摩擦问题中的边界混合变分不等式

本文以弹性力学中的摩擦问题为背景,讨论了非线性、不可微的混合变分不等式解的存在唯一性,给出相应的边界变分不等式及其解的存在唯一性。为使用边界元方法数值求解提供了理论依据。     

Cassini卵形谱包含域的改进及应用

本文给出了改进的Cassini卵形谱包含域,讨论了相应的隔离定理及边界问 题,所得结果推广了文[1-4]的相应定理.作为应用得到了M-矩阵的一个新表征.     

流形上调和函数关于无穷远边界的Dirichlet问题

本文讨论了流形上φ-调和函数在无穷远边界上的Dirichlet问题的解,并在此基础上得到了调和函数在无穷远边界上的Dirichlet问题的解,这给出了一类流形上有界非平凡的调和函数的存在性并推广了S.Y.Cheng的相应的结论.     

Petrovsky系统边界控制的正则性

讨论了具有Dirichlet边界控制和同位观测的Petrovsky系统的正则性,给出了相应的直接传输算子,证明了系统在G.Weiss意义下是正则的,且其直接传输算子为零.     

流形上调和函数关于无穷远边界的Dirichlet问题

本文讨论了流形上φ-调和函数在无穷远边界上的Dirichlet问题的解,并在此基础上得到了调和 函数在无穷远边界上的Dirichlet问题的解,这给出了一类流形上有界非平凡的调和函数的存在性并推 广了S．Y．Cheng的相应的结论．     

无界区域抛物方程自然边界元方法

本文应用自然边界元方法求解无界区域抛物型初边值问题。首先将控制方程对时间进行离散化,得到关于时间步长离散化的椭圆型问题。通过Fourier展开,导出相应问题的自然积分方程和Poisson积分公式。研究了自然积分算子的性质,并讨论了自然积分方程的数值解法,最后给出数值例子。从而解决了抛物型问题的自然边界归化和自然边界元方法。     

极性连续统的增率型运动方程和边界条件

推导出了各种偶应力张量间和它们的变率间的关系,并建立起增率型角动量方程及其相应的边界条件。于是,把这些结果和匡震邦在“非线性连续介质力学基础”中给出的经典连续统力学的相应结果组合起来即得Cauchy形式和Piola形式以及Kirchhoff形式的极性连续统的增率型运动方程和边界条件。     

单位圆到任意曲线保角变换的近似计算方法

本文讨论了将单位圆内部映射成由任意曲线(包括任意曲线割缝)边界围成的单连通域内部或外部的保角变换问题.以多边形逼近单连通域的边界,采用Schwartz-Christoffel积分建立单位圆与该多边形的映射函数.给出了确定Schwartz-Christoffel积分中未知参数的数值计算方法.     

一类边值问题的随机分析解法

本文首先利用一类偏微分方程边值问题基本解的性质,构造并讨论了R~d中有界C~3区域上反射扩散过程及相应的边界局部时.作为它们的应用,我们用随机分析的方法给出了一类偏微分方程第三边值问题唯一有界解的明显表达式.     

分片常系数电导率问题的边界元配点法

用边界元方法讨论了具有分片常系数电导率方程Δ↓(γΔ↓u)=0的Dirichlet边值问题,由于方程的基本解无法显式写出,在应用通常边界元时存在很大的困难,基于这个电导率方程的解的积分表达式,导出一个在边界和交界面上的积分方程组,并讨论了这个方程组的性质,对于这个积分方程组,用配点法进行求解,且给出其误差分析．相应的数值例子证实了算法的有效性．应该指出的是本文所用的方法也适用于具有分片常系数椭圆方程的不同边界问题。     

The convergence of the horizontal line method for Maxwell's equations

The interior initial-boundary value problem for Maxwell's equations is considered for certain homogeneous boundary conditions and currents which satisfy corresponding compatibility conditions. Under the assumption that a solution exists, it is shown by means of suitable normed spaces that the sequence of discrete solutions obtained from the horizontal line method converges discretely to this solution. A priori estimations are given both for the discrete solutions and the solution of the given problem.     

An Upwind Difference Scheme and Its Corresponding Boundary Scheme

An upwind difference scheme was given by the author in [5] for the numerical solution of steady-state problems. The present work studies this upwind scheme and its corresponding boundary scheme for the numerical solution of unsteady problems. For interior points the difference equations are approximations of the characteristic relations; for boundary points difference equatons are approximations of the characteristicrelations corresponding to the outgoing characteristics and the "non-reflecting" boundary conditions. Calculation of a Riemann problem in a finite computational region yields promising numerical results.     

Acceleration of the Process of Entering Stationary Mode for Solutions of a Linearized System of Viscous Gas Dynamics. I

The problem of construction of control Dirichlet boundary conditions accelerating the convergence of the corresponding solution to its steady state for given initial conditions is studied for the linearized system of differential equations approximately describing the dynamics of viscous gas. The algorithm is described and estimates of convergence rate are presented for the differential case.     

