An implicit RBF meshless approach for time fractional diffusion equations

This paper aims to develop an implicit meshless approach based on the radial basis function (RBF) for numerical simulation of time fractional diffusion equations. The meshless RBF interpolation is firstly briefed. The discrete equations for two-dimensional time fractional diffusion equation (FDE) are obtained by using the meshless RBF shape functions and the strong-forms of the time FDE. The stability and convergence of this meshless approach are discussed and theoretically proven. Numerical examples with different problem domains and different nodal distributions are studied to validate and investigate accuracy and efficiency of the newly developed meshless approach. It has proven that the present meshless formulation is very effective for modeling and simulation of fractional differential equations.

Novel numerical methods for time-space fractional reaction diffusion equations in two dimensions

We consider time-space fractional reaction diffusion equations in two dimensions. This equation is obtained from the standard reaction diffusion equation by replacing the first order time derivative with the Caputo fractional derivative, and the second order space derivatives with the fractional Laplacian. Using the matrix transfer technique proposed by Ilic, Liu, Turner and Anh [Fract. Calc. Appl. Anal., 9:333--349, 2006] and the numerical solution strategy used by Yang, Turner, Liu, and Ilic [SIAM J. Scientific Computing, 33:1159--1180, 2011], the solution of the time-space fractional reaction diffusion equations in two dimensions can be written in terms of a matrix function vector product $f(A)b$ at each time step, where $A$ is an approximate matrix representation of the standard Laplacian. We use the finite volume method over unstructured triangular meshes to generate the matrix $A$, which is therefore non-symmetric. However, the standard Lanczos method for approximating $f(A)b$ requires that $A$ is symmetric. We propose a simple and novel transformation in which the standard Lanczos method is still applicable to find $f(A)b$, despite the loss of symmetry. Numerical results are presented to verify the accuracy and efficiency of our newly proposed numerical solution strategy.

An analytical solution for diffusion and nonlinear uptake of oxygen in a spherical cell

We develop a new analytical solution for a reactive transport model that describes the steady-state distribution of oxygen subject to diffusive transport and nonlinear uptake in a sphere. This model was originally reported by Lin (Journal of Theoretical Biology, 1976 v60, pp449–457) to represent the distribution of oxygen inside a cell and has since been studied extensively by both the numerical analysis and formal analysis communities. Here we extend these previous studies by deriving an analytical solution to a generalized reaction-diffusion equation that encompasses Lin’s model as a particular case. We evaluate the solution for the parameter combinations presented by Lin and show that the new solutions are identical to a grid-independent numerical approximation.

Influence of physical and chemical parameters on the treatment of heavy metals in polluted stormwater using zeolite : A review

Zeolite-based technology can provide a cost effective solution for stormwater treatment for the removal of toxic heavy metals under increasing demand of safe water from alternative sources. This paper reviews the currently available knowledge relating to the effect of properties of zeolites such as pore size, surface area and Si:Al ratio and the physico-chemical conditions of the system such as pH, temperature, initial metal concentration and zeolite concentration on heavy metal removal performance. The primary aims are, to consolidate available knowledge and identify knowledge gaps. It was established that an in-depth understanding of operational issues such as, diffusion of metal ions into the zeolite pore structure, pore clogging, zeolite surface coverage by particulates in stormwater as well as the effect of pH on stormwater quality in the presence of zeolites is essential for developing a zeolite-based technology for the treatment of polluted stormwater. The optimum zeolite concentration to treat typical volumes of stormwater and initial heavy metal concentrations in stormwater should also be considered as operational issues in this regard. Additionally, leaching of aluminium and sodium ions from the zeolite structure to solution were identified as key issues requiring further research in the effort to develop cost effective solutions for the removal of heavy metals from stormwater.

An advanced meshless method for time fractional diffusion equation

Recently, because of the new developments in sustainable engineering and renewable energy, which are usually governed by a series of fractional partial differential equations (FPDEs), the numerical modelling and simulation for fractional calculus are attracting more and more attention from researchers. The current dominant numerical method for modeling FPDE is Finite Difference Method (FDM), which is based on a pre-defined grid leading to inherited issues or shortcomings including difficulty in simulation of problems with the complex problem domain and in using irregularly distributed nodes. Because of its distinguished advantages, the meshless method has good potential in simulation of FPDEs. This paper aims to develop an implicit meshless collocation technique for FPDE. The discrete system of FPDEs is obtained by using the meshless shape functions and the meshless collocation formulation. The stability and convergence of this meshless approach are investigated theoretically and numerically. The numerical examples with regular and irregular nodal distributions are used to validate and investigate accuracy and efficiency of the newly developed meshless formulation. It is concluded that the present meshless formulation is very effective for the modeling and simulation of fractional partial differential equations.

