热传导型半导体问题的配置方法及误差分析

热传导型半导体瞬态问题的数学模型是一类非线性偏微分方程的初边值问题．电子位势方程是椭圆型的，电子、空穴浓度方程及热传导方程是抛物型的．该文给出求解的配置方法，得到次优犔２模误差估计，并将配置法和Ｇａｌｅｒｋｉｎ有限元方法进行数值结果比较．     

热传导型半导体瞬态问题特征变网格有限元法及其分析

热传导型半导体器件的瞬时状态由四个方程的非线性偏微分方程组的初边值问题所决定，其中电子位势方程是椭圆型的，电子和空穴浓度方程是对流扩散型的，温度方程为热传导型的。本文提出解这类问题的特征变网格有限元法，并进行了理论分析，在一定条件下，得到了某种意义下的最佳L^2误差估计结果。     

三维热传导型半导体问题的特征混合元方法和分析

本文研究三维热传导型半导体态问题的特征混合元方法及其理论分析,其数学模型是一类非线性偏微分方程的初边值问题,对电子位势方程提出混合元逼近,对电子,空穴浓度方程笔挺表限元逼近;对热传导方程采用对时间向后差分的Ｇａｌｅｒｋｉｎ逼近,应用微分方程先验估计理论和技巧得到了最优阶Ｌ＾２误差估计。     

三维热传导型半导体问题的特征有限元方法和分析

本文研究三维热传导型半导体瞬态问题的特征有限元方法及其理论分析,其数学模型是一类非线性偏微分方程的初边值问题,对电子位势方程提出Ｇａｌｅｒｋｉｎ逼近;对电子,空穴浓度方程采用特征有限元逼近;对热传导方程采用对时间向后差分的Ｇａｌｅｒｋｉｎ逼近．应用微分方程先验估计理论和技巧得到了最优阶Ｌ＾２误差估计。     

三维热传导型半导体器件问题在局部加密网格上的有限差分格式

局部网格加密技术能很好地解决局部性很强的问题,半导体器件问题的解在半导体的p-n结附近有很强的局部性质.热传导型半导体器件瞬态问题的数学模型由四个方程组成的非线性偏微分方程组的初边值问题决定,电场位势方程是椭圆型的,电子和空穴浓度方程是抛物型的,温度方程是热传导型的.依据实际数值模拟的需要,提出了一类三维热传导型半导体问题在时间上进行局部加密的复合网格上的有限差分格式,并给出了电子、空穴浓度和温度的最大模误差估计以及数值算例.这些研究结果对半导体器件数值模拟的算法理论、实际应用和工程软件系统的研制,均具有重要的价值.     

连续初值条件下热传导方程解的存在性证明

考虑齐次热传导方程的初边值问题,通过对形式级数解的性质的研究,得到了在连续初值条件下热传导方程的古典解的存在性证明.     

解抛物型方程的高精度显式差分格式

在渗流、扩散、热传导等邻域中经常会遇到求解抛物型方程的问题.在一维情形,其模型为如下初边值问题:     

热传导方程初边值问题的注记

与一般教材不同的是,不是首先转化问题,而是通过直接运用极值原理,讨论了对热传导方程具第三类初边值问题的解的估计.接下来又通过深入分析一般教材中一端具第二类初边值问题的解的估计中辅助函数的构造,给出了两端都具第二类边界条件的解的估计的实现方法.     

二维磁Bénard方程的热传导系数消失极限

本文主要研究二维磁Bénard方程初边值问题的热传导系数消失极限,得到解的收敛速度和边界层的宽度估计,证明当热传导系数趋向于0时,原方程的解在内部区域(远离边界层)一致收敛到极限方程的解.     

关于热传导方程半离散差分格式的一个注记

本文研究了热传导方程初边值问题的半离散化差分格式直接解算法.分别从Dirichlet和Neumann边界条件出发,直接由空间差分格式导出与时间相关的一阶常微分方程组,随后通过正/余弦变换获得了原方程的半解析解,并给出了相关收敛性分析.并对中心差分格式和紧差分格式的精度差异,通过矩阵特征值理论给出了相关原因分析.另外,对于二维热传导方程初边值问题,应用矩阵张量积运算,该直接解算法可直接演变成二重正(余)弦变换.该方法由于不涉及时间上的离散,从而具有较好的计算效率.     

Justification of the collocation method for the integral equation for a mixed boundary value problem for the Helmholtz equation

The surface integral equation for a spatial mixed boundary value problem for the Helmholtz equation is considered. At a set of chosen points, the equation is replaced with a system of algebraic equations, and the existence and uniqueness of the solution of this system is established. The convergence of the solutions of this system to the exact solution of the integral equation is proven, and the convergence rate of the method is determined.     

