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.

A comparison of different methods to determine diffusion length in dye-sensitized solar cells

A new steady state method for determination of the electron diffusion length in dye-sensitized solar cells (DSCs) is described and illustrated with data obtained using cells containing three different types of electrolyte. The method is based on using near-IR absorbance methods to establish pairs of illumination intensity for which the total number of trapped electrons is the same at open circuit (where all electrons are lost by interfacial electron transfer) as at short circuit (where the majority of electrons are collected at the contact). Electron diffusion length values obtained by this method are compared with values derived by intensity modulated methods and by impedance measurements under illumination. The results indicate that the values of electron diffusion length derived from the steady state measurements are consistently lower than the values obtained by the non steady-state methods. For all three electrolytes used in the study, the electron diffusion length was sufficiently high to guarantee electron collection efficiencies greater than 90%. Measurement of the trap distributions by near-IR absorption confirmed earlier observations of much higher electron trap densities for electrolytes containing Li+ ions. It is suggested that the electron trap distributions may not be intrinsic properties of the TiO2 nanoparticles, but may be associated with electron-ion interactions.

Mobile data technologies and SME adoption and diffusion : an empirical study on barriers and facilitators

The technological environment in which Australian SMEs operate can be best described as dynamic and vital. The rate of technological change provides the SME owner/manager a complex and challenging operational context. Wireless applications are being developed that provide mobile devices with Internet content and e-business services. In Australia the adoption of e-commerce by large organisations has been relatively high, however the same cannot be said for SMEs where adoption has been slower than other developed countries. In contrast however mobile telephone adoption and diffusion is relatively high by SMEs. This exploratory study identifies attitudes, perceptions and issues for mobile data technologies by regional SME owner/managers across a range of industry sectors. The major issues include the sector the firm belongs to, the current adoption status of the firm, the level of mistrust of the IT industry, the cost of the technologies and the applications and attributes of the technologies.

Confusion with diffusion? Unravelling is diffusion and innovation literature with a focus on SMEs

The aim of this paper is to contribute to the understanding of various models used in research for the adoption and diffusion of information technology in small and medium-sized enterprises (SMEs). Starting with Rogers' diffusion theory and behavioural models, technology adoption models used in IS research are discussed. Empirical research has shown that the reasons why firms choose to adopt or not adopt technology is dependent on a number of factors. These factors can be categorised as owner/manager characteristics, firm characteristics and other characteristics. The existing models explaining IS diffusion and adoption by SMEs overlap and complement each other. This paper reviews the existing literature and proposes a comprehensive model which includes the whole array of variables from earlier models.

Mobile data technologies and small business adoption and diffusion : an empirical study of barriers and facilitators

The technological environment in which contemporary small- and medium-sized enterprises (SMEs) operate can only be described as dynamic. The exponential rate of technological change, characterised by perceived increases in the benefits associated with various technologies, shortening product life cycles and changing standards, provides for the SME a complex and challenging operational context. The primary aim of this research was to identify the needs of SMEs in regional areas for mobile data technologies (MDT). In this study a distinction was drawn between those respondents who were full-adopters of technology, those who were partial-adopters, and those who were non-adopters and these three segments articulated different needs and requirements for MDT. Overall, the needs of regional SMEs for MDT can be conceptualised into three areas where the technology will assist business practices; communication, e-commerce and security

