A novel numerical approximation for the space fractional advection-dispersion equation

In this paper, we consider a space fractional advection–dispersion equation. The equation is obtained from the standard advection–diffusion equation by replacing the first- and second-order space derivatives by the Riesz fractional derivatives of order β1 ∈ (0, 1) and β2 ∈ (1, 2], respectively. The fractional advection and dispersion terms are approximated by using two fractional centred difference schemes. A new weighted Riesz fractional finite-difference approximation scheme is proposed. When the weighting factor θ equals 12, a second-order accuracy scheme is obtained. The stability, consistency and convergence of the numerical approximation scheme are discussed. A numerical example is given to show that the numerical results are in good agreement with our theoretical analysis.

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.

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.

Numerical approximation for a variable-order nonlinear reaction–subdiffusion equation

Fractional reaction–subdiffusion equations are widely used in recent years to simulate physical phenomena. In this paper, we consider a variable-order nonlinear reaction–subdiffusion equation. A numerical approximation method is proposed to solve the equation. Its convergence and stability are analyzed by Fourier analysis. By means of the technique for improving temporal accuracy, we also propose an improved numerical approximation. Finally, the effectiveness of the theoretical results is demonstrated by numerical examples.

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.

An accurate numerical scheme for the contraction of a bubble in a Hele-Shaw cell

We report on an accurate numerical scheme for the evolution of an inviscid bubble in radial Hele-Shaw flow, where the nonlinear boundary effects of surface tension and kinetic undercooling are included on the bubble-fluid interface. As well as demonstrating the onset of the Saffman-Taylor instability for growing bubbles, the numerical method is used to show the effect of the boundary conditions on the separation (pinch-off) of a contracting bubble into multiple bubbles, and the existence of multiple possible asymptotic bubble shapes in the extinction limit. The numerical scheme also allows for the accurate computation of bubbles which pinch off very close to the theoretical extinction time, raising the possibility of computing solutions for the evolution of bubbles with non-generic extinction behaviour.

Integrated genomic characterization of endometrial carcinoma

We performed an integrated genomic, transcriptomic and proteomic characterization of 373 endometrial carcinomas using array- and sequencing-based technologies. Uterine serous tumours and ∼25% of high-grade endometrioid tumours had extensive copy number alterations, few DNA methylation changes, low oestrogen receptor/progesterone receptor levels, and frequent TP53 mutations. Most endometrioid tumours had few copy number alterations or TP53 mutations, but frequent mutations in PTEN, CTNNB1, PIK3CA, ARID1A and KRAS and novel mutations in the SWI/SNF chromatin remodelling complex gene ARID5B. A subset of endometrioid tumours that we identified had a markedly increased transversion mutation frequency and newly identified hotspot mutations in POLE. Our results classified endometrial cancers into four categories: POLE ultramutated, microsatellite instability hypermutated, copy-number low, and copy-number high. Uterine serous carcinomas share genomic features with ovarian serous and basal-like breast carcinomas. We demonstrated that the genomic features of endometrial carcinomas permit a reclassification that may affect post-surgical adjuvant treatment for women with aggressive tumours.

Numerical solution of an optimal control model of dendritic cell treatment of a growing tumour

A new optimal control model of the interactions between a growing tumour and the host immune system along with an immunotherapy treatment strategy is presented. The model is based on an ordinary differential equation model of interactions between the growing tu- mour and the natural killer, cytotoxic T lymphocyte and dendritic cells of the host immune system, extended through the addition of a control function representing the application of a dendritic cell treat- ment to the system. The numerical solution of this model, obtained from a multi species Runge–Kutta forward-backward sweep scheme, is described. We investigate the effects of varying the maximum al- lowed amount of dendritic cell vaccine administered to the system and find that control of the tumour cell population is best effected via a high initial vaccine level, followed by reduced treatment and finally cessation of treatment. We also found that increasing the strength of the dendritic cell vaccine causes an increase in the number of natural killer cells and lymphocytes, which in turn reduces the growth of the tumour.

