A preconditioned Lanczos method for space-fractional reaction-diffusion equations on two-dimensional unstructured meshes

We consider a two-dimensional space-fractional reaction diffusion equation with a fractional Laplacian operator and homogeneous Neumann boundary conditions. The finite volume method is used with the matrix transfer technique of Ilić et al. (2006) to discretise in space, yielding a system of equations that requires the action of a matrix function to solve at each timestep. Rather than form this matrix function explicitly, we use Krylov subspace techniques to approximate the action of this matrix function. Specifically, we apply the Lanczos method, after a suitable transformation of the problem to recover symmetry. To improve the convergence of this method, we utilise a preconditioner that deflates the smallest eigenvalues from the spectrum. We demonstrate the efficiency of our approach for a fractional Fisher’s equation on the unit disk.

Neutralization of acid sulfate solutions using bauxite refinery residues and its derivatives

This investigation has shown that by transforming free caustic in red mud (RM) to Bayer hydrotalcite (during the seawater neutralization (SWN) process) enables a more controlled release mechanism for the neutralization of acid sulfate soils. The formation of hydrotalcite has been confirmed by X-ray diffraction (XRD) and differential thermalgravimetric analysis (DTG), while the dissolution of hydrotalcite and sodalite has been observed through XRD, DTG, pH plots, and ICP-OES. Coupling of all techniques enabled three neutralization mechanisms to be determined: (1) free alkali, (2) hydrotalcite dissolution, and (3) sodalite dissolution. The mechanisms are determined on the basis of ICP-OES and kinetic information. When the mass of RM or SWN-RM is greater than 0.08 g/50 mL, the pH of solution increases to a suitable value for plant life with aluminum leaching kept at a minimum. To obtain a neutralization pH greater than 6 in 10 min, the following ratio of bauxite residue (g) in 50 mL with a known iron sulfate (Fe2(SO4)3) concentration can be determined as follows: 0.04 g:50 mL:0.1 g/L of Fe2(SO4)3.

Two novel numerical methods for solving the two-dimensional fractional Fitzhugh-Nagumo-monodomain model

Fractional mathematical models represent a new approach to modelling complex spatial problems in which there is heterogeneity at many spatial and temporal scales. In this paper, a two-dimensional fractional Fitzhugh-Nagumo-monodomain model with zero Dirichlet boundary conditions is considered. The model consists of a coupled space fractional diffusion equation (SFDE) and an ordinary differential equation. For the SFDE, we first consider the numerical solution of the Riesz fractional nonlinear reaction-diffusion model and compare it to the solution of a fractional in space nonlinear reaction-diffusion model. We present two novel numerical methods for the two-dimensional fractional Fitzhugh-Nagumo-monodomain model using the shifted Grunwald-Letnikov method and the matrix transform method, respectively. Finally, some numerical examples are given to exhibit the consistency of our computational solution methodologies. The numerical results demonstrate the effectiveness of the methods.

The use of a Riesz fractional differential-based approach for texture enhancement in image processing

Abstract: Texture enhancement is an important component of image processing, with extensive application in science and engineering. The quality of medical images, quantified using the texture of the images, plays a significant role in the routine diagnosis performed by medical practitioners. Previously, image texture enhancement was performed using classical integral order differential mask operators. Recently, first order fractional differential operators were implemented to enhance images. Experiments conclude that the use of the fractional differential not only maintains the low frequency contour features in the smooth areas of the image, but also nonlinearly enhances edges and textures corresponding to high-frequency image components. However, whilst these methods perform well in particular cases, they are not routinely useful across all applications. To this end, we applied the second order Riesz fractional differential operator to improve upon existing approaches of texture enhancement. Compared with the classical integral order differential mask operators and other fractional differential operators, our new algorithms provide higher signal to noise values, which leads to superior image quality.