Studies of the radiation chemistry and grafting of a fluoropolymer

The radiation chemistry and the grafting of a fluoropolymer, poly(tetrafluoroethylene-coperfluoropropyl vinyl ether) (PFA), was investigated with the aim of developing a highly stable grafted support for use in solid phase organic chemistry (SPOC). A radiation-induced grafting method was used whereby the PFA was exposed to ionizing radiation to form free radicals capable of initiating graft copolymerization of styrene. To fully investigate this process, both the radiation chemistry of PFA and the grafting of styrene to PFA were examined. Radiation alone was found to have a detrimental effect on PFA when irradiated at 303 K. This was evident from the loss in the mechanical properties due to chain scission reactions. This meant that when radiation was used for the grafting reactions, the total radiation dose needed to be kept as low as possible. The radicals produced when PFA was exposed to radiation were examined using electron spin resonance spectroscopy. Both main-chain (–CF2–C.F–CF2-) and end-chain (–CF2–C.F2) radicals were identified. The stability of the majority of the main-chain radicals when the polymer was heated above the glass transition temperature suggested that they were present mainly in the crystalline regions of the polymer, while the end-chain radicals were predominately located in the amorphous regions. The radical yield at 77 K was lower than the radical yield at 303 K suggesting that cage recombination at low temperatures inhibited free radicals from stabilizing. High-speed MAS 19F NMR was used to identify the non-volatile products after irradiation of PFA over a wide temperature range. The major products observed over the irradiation temperature 303 to 633 K included new saturated chain ends, short fluoromethyl side chains in both the amorphous and crystalline regions, and long branch points. The proportion of the radiolytic products shifted from mainly chain scission products at low irradiation temperatures to extensive branching at higher irradiation temperatures. Calculations of G values revealed that net crosslinking only occurred when PFA was irradiated in the melt. Minor products after irradiation at elevated temperatures included internal and terminal double bonds and CF3 groups adjacent to double bonds. The volatile products after irradiation at 303 K included tetrafluoromethane (CF4) and oxygen-containing species from loss of the perfluoropropyl ether side chains of PFA as identified by mass spectrometry and FTIR spectroscopy. The chemical changes induced by radiation exposure were accompanied by changes in the thermal properties of the polymer. Changes in the crystallinity and thermal stability of PFA after irradiation were examined using DSC and TGA techniques. The equilibrium melting temperature of untreated PFA was 599 K as determined using a method of extrapolation of the melting temperatures of imperfectly formed crystals. After low temperature irradiation, radiation- induced crystallization was prevalent due to scission of strained tie molecules, loss of perfluoropropyl ether side chains, and lowering of the molecular weight which promoted chain alignment and hence higher crystallinity. After irradiation at high temperatures, the presence of short and long branches hindered crystallization, lowering the overall crystallinity. The thermal stability of the PFA decreased with increasing radiation dose and temperature due to the introduction of defect groups. Styrene was graft copolymerized to PFA using -radiation as the initiation source with the aim of preparing a graft copolymer suitable as a support for SPOC. Various grafting conditions were studied, such as the total dose, dose rate, solvent effects and addition of nitroxides to create “living” graft chains. The effect of dose rate was examined when grafting styrene vapour to PFA using the simultaneous grafting method. The initial rate of grafting was found to be independent of the dose rate which implied that the reaction was diffusion controlled. When the styrene was dissolved in various solvents for the grafting reaction, the graft yield was strongly dependent of the type and concentration of the solvent used. The greatest graft yield was observed when the solvent swelled the grafted layers and the substrate. Microprobe Raman spectroscopy was used to map the penetration of the graft into the substrate. The grafted layer was found to contain both poly(styrene) (PS) and PFA and became thicker with increasing radiation dose and graft yield which showed that grafting began at the surface and progressively penetrated the substrate as the grafted layer was swollen. The molecular weight of the grafted PS was estimated by measuring the molecular weight of the non-covalently bonded homopolymer formed in the grafted layers using SEC. The molecular weight of the occluded homopolymer was an order of magnitude greater than the free homopolymer formed in the surrounding solution suggesting that the high viscosity in the grafted regions led to long PS grafts. When a nitroxide mediated free radical polymerization was used, grafting occurred within the substrate and not on the surface due to diffusion of styrene into the substrate at the high temperatures needed for the reaction to proceed. Loading tests were used to measure the capacity of the PS graft to be functionialized with aminomethyl groups then further derivatized. These loading tests showed that samples grafted in a solution of styrene and methanol had superior loading capacity over samples graft using other solvents due to the shallow penetration and hence better accessibility of the graft when methanol was used as a solvent.

Diffusion tensor of water in model articular cartilage

We used Monte Carlo simulations of Brownian dynamics of water to study anisotropic water diffusion in an idealised model of articular cartilage. The main aim was to use the simulations as a tool for translation of the fractional anisotropy of the water diffusion tensor in cartilage into quantitative characteristics of its collagen fibre network. The key finding was a linear empirical relationship between the collagen volume fraction and the fractional anisotropy of the diffusion tensor. Fractional anisotropy of the diffusion tensor is potentially a robust indicator of the microstructure of the tissue because, in the first approximation, it is invariant to the inclusion of proteoglycans or chemical exchange between free and collagen-bound water in the model. We discuss potential applications of Monte Carlo diffusion-tensor simulations for quantitative biophysical interpretation of MRI diffusion-tensor images of cartilage. Extension of the model to include collagen fibre disorder is also discussed.