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.

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.

A comparison of finite difference and finite volume methods for solving the space fractional advection-dispersion equation with variable coefficients

Transport processes within heterogeneous media may exhibit non-classical diffusion or dispersion; that is, not adequately described by the classical theory of Brownian motion and Fick's law. We consider a space fractional advection-dispersion equation based on a fractional Fick's law. The equation involves the Riemann-Liouville fractional derivative which arises from assuming that particles may make large jumps. Finite difference methods for solving this equation have been proposed by Meerschaert and Tadjeran. In the variable coefficient case, the product rule is first applied, and then the Riemann-Liouville fractional derivatives are discretised using standard and shifted Grunwald formulas, depending on the fractional order. In this work, we consider a finite volume method that deals directly with the equation in conservative form. Fractionally-shifted Grunwald formulas are used to discretise the fractional derivatives at control volume faces. We compare the two methods for several case studies from the literature, highlighting the convenience of the finite volume approach.

Spectral approximations to the fractional integral and derivative

In this paper, the spectral approximations are used to compute the fractional integral and the Caputo derivative. The effective recursive formulae based on the Legendre, Chebyshev and Jacobi polynomials are developed to approximate the fractional integral. And the succinct scheme for approximating the Caputo derivative is also derived. The collocation method is proposed to solve the fractional initial value problems and boundary value problems. Numerical examples are also provided to illustrate the effectiveness of the derived 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.

A computationally effective alternating direction method for the space and time fractional Bloch–Torrey equation in 3-D

The space and time fractional Bloch–Torrey equation (ST-FBTE) has been used to study anomalous diffusion in the human brain. Numerical methods for solving ST-FBTE in three-dimensions are computationally demanding. In this paper, we propose a computationally effective fractional alternating direction method (FADM) to overcome this problem. We consider ST-FBTE on a finite domain where the time and space derivatives are replaced by the Caputo–Djrbashian and the sequential Riesz fractional derivatives, respectively. The stability and convergence properties of the FADM are discussed. Finally, some numerical results for ST-FBTE are given to confirm our theoretical findings.

Incorporation of bioactive polyvinylpyrrolidone–iodine within bilayered collagen scaffolds enhances the differentiation and subchondral osteogenesis of mesenchymal stem cells

Polyvinylpyrrolidone–iodine (Povidone-iodine, PVP-I) is widely used as an antiseptic agent for lavation during joint surgery; however, the biological effects of PVP–I on cells from joint tissue are unknown. This study examined the biocompatibility and biological effects of PVP–I on cells from joint tissue, with the aim of optimizing cell-scaffold based joint repair. Cells from joint tissue, including cartilage derived progenitor cells (CPC), subchondral bone derived osteoblast and bone marrow derived mesenchymal stem cells (BM-MSC) were isolated. The concentration-dependent effects of PVP–I on cell proliferation, migration and differentiation were evaluated. Additionally, the efficacy and mechanism of a PVP–I loaded bilayer collagen scaffold for osteochondral defect repair was investigated in a rabbit model. A micromolar concentration of PVP–I was found not to affect cell proliferation, CPC migration or extracellular matrix production. Interestingly, micromolar concentrations of PVP–I promote osteogenic differentiation of BM-MSC, as evidenced by up-regulation of RUNX2 and Osteocalcin gene expression, as well as increased mineralization on the three-dimensional scaffold. PVP–I treatment of collagen scaffolds significantly increased fibronectin binding onto the scaffold surface and collagen type I protein synthesis of cultured BM-MSC. Implantation of PVP–I treated collagen scaffolds into rabbit osteochondral defect significantly enhanced subchondral bone regeneration at 6 weeks post-surgery compared with the scaffold alone (subchondral bone histological score of 8.80 ± 1.64 vs. 3.8 ± 2.19, p < 0.05). The biocompatibility and pro-osteogenic activity of PVP–I on the cells from joint tissue and the enhanced subchondral bone formation in PVP–I treated scaffolds would thus indicate the potential of PVP–I for osteochondral defect repair.