Numerical methods of the variable-order Rayleigh-Stokes problem for a heated generalized second grade fluid with fractional derivative

Rayleigh–Stokes problems have in recent years received much attention due to their importance in physics. In this article, we focus on the variable-order Rayleigh–Stokes problem for a heated generalized second grade fluid with fractional derivative. Implicit and explicit numerical methods are developed to solve the problem. The convergence, stability of the numerical methods and solvability of the implicit numerical method are discussed via Fourier analysis. Moreover, a numerical example is given and the results support the effectiveness of the theoretical analysis.

The analytical solution and numerical solution of the fractional diffusion-wave equation with damping

Fractional partial differential equations have been applied to many problems in physics, finance, and engineering. Numerical methods and error estimates of these equations are currently a very active area of research. In this paper we consider a fractional diffusionwave equation with damping. We derive the analytical solution for the equation using the method of separation of variables. An implicit difference approximation is constructed. Stability and convergence are proved by the energy method. Finally, two numerical examples are presented to show the effectiveness of this approximation.

Fractional models for the migration of biological cells in complex spatial domains

Travelling wave phenomena are observed in many biological applications. Mathematical theory of standard reaction-diffusion problems shows that simple partial differential equations exhibit travelling wave solutions with constant wavespeed and such models are used to describe, for example, waves of chemical concentrations, electrical signals, cell migration, waves of epidemics and population dynamics. However, as in the study of cell motion in complex spatial geometries, experimental data are often not consistent with constant wavespeed. Non-local spatial models have successfully been used to model anomalous diffusion and spatial heterogeneity in different physical contexts. In this paper, we develop a fractional model based on the Fisher-Kolmogoroff equation and analyse it for its wavespeed properties, attempting to relate the numerical results obtained from our simulations to experimental data describing enteric neural crest-derived cells migrating along the intact gut of mouse embryos. The model proposed essentially combines fractional and standard diffusion in different regions of the spatial domain and qualitatively reproduces the behaviour of neural crest-derived cells observed in the caecum and the hindgut of mouse embryos during in vivo experiments.

Computational study of methyl derivatives of ammonia borane for hydrogen storage

The structures and thermodynamic properties of methyl derivatives of ammonia–borane (BH3NH3, AB) have been studied with the frameworks of density functional theory and second-order Møller–Plesset perturbation theory. It is found that, with respect to pure AB, methyl ammonia–boranes show higher complexation energies and lower reaction enthalpies for the release of H2, together with a slight increment of the activation barrier. These results indicate that the methyl substitution can enhance the reversibility of the system and prevent the formation of BH3/NH3, but no enhancement of the release rate of H2 can be expected.

A closed-form approximation for pricing temperature-based weather derivatives

This paper develops analytical distributions of temperature indices on which temperature derivatives are written. If the deviations of daily temperatures from their expected values are modelled as an Ornstein-Uhlenbeck process with timevarying variance, then the distributions of the temperature index on which the derivative is written is the sum of truncated, correlated Gaussian deviates. The key result of this paper is to provide an analytical approximation to the distribution of this sum, thus allowing the accurate computation of payoffs without the need for any simulation. A data set comprising average daily temperature spanning over a hundred years for four Australian cities is used to demonstrate the efficacy of this approach for estimating the payoffs to temperature derivatives. It is demonstrated that expected payoffs computed directly from historical records are a particularly poor approach to the problem when there are trends in underlying average daily temperature. It is shown that the proposed analytical approach is superior to historical pricing.

AFM study of morphological changes associated with electrochemical solid--solid transformation of three-dimensional crystals of TCNQ to metal derivatives (metal= Cu, Co, Ni; TCNQ= tetracyanoquinodimethane)