Thermal performance of load bearing cold-formed steel walls under fire conditions using numerical studies

Cold–formed Light gauge Steel Frame (LSF) wall systems are increasingly used in low-rise and multi-storey buildings and hence their fire safety has become important in the design of buildings. A composite LSF wall panel system was developed recently, where a thin insulation was sandwiched between two plasterboards to improve the fire performance of LSF walls. Many experimental and numerical studies have been undertaken to investigate the fire performance of non-load bearing LSF wall under standard conditions. However, only limited research has been undertaken to investigate the fire performance of load bearing LSF walls under standard and realistic design fire conditions. Therefore in this research, finite element thermal models of both the conventional load bearing LSF wall panels with cavity insulation and the innovative LSF composite wall panel were developed to simulate their thermal behaviour under standard and realistic design fire conditions. Suitable thermal properties were proposed for plasterboards and insulations based on laboratory tests and available literature. The developed models were then validated by comparing their results with available fire test results of load bearing LSF wall. This paper presents the details of the developed finite element models of load bearing LSF wall panels and the thermal analysis results. It shows that finite element models can be used to simulate the thermal behaviour of load bearing LSF walls with varying configurations of insulations and plasterboards. Failure times of load bearing LSF walls were also predicted based on the results from finite element thermal analyses. Finite element analysis results show that the use of cavity insulation was detrimental to the fire rating of LSF walls while the use of external insulation offered superior thermal protection to them. Effects of realistic design fire conditions are also presented in this paper.

Characterization of the sulphate mineral amarantite Fe3 2 3+ (SO4)O 7H2O using infrared, Raman spectroscopy and thermogravimetry

The mineral amarantite Fe23+(SO4)O∙7H2O has been studied using a combination of techniques including thermogravimetry, electron probe analyses and vibrational spectroscopy. Thermal analysis shows decomposition steps at 77.63, 192.2, 550 and 641.4°C. The Raman spectrum of amarantite is dominated by an intense band at 1017 cm-1 assigned to the SO42- ν1 symmetric stretching mode. Raman bands at 1039, 1054, 1098, 1131, 1195 and 1233 cm-1 are attributed to the SO42- ν3 antisymmetric stretching modes. Very intense Raman band is observed at 409 cm-1 with shoulder bands at 399, 451 and 491 cm-1 are assigned to the v2 bending modes. A series of low intensity Raman bands are found at 543, 602, 622 and 650 cm-1 are assigned to the v4 bending modes. A very sharp Raman band at 3529 cm-1 is assigned to the stretching vibration of OH units. A series of Raman bands observed at 3025, 3089, 3227, 3340, 3401 and 3480 cm-1 are assigned to water bands. Vibrational spectroscopy enables aspects of the molecular structure of the mineral amarantite to be ascertained.

Preparation and characterization of TiO2/acid leached serpentinite tailings composites and their photocatalytic reduction of Chromium(VI)

Composite TiO2/acid leached serpentine tailings (AST) were synthesized through the hydrolysis–deposition method and characterized by X-ray diffraction (XRD), scanning electron microscopy (SEM), energydispersive X-ray spectrometry (EDS), Fourier-transform infrared spectroscopy (FT-IR), transmission electron microscopy (TEM), X-ray photoelectron spectroscopy (XPS), and surface area measurement (BET). The XRD analysis showed that TiO2 coated on the surface of acid leached serpentine tailings was mixed crystal phases of rutile and anatase, the grain size of which is 10–30 nm. SEM, TEM, and EDS analysis exhibited that nano-TiO2 particles were deposited on the surface and internal cavities of acid leaching serpentine tailings. The XPS and FT-IR analysis demonstrated that the coating process of TiO2 on AST was a physical adsorption process. The large specific surface area, porous structure, and plentiful surface hydroxyl group of TiO2/AST composite resulted in the high adsorption capacity of Cr(VI). The experimental results demonstrated that initial concentration of Cr(VI), the amount of the catalyst, and pH greatly influenced the removal efficiency of Cr(VI). The removal kinetics of Cr(VI) at a relative low initial concentration was fitted well with Langmuir–Hinshelwood kinetics model with R2 value of about unity. The asprepared composites exhibited strong adsorption and photocatalytic capacity for the removal of Cr(VI), and the possible photocatalytic reduction mechanism was studied. The photodecomposition of Cr(VI) was as high as 95% within 2 h, and the reusability of the photocatalysis was proven.