A combination of energy method and spectral analysis for study of equations of gas motion

There have been extensive studies on the large time behavior of solutions to systems on gas motions, such as the Navier-Stokes equations and the Boltzmann equation. Recently, an approach is introduced by combining the energy method and the spectral analysis to the study of the optimal rates of convergence to the asymptotic profiles. In this paper, we will first illustrate this method by using some simple model and then we will present some recent results on the Navier-Stokes equations and the Boltzmann equation. Precisely, we prove the stability of the non-trivial steady state for the Navier-Stokes equations with potential forces and also obtain the optimal rate of convergence of solutions toward the steady state. The same issue was also studied for the Boltzmann equation in the presence of the general time-space dependent forces. It is expected that this approach can also be applied to other dissipative systems in fluid dynamics and kinetic models such as the model system of radiating gas and the Vlasov-Poisson-Boltzmann system.      

Iterative Methods for a Linearly Perturbed Algebraic Matrix Riccati Equation Arising in Stochastic Control

We start with a discussion of coupled algebraic Riccati equations arising in the study of linear-quadratic optimal control problems for Markov jump linear systems. Under suitable assumptions, this system of equations has a unique positive semidefinite solution, which is the solution of practical interest. The coupled equations can be rewritten as a single linearly perturbed matrix Riccati equation with special structures. We study the linearly perturbed Riccati equation in a more general setting and obtain a class of iterative methods from different splittings of a positive operator involved in the Riccati equation. We prove some special properties of the sequences generated by these methods and determine and compare the convergence rates of these methods. Our results are then applied to the coupled Riccati equations of jump linear systems. We obtain linear convergence of the Lyapunov iteration and the modified Lyapunov iteration, and confirm that the modified Lyapunov iteration indeed has faster convergence than the original Lyapunov iteration.     

Large deviations for the Boussinesq equations under random influences

A Boussinesq model for the Bénard convection under random influences is considered as a system of stochastic partial differential equations. This is a coupled system of stochastic Navier–Stokes equations and the transport equation for temperature. Large deviations are proved, using a weak convergence approach based on a variational representation of functionals of infinite-dimensional Brownian motion.     

Modified integral current methods in electrodynamics of nonhomogeneous media

The main difficulty in numerical solution of integral equations of electrodynamics is associated with the need to solve a high-order system of linear equations with a dense matrix. It is therefore relevant to develop numerical methods that lead to linear equation systems of lower order at the cost of more complex evaluation of the coefficients. In this article we propose a method for solving linear equations of electrodynamics which is a modification of the integral current method. The main distinctive feature of the proposed method is double integration of the electric Green’s tensor in the process of algebraization of the original integral equation. The solutions of the system of linear equations are thus integral means of the electric field inside the anomaly constructed by the proposed transformation formula. We prove convergence and derive error bounds for both the solution of the integral equation and the electromagnetic field components evaluated from approximate transformation formulas.     

Quasi-Neutral Limit of the Euler–Poisson and Euler–Monge–Ampère Systems

ABSTRACT

This paper studies the pressureless Euler–Poisson system and its fully nonlinear counterpart, the Euler–Monge–Ampère system, where the fully nonlinear Monge–Ampère equation substitutes for the linear Poisson equation. While the first is a model of plasma physics, the second is derived as a geometric approximation to the Euler incompressible equations. Using energy estimates, convergence of both systems to the Euler incompressible equations is proved.     

Product integration methods for solving a system of nonlinear Volterra integral equations

In this paper the technique of subtracting out singularities is used to derive explicit and implicit product Euler schemes with order one convergence and a product trapezoidal scheme with order two convergence for a system of Volterra integral equations with a weakly singular kernel. The convergence proofs of the numerical schemes are presented; these are nonstandard since the nonlinear function involved in the integral equation system does not satisfy a global Lipschitz condition.     

Spectral collocation and waveform relaxation methods for nonlinear delay partial differential equations

We investigate Chebyshev spectral collocation and waveform relaxation methods for nonlinear delay partial differential equations. Waveform relaxation methods allow to replace the system of nonlinear differential equations resulting from the application of spectral collocation methods by a sequence of linear problems which can be effectively integrated in a parallel computing environment by highly stable implicit methods. The effectiveness of this approach is illustrated by numerical experiments on the Hutchinson's equation. The boundedness of waveform relaxation iterations is proved for the Hutchinson's equation. This result is used in the proof of the superlinear convergence of the iterations.     

On the Relationship between Properties of Solutions of Difference Equations and the Corresponding Differential Equations

We establish conditions under which the existence of a periodic solution of a differential equation is preserved if a solution of the corresponding difference equation possesses the same property. We prove the convergence of periodic solutions of a system of difference equations to a periodic solution of a system of differential equations. Analogous problems are considered for bounded solutions. __________ Translated from Ukrains'kyi Matematychnyi Zhurnal, Vol. 57, No. 7, pp. 989–996, July, 2005.