An analytical and numerical study of the two-dimensional Bratu equation

Bratu's problem, which is the nonlinear eigenvalue equationu+ exp(u)=0 withu=0 on the walls of the unit square and as the eigenvalue, is used to develop several themes on applications of Chebyshev pseudospectral methods. The first is the importance ofsymmetry: because of invariance under the C₄ rotation group and parity in bothx andy, one can slash the size of the basis set by a factor of eight and reduce the CPU time by three orders of magnitude. Second, the pseudospectral method is ananalytical as well as a numerical tool: the simple approximation3.2A exp(–0.64A), whereA is the maximum value ofu(x, y), is derived via collocation with but a single interpolation point, but is quantitatively accurate for small and moderateA. Third, the Newton-Kantorovich/Chebyshev pseudospectral algorithm is so efficient that it is possible to compute good numerical solutions—five decimal places—on amicrocomputer inbasic. Fourth, asymptotic estimates of the Chebyshev coefficients can be very misleading: the coefficients for moderately or strongly nonlinear solutions to Bratu's equations fall off exponentially rather than algebraically withv untilv is so large that one has already obtained several decimal places of accuracy. The corner singularities, which dominate the behavior of the Chebyshev coefficients in thelimit v, are so weak as to be irrelevant, and replacing Bratu's problem by a more complicated and realistic equation would merely exaggerate the unimportance of the corner branch points even more.     

Fourier transform infrared study of high-purity maceral types

A suite of density-differentiated macerals from several coals was analysed by Fourier transform infrared spectroscopy to obtain information on the nature and magnitude of the variations exhibited by the various maceral fractions. The most characteristic change between maceral groups was the variation in aliphatic hydrogen content, with exinite ? vitrinite ? inertinite. Since the separation technique (density gradient centrifugation) generally provided a number of fractions within a maceral group region, some of these were also analysed. In a series of density fractions from a low rank vitrinite, it was found that the aliphatic hydrogen content decreased as the density increased. The inertinites also exhibited a decrease in aliphatic hydrogen. The inertinite C—O bands had extinction coefficients different from those of vitrinites or exinites. The data suggest that quite profound variations in organic structure as determined by FT-i.r. spectroscopy can exist within a maceral group, so that for the most critical work on coals it is necessary to take this into account.     

Mössbauer studies of iron-bearing minerals in coal and coal ash

⁵⁷Fe Mössbauer spectroscopy has been used to characterize the iron-bearing minerals present in some British Columbia coals. The minerals observed in a range of samples include pyrite/marcasite, siderite, jarosite, rozenite or melanterite, and Fe²⁺ -bearing clay minerals such as illite. The presence of jarosite is confirmed through the magnetic hyperfine interaction observed at 4.2 K. The absence of magnetic ordering at 4.2 K for the ferrous sulphate component indicates the presence of rozenite, FeS0₄ · 4H₂0, or melanterite, FeS0₄ · 7H₂0, rather than szomolnokite, FeS0₄ · H₂0 or FeS0₄. The effects of ashing on two of the coals is also studied.     

Some aspects of the analysis of coal by X-ray fluorescence spectroscopy

Bituminous coal samples from 84 distinct sources were analysed by conventional British Standards (BS) methods for phosphorus, sulphur, chlorine, ash and the ash-forming elements (calcium, silicon, aluminium, iron, sodium, magnesium, potassium, titanium and manganese). In most cases four determinations were made per element per source. Samples were crushed to ?212 μm (72 BS mesh) and pressed into aluminium cups at 20 t in^?2 (309 MPa) without binder or backing. Two pellets from each sample were analysed by wave-length dispersive X-ray fluorescence techniques, using a spectrometer equipped with a rhodium rube; the X-ray count took 60 s. The X-ray results were calibrated against the conventional results using multiple regression. The accuracy obtained was comparable with routine ‘wet chemical’ techniques. The X-ray technique is suitable for the routine determination of large numbers of samples.     

Analysis of solvent extracts of bituminous coal macerals by chromatography, 1H n.m.r. and field-desorption mass spectrometry

Analysis of the carbon disulphide extracts of nine samples of UK coal-maceral concentrates by ¹H highresolution n.m.r. spectroscopy, gas-liquid chromatography and field-desorption mass spectrometry indicates the presence of components with a wide molecular weight range extending up to 1200 amu; these are attributed to n-alkanes up to ≈C₅₀ and highly condensed polynuclear aromatic compounds.     

Isobaric vapor—liquid equilibria for binary systems composed of methyl butanoate with ethanol and with propan-1-ol at 114.66 and 127.99 kpa

Vapor-liquid equilibrium data for the binary systems methyl butanoate/ethanol and methyl butanoate/propan-l-ol at pressures of 114.66 and 127.99 kPa are presented. Both mixtures yield an azeotrope, the compositions (ester) of which decrease quasi-linearly as the pressure increases. The experimental data for all the binary mixtures were fitted to a suitable equation and then used for comparison with predictions by ASOG and UNIFAC methods. These group contribution models give a good estimation of activity coefficients with an overall mean error less than 5% for all cases.     

CO2压缩机强烈振动的频谱诊断与治理对策

通过对安庆石化总厂化肥厂ＣＯ＿２压缩机多起强烈振动信号的频谱分析，找出了谱图上的一些规律性特征，并在理论上加以解释，文章还提出了现场应急处理的对策以及根治办法。     

Multidimensional ARKN methods for general oscillatory second-order initial value problems

Based on B-series theory, the order conditions of the multidimensional ARKN methods are presented for the general multi-frequency and multidimensional oscillatory second-order initial value problems by Wu et al. (2009). These multidimensional ARKN methods exactly integrate the multi-frequency and multidimensional unperturbed oscillators. In this paper, we pay attention to the analysis of the concrete multidimensional ARKN methods for the general multi-frequency oscillatory second-order initial value problems whose right-hand side functions depend on both  y$y$and  y^′$y^{\prime }$ (the class of physical problems which fall within its scope is broader). Numerical experiments are carried out to show that the new multidimensional ARKN methods are more efficient compared with some well-known methods for dealing with the oscillatory problems in the scientific literature.     

A conservative discontinuous Galerkin scheme for the 2D incompressible Navier–Stokes equations

In this paper we consider a conservative discretization of the two-dimensional incompressible Navier–Stokes equations. We propose an extension of Arakawa’s classical finite difference scheme for fluid flow in the vorticity–stream function formulation to a high order discontinuous Galerkin approximation. In addition, we show numerical simulations that demonstrate the accuracy of the scheme and verify the conservation properties, which are essential for long time integration. Furthermore, we discuss the massively parallel implementation on graphic processing units.