Vibrational spectroscopic characterization of the phosphate mineral series eosphorite–childrenite–(Mn,Fe)Al(PO4)(OH)2·(H2O)

The phosphate mineral series eosphorite–childrenite–(Mn,Fe)Al(PO4)(OH)2·(H2O) has been studied using a combination of electron probe analysis and vibrational spectroscopy. Eosphorite is the manganese rich mineral with lower iron content in comparison with the childrenite which has higher iron and lower manganese content. The determined formulae of the two studied minerals are: (Mn0.72,Fe0.13,Ca0.01)(Al)1.04(PO4, OHPO3)1.07(OH1.89,F0.02)·0.94(H2O) for SAA-090 and (Fe0.49,Mn0.35,Mg0.06,Ca0.04)(Al)1.03(PO4, OHPO3)1.05(OH)1.90·0.95(H2O) for SAA-072. Raman spectroscopy enabled the observation of bands at 970 cm−1 and 1011 cm−1 assigned to monohydrogen phosphate, phosphate and dihydrogen phosphate units. Differences are observed in the area of the peaks between the two eosphorite minerals. Raman bands at 562 cm−1, 595 cm−1, and 608 cm−1 are assigned to the �4 bending modes of the PO4, HPO4 and H2PO4 units; Raman bands at 405 cm−1, 427 cm−1 and 466 cm−1 are attributed to the �2 modes of these units. Raman bands of the hydroxyl and water stretching modes are observed. Vibrational spectroscopy enabled details of the molecular structure of the eosphorite mineral series to be determined.

Vibrational spectroscopic characterization of the phosphate mineral kulanite Ba(Fe2+,Mn2+,Mg)2(Al,Fe3+)2(PO4)3(OH)3

The mineral kulanite BaFe2Al2(PO4)3(OH)3, a barium iron aluminum phosphate, has been studied by using a combination of electron microscopy and vibrational spectroscopy. Scanning electron microscopy with EDX shows the mineral is homogenous with no other phases present. The Raman spectrum is dominated by an intense band at 1022 cm−1 assigned to the PO43-ν1 symmetric stretching mode. Low intensity Raman bands at 1076, 1110, 1146, 1182 cm−1 are attributed to the PO43-ν3 antisymmetric stretching vibrations. The infrared spectrum shows a complex spectral profile with overlapping bands. Multiple phosphate bending vibrations supports the concept of a reduction in symmetry of the phosphate anion. Raman spectrum at 3211, 3513 and 3533 cm−1 are assigned to the stretching vibrations of the OH units. Vibrational spectroscopy enables aspects on the molecular structure of kulanite to be assessed.

Numerical solutions for thin film flow down the outside and inside of a vertical cylinder

We consider a model for thin film flow down the outside and inside of a vertical cylinder. Our focus is to study the effect that the curvature of the cylinder has on the gravity-driven instability of the advancing contact line and to simulate the resulting fingering patterns that form due to this instability. The governing partial differential equation is fourth order with a nonlinear degenerate diffusion term that represents the stabilising effect of surface tension. We present numerical solutions obtained by implementing an efficient alternating direction implicit scheme. When compared to the problem of flow down a vertical plane, we find that increasing substrate curvature tends to increase the fingering instability for flow down the outside of the cylinder, whereas flow down the inside of the cylinder substrate curvature has the opposite effect. Further, we demonstrate the existence of nontrivial travelling wave solutions which describe fingering patterns that propagate down the inside of a cylinder at constant speed without changing form. These solutions are perfectly analogous to those found previously for thin film flow down an inclined plane.

Vibrational spectroscopic characterization of the phosphate mineral bermanite–Mn2+Mn23+(PO4)2(OH)2.4(H2O)

Bermanite Mn2þMn3þ2 ðPO4Þ2ðOHÞ2 � 4ðH2OÞ is a mixed valent hydrated hydroxy phosphate mineral. The mineral is reddish-brown and occurs in crystal aggregates and as lamellar masses. Bermanite is a common mineral in granitic pegmatites. The chemical composition of bermanite was obtained using EDS techniques. We have studied the molecular structure of bermanite using vibrational spectroscopy. The mineral is characterized by a Raman doublet at 991 and 999 cm-1 attributed to the phosphate stretching mode of two non-equivalent phosphate units. Raman bands at 1071, 1117 and 1142 cm-1 are assigned to the phosphate antisymmetric stretching modes. The hydroxyl stretching spectral region is complex with overlapping bands attributed to water and hydroxyl stretching vibrations. Vibrational spectroscopy proves most useful for the study of the mineral bermanite.