In situ atomic force microscopy (AFM) allows images from the upper face and sides of TCNQ crystals to be monitored during the course of the electrochemical solid–solid state conversion of 50 × 50 μm2 three-dimensional drop cast crystals of TCNQ to CuTCNQ or M[TCNQ]2(H2O)2 (M = Co, Ni). Ex situ images obtained by scanning electron microscopy (SEM) also allow the bottom face of the TCNQ crystals, in contact with the indium tin oxide or gold electrode surface and aqueous metal electrolyte solution, to be examined. Results show that by carefully controlling the reaction conditions, nearly mono-dispersed, rod-like phase I CuTCNQ or M[TCNQ]2(H2O)2 can be achieved on all faces. However, CuTCNQ has two different phases, and the transformation of rod-like phase 1 to rhombic-like phase 2 achieved under conditions of cyclic voltammetry was monitored in situ by AFM. The similarity of in situ AFM results with ex situ SEM studies accomplished previously implies that the morphology of the samples remains unchanged when the solvent environment is removed. In the process of crystal transformation, the triple phase solid∣electrode∣electrolyte junction is confirmed to be the initial nucleation site. Raman spectra and AFM images suggest that 100% interconversion is not always achieved, even after extended electrolysis of large 50 × 50 μm2 TCNQ crystals.

The Tarim picrite-basalt-rhyolite suite, a Permian flood basalt from Northwest China with contrasting rhyolites produced by fractional crystallization and anatexis

We report major and trace element composition, Sr–Nd isotopic and seismological data for a picrite–basalt–rhyolite suite from the northern Tarim uplift (NTU), northwest China. The samples were recovered from 13 boreholes at depths between 5,166 and 6,333 m. The picritic samples have high MgO (14.5–16.8 wt%, volatiles included) enriched in incompatible element and have high 87Sr/86Sr and low 143Nd/144Nd isotopic ratios (εNd (t) = −5.3; Sri = 0.707), resembling the Karoo high-Ti picrites. All the basaltic samples are enriched in TiO2 (2.1–3.2 wt%, volatiles free), have high FeOt abundances (11.27–15.75 wt%, volatiles free), are enriched in incompatible elements and have high Sr and low Nd isotopic ratios (Sri = 0.7049–0.7065; εNd (t) = −4.1 to −0.4). High Nb/La ratios (0.91–1.34) of basalts attest that they are mantle-derived magma with negligible crustal contamination. The rhyolite samples can be subdivided into two coeval groups with overlapping U–Pb zircon ages between 291 ± 4 and 272 ± 2 Ma. Group 1 rhyolites are enriched in Nb and Ta, have similar Nb/La, Nb/U, and Sr–Nd isotopic compositions to the associated basalts, implying that they are formed by fractional crystallization of the basalts. Group 2 rhyolites are depleted in Nb and Ta, have low Nb/La ratios, and have very high Sr and low Nd isotopic ratios, implying that crustal materials have been extensively, if not exclusively, involved in their source. The picrite–basalt–rhyolite suite from the NTU, together with Permian volcanic rocks from elsewhere Tarim basin, constitute a Large Igneous Province (LIP) that is characterized by large areal extent, rapid eruption, OIB-type chemical composition, and eruption of high temperature picritic magma. The Early Permian magmatism, which covered an area >300,000 km2, is therefore named the Tarim Flood Basalt.

Rapid differentiation of isomeric lipids by photodissociation mass spectrometry of fatty acid derivatives

RATIONALE Both traditional electron ionization and electrospray ionization tandem mass spectrometry have demonstrated limitations in the unambiguous identification of fatty acids. In the former case, high electron energies lead to extensive dissociation of the radical cations from which little specific structural information can be obtained. In the latter, conventional collision-induced dissociation (CID) of even-electron ions provides little intra-chain fragmentation and thus few structural diagnostics. New approaches that harness the desirable features of both methods, namely radical-driven dissociation with discrete energy deposition, are thus required. METHODS Herein we describe the derivatization of a structurally diverse suite of fatty acids as 4-iodobenzyl esters (FAIBE). Electrospray ionization of these derivatives in the presence of sodium acetate yields abundant [M+Na]+ ions that can be mass-selected and subjected to laser irradiation (=266nm) on a modified linear ion-trap mass spectrometer. RESULTS Photodissociation (PD) of the FAIBE derivatives yields abundant radical cations by loss of atomic iodine and in several cases selective dissociation of activated carboncarbon bonds (e.g., at allylic positions) are also observed. Subsequent CID of the [M+NaI]center dot+ radical cations yields radical-directed dissociation (RDD) mass spectra that reveal extensive carboncarbon bond dissociation without scrambling of molecular information. CONCLUSIONS Both PD and RDD spectra obtained from derivatized fatty acids provide a wealth of structural information including the position(s) of unsaturation, chain-branching and hydroxylation. The structural information obtained by this approach, in particular the ability to rapidly differentiate isomeric lipids, represents a useful addition to the lipidomics tool box. Copyright (c) 2013 John Wiley & Sons, Ltd.