Physical characteristics of small ions and their interaction with ultrafine particles in the urban environment

In recent years, the effect of ions and ultrafine particles on ambient air quality and human health has been well documented, however, knowledge about their sources, concentrations and interactions within different types of urban environments remains limited. This thesis presents the results of numerous field studies aimed at quantifying variations in ion concentration with distance from the source, as well as identifying the dynamics of the particle ionisation processes which lead to the formation of charged particles in the air. In order to select the most appropriate measurement instruments and locations for the studies, a literature review was also conducted on studies that reported ion and ultrafine particle emissions from different sources in a typical urban environment. The initial study involved laboratory experiments on the attachment of ions to aerosols, so as to gain a better understanding of the interaction between ions and particles. This study determined the efficiency of corona ions at charging and removing particles from the air, as a function of different particle number and ion concentrations. The results showed that particle number loss was directly proportional to particle charge concentration, and that higher small ion concentrations led to higher particle deposition rates in all size ranges investigated. Nanoparticles were also observed to decrease with increasing particle charge concentration, due to their higher Brownian mobility and subsequent attachment to charged particles. Given that corona discharge from high voltage powerlines is considered one of the major ion sources in urban areas, a detailed study was then conducted under three parallel overhead powerlines, with a steady wind blowing in a perpendicular direction to the lines. The results showed that large sections of the lines did not produce any corona at all, while strong positive emissions were observed from discrete components such as a particular set of spacers on one of the lines. Measurements were also conducted at eight upwind and downwind points perpendicular to the powerlines, spanning a total distance of about 160m. The maximum positive small and large ion concentrations, and DC electric field were observed at a point 20 m downwind from the lines, with median values of 4.4×103 cm-3, 1.3×103 cm-3 and 530 V m-1, respectively. It was estimated that, at this point, less than 7% of the total number of particles was charged. The electrical parameters decreased steadily with increasing downwind distance from the lines but remained significantly higher than background levels at the limit of the measurements. Moreover, vehicles are one of the most prevalent ion and particle emitting sources in urban environments, and therefore, experiments were also conducted behind a motor vehicle exhaust pipe and near busy motorways, with the aim of quantifying small ion and particle charge concentration, as well as their distribution as a function of distance from the source. The study found that approximately equal numbers of positive and negative ions were observed in the vehicle exhaust plume, as well as near motorways, of which heavy duty vehicles were believed to be the main contributor. In addition, cluster ion concentration was observed to decrease rapidly within the first 10-15 m from the road and ion-ion recombination and ion-aerosol attachment were the most likely cause of ion depletion, rather than dilution and turbulence related processes. In addition to the above-mentioned dominant ion sources, other sources also exist within urban environments where intensive human activities take place. In this part of the study, airborne concentrations of small ions, particles and net particle charge were measured at 32 different outdoor sites in and around Brisbane, Australia, which were classified into seven different groups as follows: park, woodland, city centre, residential, freeway, powerlines and power substation. Whilst the study confirmed that powerlines, power substations and freeways were the main ion sources in an urban environment, it also suggested that not all powerlines emitted ions, only those with discrete corona discharge points. In addition to the main ion sources, higher ion concentrations were also observed environments affected by vehicle traffic and human activities, such as the city centre and residential areas. A considerable number of ions were also observed in a woodland area and it is still unclear if they were emitted directly from the trees, or if they originated from some other local source. Overall, it was found that different types of environments had different types of ion sources, which could be classified as unipolar or bipolar particle sources, as well as ion sources that co-exist with particle sources. In general, fewer small ions were observed at sites with co-existing sources, however particle charge was often higher due to the effect of ion-particle attachment. In summary, this study quantified ion concentrations in typical urban environments, identified major charge sources in urban areas, and determined the spatial dispersion of ions as a function of distance from the source, as well as their controlling factors. The study also presented ion-aerosol attachment efficiencies under high ion concentration conditions, both in the laboratory and in real outdoor environments. The outcomes of these studies addressed the aims of this work and advanced understanding of the charge status of aerosols in the urban environment.