On the solution of the non-local parabolic partial differential equations via radial basis functions

In this paper, the problem of solving the one-dimensional parabolic partial differential equation subject to given initial and non-local boundary conditions is considered. The approximate solution is found using the radial basis functions collocation method. There are some difficulties in computing the solution of the time dependent partial differential equations using radial basis functions. If time and space are discretized using radial basis functions, the resulted coefficient matrix will be very ill-conditioned and so the corresponding linear system cannot be solved easily. As an alternative method for solution, we can use finite-difference methods for discretization of time and radial basis functions for discretization of space. Although this method is easy to use but an accurate solution cannot be provided. In this work an efficient collocation method is proposed for solving non-local parabolic partial differential equations using radial basis functions. Numerical results are presented and are compared with some existing methods.     

Stability of difference schemes for two-dimensional parabolic equations with non-local boundary conditions

The stability of difference schemes for one-dimensional and two-dimensional parabolic equations, subject to non-local (Bitsadze-Samarskii type) boundary conditions is dealt with. To analyze the stability of difference schemes, the structure of the spectrum of the matrix that defines the linear system of difference equations for a respective stationary problem is studied. Depending on the values of parameters in non-local conditions, this matrix can have one zero, one negative or complex eigenvalues. The stepwise stability is proved and the domain of stability of difference schemes is found.     

An asymptotic method for calculating limits of maximal means

For a continuous almost periodic function , we show that the function

An implicit-gradient-enhanced incremental-secant mean-field homogenization scheme for elasto-plastic composites with damage

This paper presents an incremental-secant mean-field homogenization (MFH) procedure for composites made of elasto-plastic constituents exhibiting damage. During the damaging process of one phase, the proposed method can account for the resulting unloading of the other phase, ensuring an accurate prediction of the scheme. When strain softening of materials is involved, classical finite element formulations lose solution uniqueness and face the strain localization problem. To avoid this issue the model is formulated in a so-called implicit gradient-enhanced approach, with a view toward macro-scale simulations. The method is then used to predict the behavior of composites whose matrix phases exhibit strain softening, and is shown to be accurate compared to unit cell simulations and experimental results. Then the convergence of the method upon strain softening, with respect to the mesh size, is demonstrated on a notched composite ply. Finally, applications consisting in a stacking plate, successively without and with a hole, are given as illustrations of the possibility of the method to be used in a multiscale framework.     

Versatile code DLAYZ for investigating population kinetics and radiative properties of plasmas in non-local thermodynamic equilibrium

A versatile code DLAYZ based on collisional-radiative model is developed for investigating the population kinetics and radiative properties of plasmas in non-local thermodynamic equilibrium. DLAYZ is implemented on the detailed level accounting (DLA) approach and can be extended to detailed configuration accounting (DCA) and hybrid DLA/DCA approaches. The code can treat both steady state and time-dependent problems. The implementation of the main modules of DLAYZ is discussed in detail including atomic data, rates, population distributions and radiative properties modules. The complete set of basic atomic data is obtained using relativistic quantum mechanics. For dense plasmas, the basic atomic data with plasma screening effects can be obtained. The populations are obtained by solving the coupled rate equations, which are used to calculate the radiative properties. A parallelized version is implemented in the code to treat the large-scale rate equations. Two illustrative examples of a steady state case for carbon plasmas and a time-dependent case for the relaxation of a K-shell excited argon are employed to show the main features of the present code.     

近红外光谱技术对闽南乌龙茶品种的识别研究

采用近红外光谱技术建立了一种快速无损的乌龙茶品种识别方法。收集闽南地区不同茶场中铁观音、黄金桂、本山、毛蟹与梅占等5个品种共210份具有代表性的乌龙茶样品,采集近红外光谱数据,选用1 100~1 300nm,1 640~2 498nm作为检测波长范围,利用主成分分析法(principal component analysis,PCA)建立模型,并在实验过程中比较多元散射校正(multiplicative scatter correction,MSC)与标准正态变量校正(standard normal variate,SNV)两种数据预处理方法对模型的影响。实验结果表明,多元散射校正对模型的影响优于标准正态变量校正,对校正集的识别准确率达到了96%,对预测集中样品的识别准确率达到了90%。实验结果证明了采用近红外光谱技术可以快速无损识别闽南地区乌龙茶,具有较强的实用价值和推广价值。     

Positronium annihilation in liquids in the framework of non-local interaction

In the bubble model of ortho positronium (o-Ps) annihilation in liquid the origin of the trapping of o-Ps is the electron-exchange repulsive interaction between the electron of o-Ps and the electron of the medium. The corresponding effective interaction is non-local in nature. However, in the prevalent bubble model, this effective interaction is usually treated as local (model) potential (sharp or smooth). In the present study, we have taken an approach to consider this trapping interaction as non-local in nature, which is included through a model separable non-local function to tackle the problem in analytically solvable manner. The analytical calculations show that this non-local interaction effectively acts as a gauge potential in the energy of the Ps atom in parameter (bubble radius) space. The computed bubble variables obtained using experimental Ps annihilation data are shown. A comparison between the present data with the calculated results using prevalent bubble model has been presented. Discussions have been made on the input parameter dependencies of the computed data.     

Radial basis functions method for solving of a non-local boundary value problem with Neumann’s boundary conditions

In this paper, the problem of solving the two-dimensional diffusion equation subject to a non-local condition involving a double integral in a rectangular region is considered. The solution of this type of problems are complicated. Therefore, a simple meshless method using the radial basis functions is constructed for the non-local boundary value problem with Neumann’s boundary conditions. Numerical examples are included to demonstrate the reliability and efficiency of this method. Also N_e and root mean square errors are obtained to show the convergence of the method.     

On multiple Fourier integrals of piecewise smooth functions with discontinuities of the second kind

It is well known that the Riesz means of eigenfunction expansions of piecewise smooth functions of order s>(n−3)/2 converge uniformly on compacts where these functions are smooth. In 2000 L. Brandolini and L. Colzani considered eigenfunction expansions of piecewise smooth functions with discontinuities of the second kind across smooth surfaces. They showed that the Riesz means of these functions of order s>(n−3)/2 may diverge even at certain points where these functions are smooth. Here it is argued that this effect depends on the measure of the singularity area, i.e. we consider functions with singularities across more limited areas and prove that the Riesz means of their eigenfunction expansions of order s>(n−3)/2 converge uniformly on compacts where these functions are continuous.