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.

Analytical solutions for the multi-term time-space Caputo-Riesz fractional advection-diffusion equations on a finite domain

Generalized fractional partial differential equations have now found wide application for describing important physical phenomena, such as subdiffusive and superdiffusive processes. However, studies of generalized multi-term time and space fractional partial differential equations are still under development. In this paper, the multi-term time-space Caputo-Riesz fractional advection diffusion equations (MT-TSCR-FADE) with Dirichlet nonhomogeneous boundary conditions are considered. The multi-term time-fractional derivatives are defined in the Caputo sense, whose orders belong to the intervals [0, 1], [1, 2] and [0, 2], respectively. These are called respectively the multi-term time-fractional diffusion terms, the multi-term time-fractional wave terms and the multi-term time-fractional mixed diffusion-wave terms. The space fractional derivatives are defined as Riesz fractional derivatives. Analytical solutions of three types of the MT-TSCR-FADE are derived with Dirichlet boundary conditions. By using Luchko's Theorem (Acta Math. Vietnam., 1999), we proposed some new techniques, such as a spectral representation of the fractional Laplacian operator and the equivalent relationship between fractional Laplacian operator and Riesz fractional derivative, that enabled the derivation of the analytical solutions for the multi-term time-space Caputo-Riesz fractional advection-diffusion equations. © 2012.

Numerical techniques for the variable order time fractional diffusion equation

In this paper we consider the variable order time fractional diffusion equation. We adopt the Coimbra variable order (VO) time fractional operator, which defines a consistent method for VO differentiation of physical variables. The Coimbra variable order fractional operator also can be viewed as a Caputo-type definition. Although this definition is the most appropriate definition having fundamental characteristics that are desirable for physical modeling, numerical methods for fractional partial differential equations using this definition have not yet appeared in the literature. Here an approximate scheme is first proposed. The stability, convergence and solvability of this numerical scheme are discussed via the technique of Fourier analysis. Numerical examples are provided to show that the numerical method is computationally efficient. Crown Copyright © 2012 Published by Elsevier Inc. All rights reserved.

Analytical solutions for the multi-term time-fractional diffusion-wave/diffusion equations in a finite domain

Multi-term time-fractional differential equations have been used for describing important physical phenomena. However, studies of the multi-term time-fractional partial differential equations with three kinds of nonhomogeneous boundary conditions are still limited. In this paper, a method of separating variables is used to solve the multi-term time-fractional diffusion-wave equation and the multi-term time-fractional diffusion equation in a finite domain. In the two equations, the time-fractional derivative is defined in the Caputo sense. We discuss and derive the analytical solutions of the two equations with three kinds of nonhomogeneous boundary conditions, namely, Dirichlet, Neumann and Robin conditions, respectively.

Intrinsic kinetics of titania photocatalysis : simplified models for their investigation

