A simplified quantum energy-transport model for semiconductors

The existence of global-in-time weak solutions to a quantum energy-transport model for semiconductors is proved. The equations are formally derived from the quantum hydrodynamic model in the large-time and small-velocity regime. They consist of a nonlinear parabolic fourth-order equation for the electron density, including temperature gradients; an elliptic nonlinear heat equation for the electron temperature; and the Poisson equation for the electric potential. The equations are solved in a bounded domain with periodic boundary conditions. The existence proof is based on an entropy-type estimate, exponential variable transformations, and a fixed-point argument. Furthermore, we discretize the equations by central finite differences and present some numerical simulations of a one-dimensional ballistic diode.     

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．     

半导体问题的向后差分配置法及误差分析

Collocation method is put forward to solve the semiconductor problem with heat-conduction, whose mathematical model is described by an initial and boundary problem for a nonlinear partial differential equation system. One elliptic equation is for the electric potential, and three parabolic equations are for the electron concentration, hole concentration and heat-conduction. Using the prior estimate and technique of differential equations, we obtained almost optimal error estimates in L2.     

A finite difference scheme for two-dimensional semiconductor device of heat conduction on composite triangular grids

The momentary state of a semiconductor device of heat conduction is described by a system of four nonlinear partial differential equations. One elliptic equation is for the electrostatic, two parabolic equations are for the electron concentration and the hole concentration, and one heat exchange equation is for the temperature. According to the necessary of practical numerical simulations and based on the balance equation, finite difference schemes for two-dimensional transient behavior of a semiconductor device of heat conduction on composite triangular grids are constructed. Studying their stability and convergence properties, the error estimate in the energy norm is obtained. Finally, a numerical example is given.     

Analysis of a bipolar energy-transport model for a metal-oxide-semiconductor diode

A simplified bipolar energy-transport model for a metal-oxide-semiconductor diode (MOS) with nonconstant lattice temperature is considered. The electron and hole current densities vanish in the diode but the particle temperature may be large. The existence of weak solutions to the system of quasilinear elliptic equations with nonlinear boundary conditions is proved using a Stampacchia trunction technique and maximum principle arguments. Further, an asymptotic analysis for the one-dimensional MOS diode is presented, which shows that only the boundary temperature influences the capacitance-voltage characteristics of the device. The analytical results are underlined by numerical experiments.     

半导体问题的交替方向有限元方法

该文用交替方向有限元方法求解半导体问题的Energy Trans port (ET)模型。对模型中椭圆型的电子位势方程采用交替方向迭代法，对流占优扩散的电子浓度和空穴浓度方程采用特征交替方向有限元方法，热传导方程利用Patch逼近采用交替方向有限元方法求解。利用微分方程的先验估计理论和技巧，分别得到了椭圆型方程和抛物型方程的最优H+1和L+2误差估计。     

Periodic solutions of a derivative nonlinear Schrödinger equation: Elliptic integrals of the third kind

The nonlinear Schrödinger equation (NLSE) is an important model for wave packet dynamics in hydrodynamics, optics, plasma physics and many other physical disciplines. The ‘derivative’ NLSE family usually arises when further nonlinear effects must be incorporated. The periodic solutions of one such member, the Chen-Lee-Liu equation, are studied. More precisely, the complex envelope is separated into the absolute value and the phase. The absolute value is solved in terms of a polynomial in elliptic functions while the phase is expressed in terms of elliptic integrals of the third kind. The exact periodicity condition will imply that only a countable set of elliptic function moduli is allowed. This feature contrasts sharply with other periodic solutions of envelope equations, where a continuous range of elliptic function moduli is permitted.     

UPWIND FINITE VOLUME SCHEMES FOR SEMICONDUCTOR DEVICE

The mathematical model of semiconductor devices is described by the initial boundary value problem of a system of three nonlinear partial differential equations. One equation in elliptic form is for the electrostatic potential; two equations of convection-dominated diffusion type are for the electron and hole concentrations. Finite volume element procedure are put forward for the electrostatic potential, while upwind     

Dynamic thermoviscoelastic thermistor problem with contact and nonmonotone friction

Abstract

The paper studies the evolution of the thermomechanical and electric state of a thermoviscoelastic thermistor that is in frictional contact with a reactive foundation. The mechanical process is dynamic, while the electric process is quasistatic. Friction is modeled with a nonmonotone relation between the tangential traction and tangential velocity. Frictional heat generation is taken into account and so is the strong dependence of the electric conductivity on the temperature. The mathematical model for the process is in the form of a system that consists of dynamic hyperbolic subdifferential inclusion for the mechanical state coupled with a nonlinear parabolic equation for the temperature and an elliptic equation for the electric potential. The paper establishes the existence of a weak solution to the problem by using time delays, a priori estimates and a convergence method.     