Solution for a problem of linear plane elasticity with mixed boundary conditions on an ellipse by the method of boundary integrals

A numerical boundary integral scheme is proposed for the solution of the system of field equations of plane, linear elasticity in stresses for homogeneous, isotropic media in the domain bounded by an ellipse under mixed boundary conditions. The stresses are prescribed on one half of the ellipse, while the displacements are given on the other half. The method relies on previous analytical work within the Boundary Integral Method [1], [2].The considered problem with mixed boundary conditions is replaced by two subproblems with homogeneous boundary conditions, one of each type, having a common solution. The equations are reduced to a system of boundary integral equations, which is then discretized in the usual way and the problem at this stage is reduced to the solution of a rectangular linear system of algebraic equations. The unknowns in this system of equations are the boundary values of four harmonic functions which define the full elastic solution inside the domain, and the unknown boundary values of stresses or displacements on proper parts of the boundary.On the basis of the obtained results, it is inferred that the tangential stress component on the fixed part of the boundary has a singularity at each of the two separation points, thought to be of logarithmic type. A tentative form for the singular solution is proposed to calculate the full solution in bulk directly from the given boundary conditions using the well-known Boundary Collocation Method. It is shown that this addition substantially decreases the error in satisfying the boundary conditions on some interval not containing the singular points.The obtained results are discussed and boundary curves for unknown functions are provided, as well as three-dimensional plots for quantities of practical interest. The efficiency of the used numerical schemes is discussed, in what concerns the number of boundary nodes needed to calculate the approximate solution.     

Positive solutions and dynamics of a finite difference reaction–diffusion system

We investigate the dynamics and methods of computation for some nonlinear finite difference systems that are the discretized equations of a time-dependent and a steady-state reaction–diffusion problem. The formulation of the discrete equations for the time-dependent problem is based on the implicit method for parabolic equations, and the computational algorithm is based on the method of monotone iterations using upper and lower solutions as the initial iterations. The monotone iterative method yields improved upper and lower bounds of the solution in each iteration, and the sequence of iterations converges monotonically to a solution for both the time-dependent and the steady-state problems. An important consequence of this method is that it leads to a bifurcation point that determines the dynamic behavior of the time-dependent problem in relation to the corresponding steady-state problem. This bifurcation point also determines whether the steady-state problem has one or two non-negative solutions, and is explicitly given in terms of the physical parameters of the system and the type of boundary conditions. Numerical results are presented for both the time-dependent and the steady-state problems under various boundary conditions, including a test problem with known analytical solution. These numerical results exhibit the predicted dynamic behavior of the time-dependent solution given by the theoretical analysis. Also discussed are the numerical stability of the computational algorithm and the convergence of the finite difference solution to the corresponding continuous solution of the reaction–diffusion problem. © 1993 John Wiley & Sons, Inc.     

An IMEX scheme for reaction-diffusion equations: application for a PEM fuel cell model

An implicit-explicit (IMEX) method is developed for the numerical solution of reaction-diffusion equations with pure Neumann boundary conditions. The corresponding method of lines scheme with finite differences is analyzed: explicit conditions are given for its convergence in the ‖·‖_∞ norm. The results are applied to a model for determining the overpotential in a proton exchange membrane (PEM) fuel cell.     

Semi‐analytical solution of MHD flow through boundary integrals on the pipe wall

A mathematical model is given for the magnetohydrodynamic (MHD) pipe flow as an inner Dirichlet problem in a 2D circular cross section of the pipe, coupled with an outer Dirichlet or Neumann magnetic problem. Inner Dirichlet problem is given as the coupled convection‐diffusion equations for the velocity and the induced current of the fluid coupling also to the outer problem, which is defined with the Laplace equation for the induced magnetic field of the exterior region with either Dirichlet or Neumann boundary condition. Unique solution of inner Dirichlet problem is obtained theoretically reducing it into two boundary integral equations defined on the boundary by using the corresponding fundamental solutions. Exterior solution is also given theoretically on the pipe wall with Poisson integral, and it is unique with Dirichlet boundary condition but exists with an additive constant obtained through coupled boundary and solvability conditions in Neumann wall condition. The collocation method is used to discretize these boundary integrals on the pipe wall. Thus, the proposed procedure is an improved theoretical analysis for combining the solution methods for the interior and exterior regions, which are consolidated numerically showing the flow behavior. The solution is simulated for several values of problem parameters, and the well‐known MHD characteristics are observed inside the pipe for increasing values of Hartmann number maintaining the continuity of induced currents on the pipe wall.     

On some generalizations of the classical Riemann problem

The special class of the homogeneous vector boundary Riemann problems on the finite sequence of algebraic surfaces is investigated completely. Its coefficients are the noncommutative permutative matrices of the arbitrary but not prime order, and boundary conditions are given on the system of open contours. The constructive solution procedure and definite structure of the canonical solution matrix are obtained and present some generalizations of the classical Riemann problem. Simultaneously the corresponding class of algebraic equations for the appropriate covering surfaces is formed explicitly too. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Generalized impedance boundary conditions for the maxwell equations as singular perturbations problems

In this paper the Maxwell equations in an exterior domain with generalaized impedance boundary conditions of Engquist-Nédélec are considered. The particular form of the assumed boundary conditions can be considered to be a singular perturbation of the Dirichlet boundary conditions. The convergence of the solution of the Maxwell equations with these generalized impedance boundary conditions to that of the corresponding Dirichlet problem is proven. The proof uses a new integral equations method combined with results dealing with singular perturbation problems of a class of pseudo-differential operators.     

Uniqueness of analytic solutions for stationary plate oscillations in an annulus

The equations governing the harmonic oscillations of a plate with transverse shear deformation are considered in an annular domain. It is shown that under nonstandard boundary conditions where both the displacements and tractions are zero on the internal boundary curve, the corresponding analytic solution is zero in the entire domain. This property is then used to prove that a boundary value problem with Dirichlet or Neumann conditions on the external boundary and Robin conditions on the internal boundary has at most one analytic solution.