Improved estimates of percolation and anisotropic permeability from 3-D X-ray microtomography using stochastic analyses and visualization

X-ray microtomography (micro-CT) with micron resolution enables new ways of characterizing microstructures and opens pathways for forward calculations of multiscale rock properties. A quantitative characterization of the microstructure is the first step in this challenge. We developed a new approach to extract scale-dependent characteristics of porosity, percolation, and anisotropic permeability from 3-D microstructural models of rocks. The Hoshen-Kopelman algorithm of percolation theory is employed for a standard percolation analysis. The anisotropy of permeability is calculated by means of the star volume distribution approach. The local porosity distribution and local percolation probability are obtained by using the local porosity theory. Additionally, the local anisotropy distribution is defined and analyzed through two empirical probability density functions, the isotropy index and the elongation index. For such a high-resolution data set, the typical data sizes of the CT images are on the order of gigabytes to tens of gigabytes; thus an extremely large number of calculations are required. To resolve this large memory problem parallelization in OpenMP was used to optimally harness the shared memory infrastructure on cache coherent Non-Uniform Memory Access architecture machines such as the iVEC SGI Altix 3700Bx2 Supercomputer. We see adequate visualization of the results as an important element in this first pioneering study.

First steps towards modeling a multi-scale Earth system

Recent advances in computational geodynamics are applied to explore the link between Earth’s heat, its chemistry and its mechanical behavior. Computational thermal-mechanical solutions are now allowing us to understand Earth patterns by solving the basic physics of heat transfer. This approach is currently used to solve basic convection patterns of terrestrial planets. Applying the same methodology to smaller scales delivers promising similarities between observed and predicted structures which are often the site of mineral deposits. The new approach involves a fully coupled solution to the energy, momentum and continuity equations of the system at all scales, allowing the prediction of fractures, shear zones and other typical geological patterns out of a randomly perturbed initial state. The results of this approach are linking a global geodynamic mechanical framework over regional-scale mineral deposits down to the underlying micro-scale processes. Ongoing work includes the challenge of incorporating chemistry into the formulation.

Rapid microwave-assisted synthesis of Mn3O4–graphene nanocomposite and its lithium storage properties

A nanocomposite of Mn3O4 wrapped in graphene sheets (GSs) was successfully synthesized via a facile, effective, energy-saving, and scalable microwave hydrothermal technique. The morphology and microstructures of the fabricated GS–Mn3O4 nanocomposite were characterized using various techniques. The results indicate that the particle size of the Mn3O4 particles in the nanocomposite markedly decreased to nearly 20 nm, significantly smaller than that for the bare Mn3O4. Electrochemical measurements demonstrated a high specific capacity of more than 900 mA h g−1 at 40 mA g−1, and excellent cycling stability with no capacity decay can be observed up to 50 cycles. All of these properties are also interpreted by experimental studies and theoretical calculations.

Coordinated probabilistic optimal decision-making model for multi-area ATC with risk control

In the decision-making of multi-area ATC (Available Transfer Capacity) in electricity market environment, the existing resources of transmission network should be optimally dispatched and coordinately employed on the premise that the secure system operation is maintained and risk associated is controllable. The non-sequential Monte Carlo simulation is used to determine the ATC probability density distribution of specified areas under the influence of several uncertainty factors, based on which, a coordinated probabilistic optimal decision-making model with the maximal risk benefit as its objective is developed for multi-area ATC. The NSGA-II is applied to calculate the ATC of each area, which considers the risk cost caused by relevant uncertainty factors and the synchronous coordination among areas. The essential characteristics of the developed model and the employed algorithm are illustrated by the example of IEEE 118-bus test system. Simulative result shows that, the risk of multi-area ATC decision-making is influenced by the uncertainties in power system operation and the relative importance degrees of different areas.

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.