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.

Living characteristics of the free-radical ring-closing polymerization of diallyldimethylammonium chloride

In this communication we provide the most recent results on RAFT-mediated ring-closing polymerization of diallyldimethylammonium chloride (DADMAC). The polymerization was carried out in aqueous solution employing 2,2′-azobis(2-methylpropionamidine)-dihydrochloride as the free radical initiator and trithiocarbonate RAFT agent (2-{[(dodecylsulfanyl)carbonothioyl sulfanyl]}propanoic acid, DoPAT) as the controlling RAFT agent. The results show that – while the system is not as completely controlled as previously described – it is nevertheless possible to mediate the polymerization of DADMAC and impart some living characteristics onto the system. The initial study on the RAFT-mediated polymerization of DADMAC may have overestimated the degree of livingness within this reaction. However, it is possible – at low conversions – for some living characteristics to be observed, as the evolution of molecular weight with conversion is linear. In addition, polymers with a reasonably narrow polydispersity can be isolated.

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.

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.

Models of collective cell motion for cell populations with different aspect ratio : diffusion, proliferation and travelling waves

Continuum, partial differential equation models are often used to describe the collective motion of cell populations, with various types of motility represented by the choice of diffusion coefficient, and cell proliferation captured by the source terms. Previously, the choice of diffusion coefficient has been largely arbitrary, with the decision to choose a particular linear or nonlinear form generally based on calibration arguments rather than making any physical connection with the underlying individual-level properties of the cell motility mechanism. In this work we provide a new link between individual-level models, which account for important cell properties such as varying cell shape and volume exclusion, and population-level partial differential equation models. We work in an exclusion process framework, considering aligned, elongated cells that may occupy more than one lattice site, in order to represent populations of agents with different sizes. Three different idealizations of the individual-level mechanism are proposed, and these are connected to three different partial differential equations, each with a different diffusion coefficient; one linear, one nonlinear and degenerate and one nonlinear and nondegenerate. We test the ability of these three models to predict the population level response of a cell spreading problem for both proliferative and nonproliferative cases. We also explore the potential of our models to predict long time travelling wave invasion rates and extend our results to two dimensional spreading and invasion. Our results show that each model can accurately predict density data for nonproliferative systems, but that only one does so for proliferative systems. Hence great care must be taken to predict density data for with varying cell shape.

1,1,3,3-Tetraethylisoindolin-2-ium chloride

In the structure of the title compound C16H26N+ Cl-, the salt of a precursor in the synthesis of an isoindolin-2-yloxyl free-radical trapping agent, the cations and anions form discrete centrosymetric cyclic dimers through N---H...Cl hydrogen-bonding associations [graph set R2/4(8)].

Diffusion determinants for passive building technologies

This summary is based on an international review of leading peer reviewed journals, in both technical and management fields. It draws on highly cited articles published between 2000 and 2009 to investigate the research question, "What are the diffusion determinants for passive building technologies in Australia?". Using a conceptual framework drawn from the innovation systems literature, this paper synthesises and interprets the literature to map the current state of passive building technologies in Australia and to analyse the drivers for, and obstacles to, their optimal diffusion. The paper concludes that the government has a key role to play through its influence over the specification of building codes.

Computationally efficient methods for solving time-variable-order time-space fractional reaction-diffusion equation

Fractional differential equations are becoming more widely accepted as a powerful tool in modelling anomalous diffusion, which is exhibited by various materials and processes. Recently, researchers have suggested that rather than using constant order fractional operators, some processes are more accurately modelled using fractional orders that vary with time and/or space. In this paper we develop computationally efficient techniques for solving time-variable-order time-space fractional reaction-diffusion equations (tsfrde) using the finite difference scheme. We adopt the Coimbra variable order time fractional operator and variable order fractional Laplacian operator in space where both orders are functions of time. Because the fractional operator is nonlocal, it is challenging to efficiently deal with its long range dependence when using classical numerical techniques to solve such equations. The novelty of our method is that the numerical solution of the time-variable-order tsfrde is written in terms of a matrix function vector product at each time step. This product is approximated efficiently by the Lanczos method, which is a powerful iterative technique for approximating the action of a matrix function by projecting onto a Krylov subspace. Furthermore an adaptive preconditioner is constructed that dramatically reduces the size of the required Krylov subspaces and hence the overall computational cost. Numerical examples, including the variable-order fractional Fisher equation, are presented to demonstrate the accuracy and efficiency of the approach.

Efficient preconditioning of the method of lines for solving nonlinear two-sided space-fractional diffusion equations

A standard method for the numerical solution of partial differential equations (PDEs) is the method of lines. In this approach the PDE is discretised in space using �finite di�fferences or similar techniques, and the resulting semidiscrete problem in time is integrated using an initial value problem solver. A significant challenge when applying the method of lines to fractional PDEs is that the non-local nature of the fractional derivatives results in a discretised system where each equation involves contributions from many (possibly every) spatial node(s). This has important consequences for the effi�ciency of the numerical solver. First, since the cost of evaluating the discrete equations is high, it is essential to minimise the number of evaluations required to advance the solution in time. Second, since the Jacobian matrix of the system is dense (partially or fully), methods that avoid the need to form and factorise this matrix are preferred. In this paper, we consider a nonlinear two-sided space-fractional di�ffusion equation in one spatial dimension. A key contribution of this paper is to demonstrate how an eff�ective preconditioner is crucial for improving the effi�ciency of the method of lines for solving this equation. In particular, we show how to construct suitable banded approximations to the system Jacobian for preconditioning purposes that permit high orders and large stepsizes to be used in the temporal integration, without requiring dense matrices to be formed. The results of numerical experiments are presented that demonstrate the effectiveness of this approach.