Bisresorcinols and arbutin derivatives from Grevillea banksii R. Br

This work is part of a series of chemical investigations of the genus Grevillea. Two new arbutin derivatives, seven new bisresorcinols, including a mixture of two isomers, three known flavonol glycosides, and four known resorcinols, including a mixture of two homologous compounds, were isolated from the ethyl acetate extract of the leaves and methanol extract of the stems of Grevillea banksii. The new compounds were identified, on the basis of spectroscopic data, as 6'-O-(3-(2(hydroxymethyl)acryloyloxy)-2-methylpropanoyl)arbutin (1), 6'-O-(2-methylacryloyl)arbutin (2), 5,5'-(4(Z)-dodecen-1,12diyl)bisresorcinol (6), 2'-methyl-5,5'-(4(Z)-tetradecen-1,14-diyl)bisresorcinol (8), 2,2'-di(4-hydroxyprenyl)-5,5'-(6(Z)-tetradecen-1,14-diyl)bisresorcinol (9), 2-(4-acetoxyprenyl)-2'-(4-hydroxyprenyl) 5,5'-(6(Z)-tetradecen-1,14-diyl)bisresorcinol (10), 2-(4-acetoxyprenyl)-2'-(4-hydroxyprenyl)5,5'-(8(Z)-tetradecen-l,14-diyl)bisresorcinol (11), 5,5'-(10(Z)-tetradecen-1-on-diyl)bisresorcinol (12) and 5,5'-(4(Z)-tetradecen-1-on-diyl)bisresorcinol (13).

Bisresorcinol Derivatives from Grevillea glauca

Seven new and three known bisresorcinols, grevirobstol A(=5,5'-((6Z,9Z)-hexadeca-6,9-diene-1,16-diyl)bisresorcinol; 8), 5,5'-[(8Z)-hexadec-8-ene-1,16-diyl]bisresorcinol (9), and 2-methyl-5,5'-[8Z)-hexadec-8-ene-1,16-diyl] bisresorcinol (10) were isolated from the stems of Grevillea glauca. The new compounds were identified on the basis of spectroscopic data as (Z)-6,7-didehydroglaucone A (1), glaucones A and B (2 and 3, resp.), 2-(3-hydroxyisopentyl)bisnorstriatol (4), 2-(3-methylbut-2-en-1-yl)bisnorstriatol (5), 2'-methylgrebustol A (6), and glaucane (7).

Antibacterial anthranilic acid derivatives from Geijera parviflora

Five anthranilic acid derivatives, a mixture I of three new compounds 11′-hexadecenoylanthranilic acid (1), 9′-hexadecenoylanthranilic acid (2), and 7′-hexadecenoylanthranilic acid (3), as well as a new compound 9,12,15-octadecatrienoylanthranilic acid (4) together with a new natural product, hexadecanoylanthranilic acid (5), were isolated from Geijera parviflora Lindl. (Rutaceae). Their structures were elucidated by extensive spectroscopic measurements, and the positions of the double bonds in compounds 1-3 of the mixture I were determined by tandem mass spectrometry employing ozone-induced dissociation. The mixture I and compound 5 showed good antibacterial activity against several Gram-positive strains. © 2013 Elsevier B.V.