Numerical schemes and multivariate extrapolation of a two-dimensional anomalous sub-diffusion equation

Anomalous dynamics in complex systems have gained much interest in recent years. In this paper, a two-dimensional anomalous subdiffusion equation (2D-ASDE) is considered. Two numerical methods for solving the 2D-ASDE are presented. Their stability, convergence and solvability are discussed. A new multivariate extrapolation is introduced to improve the accuracy. Finally, numerical examples are given to demonstrate the effectiveness of the schemes and confirm the theoretical analysis.

Stability and convergence of a new explicit finite-difference approximation for the variable-order nonlinear fractional diffusion equation

In this paper, we consider the variable-order nonlinear fractional diffusion equation View the MathML source where xRα(x,t) is a generalized Riesz fractional derivative of variable order View the MathML source and the nonlinear reaction term f(u,x,t) satisfies the Lipschitz condition |f(u1,x,t)-f(u2,x,t)|less-than-or-equals, slantL|u1-u2|. A new explicit finite-difference approximation is introduced. The convergence and stability of this approximation are proved. Finally, some numerical examples are provided to show that this method is computationally efficient. The proposed method and techniques are applicable to other variable-order nonlinear fractional differential equations.

Nanoparticle melting as a Stefan moving boundary problem

The melting of spherical nanoparticles is considered from the perspective of heat flow in a pure material and as a moving boundary (Stefan) problem. The dependence of the melting temperature on both the size of the particle and the interfacial tension is described by the Gibbs-Thomson effect, and the resulting two-phase model is solved numerically using a front-fixing method. Results show that interfacial tension increases the speed of the melting process, and furthermore, the temperature distribution within the solid core of the particle exhibits behaviour that is qualitatively different to that predicted by the classical models without interfacial tension.

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.

Near infrared spectroscopy of stearic acid adsorbed on montmorillonite

The adsorption of stearic acid on both sodium montmorillonites and calcium montmorillonites has been studied by near infrared spectroscopy complimented with infrared spectroscopy. Upon adsorption of stearic acid on Ca-Mt additional near infrared bands are observed at 8236 cm-1 and is assigned to an interaction of stearic acid with the water of hydration. Upon adsorption of the stearic acid on Na-Mt, the NIR bands are now observed at 5671, 5778, 5848 and 5912 cm-1 and are assigned to the overtone and combination bands of the CH fundamentals. Additional bands at 4177, 4250, 4324, 4337, 4689 and 4809 cm-1 are attributed to CH combination bands resulting from the adsorption of the stearic acid. Stearic acid is used as a model molecule for adsorption studies. The application of near infrared spectroscopy to the study of this adsorption proved most useful.

An application of near-infrared and mid-infrared spectroscopy to the study of selected minerals

Near-infrared spectroscopy is a somewhat unutilised technique for the study of minerals. The technique has the ability to determine water content, hydroxyl groups and transition metals. In this paper we show the application of NIR spectroscopy to the study of selected minerals. The structure and spectral properties of two Cu-tellurite minerals graemite and teineite are compared with bismuth containing tellurite mineral smirnite by the application of NIR and IR spectroscopy. The position of Cu2+ bands and their splitting in the electronic spectra of tellurites are in conformity with octahedral geometry distortion. The spectral pattern of smirnite resembles graemite and the observed band at 10855 cm-1 with a weak shoulder at 7920 cm-1 is identified as due to Cu2+ ion. Any transition metal impurities may be identified by their bands in this spectral region. Three prominent bands observed in the region of 7200-6500 cm-1 are the overtones of water whilst the weak bands observed near 6200 cm-1in tellurites may be attributed to the hydrogen bonding between (TeO3)2- and H2O. The observation of a number of bands centred at around 7200 cm-1 confirms molecular water in tellurite minerals. A number of overlapping bands in the low wavenumbers 4500-4000 cm-1 is the result of combinational modes of (TeO3)2−ion. The appearance of the most intense peak at 5200 cm-1 with a pair of weak bands near 6000 cm-1 is a common feature in all the spectra and is related to the combinations of OH vibrations of water molecules, and bending vibrations ν2 (δ H2O). Bending vibrations δ H2O observed in the IR spectra shows a single band for smirnite at 1610 cm-1. The resolution of this band into number of components is evidenced for non-equivalent types of molecular water in graemite and teineite. (TeO3)2- stretching vibrations are characterized by three main absorptions at 1080, 780 and 695 cm-1.

Vibrational spectroscopy of intercalated kaolinites. Part I

The industrial application of kaolinite is closely related to its reactivity and surface properties. The reactivity of kaolinite can be tested by intercalation, i.e. via the insertion of low molecular weight organic compounds between the kaolinite layers resulting in the formation of a nano-layered organo-complex. Although intercalation of kaolinite is an old and ongoing research topic, there is a limited knowledge available on the reactivity of different kaolinites, the mechanism of complex formation as well as on the structure of the complexes formed. Grafting and incorporation of exfoliated kaolinite in polymer matrices and other potential applications can open new horizons in the study of kaolinite intercalation. This paper attempts to summarize (without completion) the most recent achievements in the study of kaolinite organo-complexes obtained with the most common intercalating compounds like urea, potassium acetate, dimethyl sulphoxide, formamide and hydrazine using vibrational spectroscopy combined with X-ray powder diffraction and thermal analysis.