Gold coating of silica and zinc oxide nanoparticles by the surface reduction of gold(I) chloride

The possibility of a surface inner sphere electron transfer mechanism leading to the coating of gold via the surface reduction of gold(I) chloride on metal and semi-metal oxide nanoparticles was investigated. Silica and zinc oxide nanoparticles are known to have very different surface chemistry, potentially leading to a new class of gold coated nanoparticles. Monodisperse silica nanoparticles were synthesised by the well known Stöber protocol in conjunction with sonication. The nanoparticle size was regulated solely by varying the amount of ammonia solution added. The presence of surface hydroxyl groups was investigated by liquid proton NMR. The resultant nanoparticle size was directly measured by the use of TEM. The synthesised silica nanoparticles were dispersed in acetonitrile (MeCN) and added to a bis acetonitrile gold(I) co-ordination complex [Au(MeCN)2]+ in MeCN. The silica hydroxyl groups were deprotonated in the presence of MeCN generating a formal negative charge on the siloxy groups. This allowed the [Au(MeCN)2]+ complex to undergo ligand exchange with the silica nanoparticles, which formed a surface co-ordination complex with reduction to gold(0), that proceeded by a surface inner sphere electron transfer mechanism. The residual [Au(MeCN)2]+ complex was allowed to react with water, disproportionating into gold(0) and gold(III) respectively, with gold(0) being added to the reduced gold already bound on the silica surface. The so-formed metallic gold seed surface was found to be suitable for the conventional reduction of gold(III) to gold(0) by ascorbic acid. This process generated a thin and uniform gold coating on the silica nanoparticles. This process was modified to include uniformly gold coated composite zinc oxide nanoparticles (Au@ZnO NPs) using surface co-ordination chemistry. AuCl dissolved in acetonitrile (MeCN) supplied chloride ions which were adsorbed onto ZnO NPs. The co-ordinated gold(I) was reduced on the ZnO surface to gold(0) by the inner sphere electron transfer mechanism. Addition of water disproportionated the remaining gold(I) to gold(0) and gold(III). Gold(0) bonded to gold(0) on the NP surface with gold(III) was reduced to gold(0) by ascorbic acid (ASC), which completed the gold coating process. This gold coating process of Au@ZnO NPs was modified to incorporate iodide instead of chloride. ZnO NPs were synthesised by the use of sodium oxide, zinc iodide and potassium iodide in refluxing basic ethanol with iodide controlling the presence of chemisorbed oxygen. These ZnO NPs were treated by the addition of gold(I) chloride dissolved in acetonitrile leaving chloride anions co-ordinated on the ZnO NP surface. This allowed acetonitrile ligands in the added [Au(MeCN)2]+ complex to surface exchange with adsorbed chloride from the dissolved AuCl on the ZnO NP surface. Gold(I) was then reduced by the surface inner sphere electron transfer mechanism. The presence of the reduced gold on the ZnO NPs allowed adsorption of iodide to generate a uniform deposition of gold onto the ZnO NP surface without the use of additional reducing agents or heat.

Numerical methods for solving the multi-term time-fractional wave-diffusion equations

In this paper, the multi-term time-fractional wave diffusion equations are considered. The multiterm time fractional derivatives are defined in the Caputo sense, whose orders belong to the intervals [0,1], [1,2), [0,2), [0,3), [2,3) and [2,4), respectively. Some computationally effective numerical methods are proposed for simulating the multi-term time-fractional wave-diffusion equations. The numerical results demonstrate the effectiveness of theoretical analysis. These methods and techniques can also be extended to other kinds of the multi-term fractional time-space models with fractional Laplacian.

The chemistry of acid catalyzed delignification of sugarcane bagasse in the ionic liquid trihexyl tetradecyl phosphonium chloride

This article reports on the cleavage of lignin ß-aryl ether bonds in sugarcane bagasse by the ionic liquid (IL) trihexyl tetradecyl phosphonium chloride [P66614] Cl, in the presence of catalytic amounts of mineral acid fca. 0.4%). The deligniflcation process of bagasse was studied over a range of temperatures (120°C to 150°C) by monitoring the production of ß-ketones (indicative of cleavage of ß-aryl ethers) using FTIR spectroscopy and by compositional analysis of the undissolved fractions. Maximum deligniflcation was obtained at 150°C, with 52% of lignin removed from the original lignin content of bagasse. No deligniflcation was observed in the absence of acid, which suggests that the reaction is acid catalyzed with the IL solubilizing the lignin fragments. The rate of deligniflcation was significantly higher at 150°C, suggesting that crossing the glass transition temperature of lignin effects greater freedom of rotation about the propanoid carbon-carbon bonds and leads to increased cleavage of ß-aryl ethers. An attempt has been made to propose a probable mechanism of deligniflcation of bagasse with the phosphonuim IL. © Taylor & Francis Group, LLC.