二维热传导型半导体问题的一个二阶格式

热传导型半导体器件瞬态问题的数学模型由四个拟线性偏微分方程所组成的方程组的初边值问题来描述。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。本文对二维热传导型半导体的一类混合初边值问题利用降阶法给出了一个二阶差分格式,并对其进行了详细的理论分析,得到了离散的犾２ 误差估计结果。     

A note on the elliptic equation method

The elliptic equation method is improved for constructing exact travelling wave solutions of nonlinear partial differential equations (PDEs). The rational forms of Jacobi elliptic functions are presented. By using new Jacobi elliptic function solutions of the elliptic equation, new doubly periodic solutions are obtained for some important PDEs. This method can be applied to many other nonlinear PDEs.     

The mixed problem for a nonlinear degenerate elliptic equation of second order

The present paper deals with the mixed boundary value problem for a nonlinear elliptic equation with degenerate rank 0. We first give the formulation of the problem and estimates of solutions of the problem, and then prove the uniqueness and existence of solutions of the above problem for the nonlinear elliptic equation by the extremum principle and the method of parameter extension. The complex method is used to discuss the corresponding problem for degenerate elliptic complex equation of first order and then that of second order.     

Lp-solvability of a full superconductive model

In this article the mean-field vortex model arising from the II-type superconductivity is investigated. The vortex model is reduced to a nonlinear hyperbolic–elliptic system of PDEs in a bounded domain. Motivated by experiments, we consider physical boundary conditions, which describe a flux of superconducting vortices through the boundary of the domain. We prove the global solvability for the system. To show the solvability result we take a vanishing “viscosity” limit in an approximated parabolic–elliptic system. Since the approximated solutions do not have a compactness property, we justify this limit transition, using a kinetic formulation of our problem. The main trick is that instead of the nonlinear system, we have to investigate a linear transport equation.     

五阶变系数模型方程新的Jacobi椭圆函数解

我们给出了一种统一的Jacobi椭圆函数方法来构造非线性偏微分方程精确行波解的新方法．借助于Mathematica，我们获得了五阶变系数模型方程的24种Jacobi椭圆函数解．     

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.     

On the coupling of boundary integral and finite element methods with nonlinear transmission conditions

We apply the coupling of boundary integral and finite element methods to study the weak solvability of certain exterior boundary value problems with nonlinear transmission conditions. As a model we consider a nonlinear second order elliptic equation in divergence form in a bounded inner region of the plane coupled with the Laplace equation in the corresponding exterior domain. The flux jump across the common nonlinear-linear interface is unknown and assumed to depend nonlinearly on the Dirichlet data. We establish the associated variational formulation in an operator equation setting and provide existence, uniqueness and approximation results.     

Two-gap elliptic solutions to integrable nonlinear equations

We study spectral surfaces associated with elliptic two-gap solutions to the nonlinear Schrödinger equation (NLS), the Korteweg-de Vries equation (KdV), and the sine-Gordon equation (SG). It is shown that elliptic solutions to the NLS and SG equations, as well as solutions to the KdV equation elliptic with respect tot, can be assigned to any hyperelliptic surface of genus 2 that forms a covering over an elliptic surface.     

Existence and regularity of the solution of a mixed boundary value problem for the Keldysh equation with a nonlinear absorption term

The Keldysh equation is a more general form of the classic Tricomi equation from fluid dynamics. Its well-posedness and the regularity of its solution are interesting and important. The Keldysh equation is elliptic in y>0 and is degenerate at the line y=0 in R². Adding a special nonlinear absorption term, we study a nonlinear degenerate elliptic equation with mixed boundary conditions in a piecewise smooth domain—similar to the potential fluid shock reflection problem. By means of an elliptic regularization technique, a delicate a priori estimate and compact argument, we show that the solution of a mixed boundary value problem of the Keldysh equation is smooth in the interior and Lipschitz continuous up to the degenerate boundary under some conditions. We believe that this kind of regularity result for the solution will be rather useful.     

一类四阶非线性椭圆问题变号解的存在性

通过建立一个新空间,在新空间中讨论了一个含Hardy位势的四阶非线性椭圆问题变号解的存在性.在一个环绕定理下,得到了问题变号解的存在性.     

Quasilinear parabolic and elliptic equations with nonlinear boundary conditions

This paper is concerned with a class of quasilinear parabolic and elliptic equations in a bounded domain with both Dirichlet and nonlinear Neumann boundary conditions. The equation under consideration may be degenerate or singular depending on the property of the diffusion coefficient. The consideration of the class of equations is motivated by some heat-transfer problems where the heat capacity and thermal conductivity are both temperature dependent. The aim of the paper is to show the existence and uniqueness of a global time-dependent solution of the parabolic problem, existence of maximal and minimal steady-state solutions of the elliptic problem, including conditions for the uniqueness of a solution, and the asymptotic behavior of the time-dependent solution in relation to the steady-state solutions. Applications are given to some heat-transfer problems and an extended logistic reaction–diffusion equation.