Even though titanium dioxide photocatalysis has been promoted as a leading green technology for water purification, many issues have hindered its application on a large commercial scale. For the materials scientist the main issues have centred the synthesis of more efficient materials and the investigation of degradation mechanisms; whereas for the engineers the main issues have been the development of appropriate models and the evaluation of intrinsic kinetics parameters that allow the scale up or re-design of efficient large-scale photocatalytic reactors. In order to obtain intrinsic kinetics parameters the reaction must be analysed and modelled considering the influence of the radiation field, pollutant concentrations and fluid dynamics. In this way, the obtained kinetic parameters are independent of the reactor size and configuration and can be subsequently used for scale-up purposes or for the development of entirely new reactor designs. This work investigates the intrinsic kinetics of phenol degradation over titania film due to the practicality of a fixed film configuration over a slurry. A flat plate reactor was designed in order to be able to control reaction parameters that include the UV irradiance, flow rates, pollutant concentration and temperature. Particular attention was paid to the investigation of the radiation field over the reactive surface and to the issue of mass transfer limited reactions. The ability of different emission models to describe the radiation field was investigated and compared to actinometric measurements. The RAD-LSI model was found to give the best predictions over the conditions tested. Mass transfer issues often limit fixed film reactors. The influence of this phenomenon was investigated with specifically planned sets of benzoic acid experiments and with the adoption of the stagnant film model. The phenol mass transfer coefficient in the system was calculated to be km,phenol=8.5815x10-7Re0.65(ms-1). The data obtained from a wide range of experimental conditions, together with an appropriate model of the system, has enabled determination of intrinsic kinetic parameters. The experiments were performed in four different irradiation levels (70.7, 57.9, 37.1 and 20.4 W m-2) and combined with three different initial phenol concentrations (20, 40 and 80 ppm) to give a wide range of final pollutant conversions (from 22% to 85%). The simple model adopted was able to fit the wide range of conditions with only four kinetic parameters; two reaction rate constants (one for phenol and one for the family of intermediates) and their corresponding adsorption constants. The intrinsic kinetic parameters values were defined as kph = 0.5226 mmol m-1 s-1 W-1, kI = 0.120 mmol m-1 s-1 W-1, Kph = 8.5 x 10-4 m3 mmol-1 and KI = 2.2 x 10-3 m3 mmol-1. The flat plate reactor allowed the investigation of the reaction under two different light configurations; liquid and substrate side illumination. The latter of particular interest for real world applications where light absorption due to turbidity and pollutants contained in the water stream to be treated could represent a significant issue. The two light configurations allowed the investigation of the effects of film thickness and the determination of the catalyst optimal thickness. The experimental investigation confirmed the predictions of a porous medium model developed to investigate the influence of diffusion, advection and photocatalytic phenomena inside the porous titania film, with the optimal thickness value individuated at 5 ìm. The model used the intrinsic kinetic parameters obtained from the flat plate reactor to predict the influence of thickness and transport phenomena on the final observed phenol conversion without using any correction factor; the excellent match between predictions and experimental results provided further proof of the quality of the parameters obtained with the proposed method.

Heating effects in nanofocusing metal wedges

This paper investigates theoretically and numerically local heating effects in plasmon nanofocusing structures with a particular focus on the sharp free-standing metal wedges. The developed model separates plasmon propagation in the wedge from the resultant heating effects. Therefore, this model is only applicable where the temperature increments in a nanofocusing structure are sufficiently small not to result in significant variations of the metal permittivity in the wedge. The problem is reduced to a one-dimensional heating model with a distributed heat source resulting from plasmon dissipation in the metal wedge. A simple heat conduction equation governing the local heating effects in a nanofocusing structure is derived and solved numerically for plasmonic pulses of different lengths and reasonable energies. Both the possibility of achieving substantial local temperature increments in the wedge (with a significant self-influence of the heating plasmonic pulses), and the possibility of relatively weak heating (to ensure the validity of the previously developed nanofocusing theory) are demonstrated and discussed, including the future applications of the obtained results. Applicability conditions for the developed model are also derived and discussed.

A two-metal ion mechanism operates in the hammerhead ribozyme-mediated cleavage of an RNA substrate

Evidence for a two-metal ion mechanism for cleavage of the HH16 hammerhead ribozyme is provided by monitoring the rate of cleavage of the RNA substrate as a function of La3+ concentration in the presence of a constant concentration of Mg2+. We show that a bell-shaped curve of cleavage activation is obtained as La3+ is added in micromolar concentrations in the presence of 8 mM Mg2+, with a maximal rate of cleavage being attained in the presence of 3 microM La3+. These results show that two-metal ion binding sites on the ribozyme regulate the rate of the cleavage reaction and, on the basis of earlier estimates of the Kd values for Mg2+ of 3.5 mM and > 50 mM, that these sites bind La3+ with estimated Kd values of 0.9 and > 37.5 microM, respectively. Furthermore, given the very different effects of these metal ions at the two binding sites, with displacement of Mg2+ by La3+ at the stronger (relative to Mg2+) binding site activating catalysis and displacement of Mg2+ by La3+ at the weaker (relative to Mg2+) (relative to Mg2+) binding site inhibiting catalysis, we show that the metal ions at these two sites play very different roles. We argue that the metal ion at binding site 1 coordinates the attacking 2'-oxygen species in the reaction and lowers the pKa of the attached proton, thereby increasing the concentration of the attacking alkoxide nucleophile in an equilibrium process. In contrast, the role of the metal ion at binding site 2 is to catalyze the reaction by absorbing the negative charge that accumulates at the leaving 5'-oxygen in the transition state. We suggest structural reasons why the Mg(2+)-La3+ ion combination is particularly suited to demonstrating these different roles of the two-metal ions in the ribozyme cleavage reaction.