The first is the Lattice Path Count (LPC). Its physical essence is that the restricted diffusion propagator connecting points A and B in time t is proportional to the number of distinct length-t paths from A to B. By using a discrete lattice, the number of such paths can be counted exactly. The second technique is the Markov transition matrix (MTM). The matrix represents the probabilities of jumps between every pair of lattice nodes within a single timestep. The propagator for an arbitrary diffusion time can be calculated as the appropriate matrix power. For periodic geometries, the transition matrix needs to be defined only for a single unit cell. This makes MTM ideally suited for periodic systems. Both LPC and MTM are closely related to existing computational techniques: LPC, to combinatorial techniques; and MTM, to the Fokker-Planck master equation. The relationship between LPC, MTM and other computational techniques is briefly discussed in the paper. Both LPC and MTM perform favourably compared to Monte Carlo sampling, yielding highly accurate and almost noiseless restricted diffusion propagators. Initial tests indicate that their computational performance is comparable to that of finite element methods. Both LPC and MTM can be applied to complicated pore-space geometries with no analytic solution. We discuss the new methods in the context of diffusion propagator calculation in porous materials and model biological tissues.

A meshless method based on Point Interpolation Method (PIM) for the space fractional diffusion equation

This paper aims to develop a meshless approach based on the Point Interpolation Method (PIM) for numerical simulation of a space fractional diffusion equation. Two fully-discrete schemes for the one-dimensional space fractional diffusion equation are obtained by using the PIM and the strong-forms of the space diffusion equation. Numerical examples with different nodal distributions are studied to validate and investigate the accuracy and efficiency of the newly developed meshless approach.

A fast semi-implicit difference method for a nonlinear two-sided space-fractional diffusion equation with variable diffusivity coefficients

In this paper, we derive a new nonlinear two-sided space-fractional diffusion equation with variable coefficients from the fractional Fick’s law. A semi-implicit difference method (SIDM) for this equation is proposed. The stability and convergence of the SIDM are discussed. For the implementation, we develop a fast accurate iterative method for the SIDM by decomposing the dense coefficient matrix into a combination of Toeplitz-like matrices. This fast iterative method significantly reduces the storage requirement of O(n2)O(n2) and computational cost of O(n3)O(n3) down to n and O(nlogn)O(nlogn), where n is the number of grid points. The method retains the same accuracy as the underlying SIDM solved with Gaussian elimination. Finally, some numerical results are shown to verify the accuracy and efficiency of the new method.

A Crank--Nicolson ADI spectral method for a two-dimensional riesz space fractional nonlinear reaction-diffusion equation

In this paper, a new alternating direction implicit Galerkin--Legendre spectral method for the two-dimensional Riesz space fractional nonlinear reaction-diffusion equation is developed. The temporal component is discretized by the Crank--Nicolson method. The detailed implementation of the method is presented. The stability and convergence analysis is strictly proven, which shows that the derived method is stable and convergent of order $2$ in time. An optimal error estimate in space is also obtained by introducing a new orthogonal projector. The present method is extended to solve the fractional FitzHugh--Nagumo model. Numerical results are provided to verify the theoretical analysis.