Method of calculating the collision integral and solution of the Boltzmann kinetic equation for simple gases,gas mixtures and gases with rotational degrees of freedom

The conservative method of calculating the Boltzmann collision integral for simple gases, gas mixtures and gases with rotational degrees of freedom of molecules is presented. In all cases the common approach based on the projection technique of summing up the contributions in the collision integral is used. The method is applied for solving the Boltzmann kinetic equation for two fundamental rarefied gas flow problems: the heat transfer problem and the problem of shock wave structure. A comparison with experimental and numerical data of other authors is reported. It is shown that the considered numerical method allows one to solve the Boltzmann equation for real gases with high accuracy.     

A new high accuracy method for two-dimensional biharmonic equation with nonlinear third derivative terms: application to Navier–Stokes equations of motion

In this paper, we propose a new compact fourth-order accurate method for solving the two-dimensional fourth-order elliptic boundary value problem with third-order nonlinear derivative terms. We use only 9-point single computational cell in the scheme. The proposed method is then employed to solve Navier–Stokes equations of motion in terms of streamfunction–velocity formulation, and the lid-driven square cavity problem. We describe the derivation of the method in details and also discuss how our streamfunction–velocity formulation is able to handle boundary conditions in terms of normal derivatives. Numerical results show that the proposed method enables us to obtain oscillation-free high accuracy solution.     

Compact difference method for solving the fractional reaction–subdiffusion equation with Neumann boundary value condition

In this paper, we derive a high-order compact finite difference scheme for solving the reaction–subdiffusion equation with Neumann boundary value condition. The L1 method is used to approximate the temporal Caputo derivative, and the compact difference operator is applied for spatial discretization. We prove that the compact finite difference method is unconditionally stable and convergent with order O(τ^2?α+h⁴) in L₂ norm, where τ, α, and h are the temporal step size, the order of time fractional derivative and the spatial step size, respectively. Finally, some numerical experiments are carried out to show the effectiveness of the proposed difference scheme.     

Numerical solution of nonlinear system of Klein–Gordon equations by cubic B-spline collocation method

A technique to approximate the solutions of nonlinear Klein–Gordon equation and Klein–Gordon-Schrödinger equations is presented separately. The approach is based on collocation of cubic B-spline functions. The above-mentioned equations are decomposed into a system of partial differential equations, which are further converted to an amenable system of ODEs. The obtained system has been solved by SSP-RK54 scheme. Numerical solutions are presented for five examples, to show the accuracy and usefulness of proposed approach. The approximate solutions of both the equations are computed without using any transformation and linearization. The technique can be applied with ease to solve linear and nonlinear PDEs and also reduces the computational work.     

Continuous-time orbit problems are decidable in polynomial-time

We place the continuous-time orbit problem in P$\mathsf{P}$, sharpening the decidability result shown by Hainry [7].     

球对称n维反应扩散方程的边界观测与输出反馈控制

许多实际系统可用n 维超球坐标系来描述, 并且系统有球对称的性质, 因而可通过研究半径方向的状态变化, 得到系统的全局动态过程. 通过将高维的对称系统转化为等价的径向一维方程, 本文采用边界Backstepping 方法设计了球对称反应扩散方程的输出反馈控制器. 使用容易测量的边界状态值, 设计了状态观测器来估计系统在空间域的所有状态, 从而实现输出反馈控制. 本文扩展了连续Backstepping 方法,提出了n维球坐标的Volterra 积分映射, 从而求出了显式表达的控制器和状态观测器. 论文用Lyapunov 函数法证明了输出反馈系统在H1范数下指数稳定, 表明状态对初值的连续依赖, 确保控制系统具有较好的性质, 不会在空间某点发散. 最后进行了数值仿真, 仿真结果表明系统在输出反馈控制律的作用下收敛到稳态值.     

一类有限域丢番图方程的解及其应用

对一类有限域线性丢番图方程c≡x+by(mod N)进行了研究,求出了其通解及域中有效解的对数,并证明其能将部分曲线密码方案求解用户私钥的计算量降低为N/z, z为子群<-b>的最小非零元.指出了5个应用该类型方程曲线密码方案,最后以一个环Z_n上广义圆锥曲线多重数字签名方案私钥的求解为例进行说明.     

基于Landsat 8影像和能量平衡的西安市热场格局研究!

基于2014年8月15日的Landsat 8影像,通过劈窗算法反演西安中心城区地表温度,定量测算热岛中心范围。估算多种地表能量分量,分析热环境格局与地表能量分量的关系。结果表明:(1)西安中心城区城市热岛集中分布在人口、居住、商业密集区、经济技术开发区以及植被覆盖较差的区域;(2)感热、波文比与地表温度呈正相关,人为热与温度呈不显著正相关,净辐射、潜热与地表温度呈显著负相关;(3)城市热岛的地表能量结构中感热与潜热差异是构成城市热岛差异的主要原因。     

Fast local image inpainting based on the Allen–Cahn model

In this paper, we propose a fast local image inpainting algorithm based on the Allen–Cahn model. The proposed algorithm is applied only on the inpainting domain and has two features. The first feature is that the pixel values in the inpainting domain are obtained by curvature-driven diffusions and utilizing the image information from the outside of the inpainting region. The second feature is that the pixel values outside of the inpainting region are the same as those in the original input image since we do not compute the outside of the inpainting region. Thus the proposed method is computationally efficient. We split the governing equation into one linear equation and one nonlinear equation by using an operator splitting technique. The linear equation is discretized by using a fully implicit scheme and the nonlinear equation is solved analytically. We prove the unconditional stability of the proposed scheme. To demonstrate the robustness and accuracy of the proposed method, various numerical results on real and synthetic images are presented.