Numerical solution of the two-dimensional Helmholtz equation with variable coefficients by the radial integration boundary integral and integro-differential equation methods

This paper presents new formulations of the boundary–domain integral equation (BDIE) and the boundary–domain integro-differential equation (BDIDE) methods for the numerical solution of the two-dimensional Helmholtz equation with variable coefficients. When the material parameters are variable (with constant or variable wave number), a parametrix is adopted to reduce the Helmholtz equation to a BDIE or BDIDE. However, when material parameters are constant (with variable wave number), the standard fundamental solution for the Laplace equation is used in the formulation. The radial integration method is then employed to convert the domain integrals arising in both BDIE and BDIDE methods into equivalent boundary integrals. The resulting formulations lead to pure boundary integral and integro-differential equations with no domain integrals. Numerical examples are presented for several simple problems, for which exact solutions are available, to demonstrate the efficiency of the proposed methods.     

Estimations of the errors between the evolving states generated by two Hamiltonians with the same initial state

Time evolution of a quantum system is described by Schrödinger equation with initial pure state, or von Neumann equation with initial mixed state. In this paper, we estimate the error between the evolving states generated by two Hamiltonians with the same initial pure state. Secondly, according to the method of operator–vector correspondence, we give a relation of the Schrödinger equation and von Neumann equation and then estimate the error between the evolving states generated by two Hamiltonians with the same initial mixed state.     

Neumann dynamic stochastic finite element method of vibration for structures with stochastic parameters to random excitation

This paper presents a method for the dynamic analysis of structures with stochastic parameters to random excitation. A procedure to derive the statistical characteristics of the dynamic response for structure is proposed by using dynamic Neumann stochastic finite element method presented herein. Random equation of motion for structure is transformed into a quasi-static equilibrium equation for the solution of displacement in time domain. Neumann expansion method is developed and applied to the equation for deriving the statistical solution of the dynamic response of such a random structure system, within the framework of Monte Carlo simulation. Then, the results from Neumann dynamic stochastic finite element method are compared with those from the first- and second-order perturbation stochastic finite element methods and the direct Monte Carlo simulation with respect to accuracy, convergence and computational efficiency. Numerical examples are examined to show that the approach proposed in this paper has a very high accuracy and efficiency in the analysis of compound random vibration.     

ODE–热方程级联系统的事件触发控制

本文针对常微分方程(ODE)耦合偏微分方程(PDE)建模的分布式参数多智能体系统进行研究, 针对一致性同步问题, 提出了事件触发的网络化ODE–热方程级联系统多智能体一致性边界交互协议. 本文考虑的热方程左边界为Neumann边界条件, 并且与ODE系统耦合, 右边界为绝热边界条件. 假设网络化多智能体系统的连接方式为全联通有向拓扑图, 给出ODE–热方程级联系统的多智能体的一致性控制协议. 另外针对现有数字式控制器, 设计了事件触发的一致性控制协议, 并利用李雅普诺夫函数验证了在事件触发条件下ODE–热方程级联系统的稳定性. 最后给出了由5个ODE–热方程级联的多智能体系统的仿真结果, 验证了事件触发控制器的有效性.     

Boundary feedback stabilization of the telegraph equation: Decay rates for vanishing damping term

We study a semilinear mildly damped wave equation that contains the telegraph equation as a special case. We consider Neumann velocity boundary feedback and prove the exponential stability of the closed loop system. We show that for vanishing damping term in the partial differential equation, the decay rate of the system approaches the rate for the system governed by the wave equation without damping term. In particular, this implies that arbitrarily large decay rates can occur if the velocity damping in the partial differential equation is sufficiently small.     

Stochastic differential equations and their application to randomly time-varying control systems

Let a control system be represented by a system of n first-order stochastic differential equations

where the coefficient matrix α(t, ω) is a sum of a deterministic matrix β(t) and a stochastic matrix α(t, ω). It is assumed that the deterministic part of the system is stable in the sense of Lyapunov. The stochastic differential equation becomes now an integral equation with a stochastic kernel. This integral equation is solved by constructing a resolvent kernel by means of a Neumann series expansion. Sufficient conditions for almost sure uniform convergence of the Neumann aeries expansion are given. Integral expressions for the expectation and covariance matrix of the stochastic vector y(t, ω) are found. These results have many applications in control systems theory.     

More accurate results for two-dimensional heat equation with Neumann’s and non-classical boundary conditions

In this article, recently proposed spectral meshless radial point interpolation (SMRPI) method is applied to the two-dimensional diffusion equation with a mixed group of Dirichlet’s and Neumann’s and non-classical boundary conditions. The present method is based on meshless methods and benefits from spectral collocation ideas. The point interpolation method with the help of radial basis functions is proposed to construct shape functions which have Kronecker delta function property. Evaluation of high-order derivatives is possible by constructing and using operational matrices. The computational cost of the method is modest due to using strong form equation and collocation approach. A comparison study of the efficiency and accuracy of the present method and other meshless methods is given by applying on mentioned diffusion equation. Stability and convergence of this meshless approach are discussed and theoretically proven. Convergence studies in the numerical examples show that SMRPI method possesses excellent rates of convergence.     

Maximum-norm stability of the finite element Ritz projection under mixed boundary conditions

As a model of the second order elliptic equation with non-trivial boundary conditions, we consider the Laplace equation with mixed Dirichlet and Neumann boundary conditions on convex polygonal domains. Our goal is to establish that finite element discrete harmonic functions with mixed Dirichlet and Neumann boundary conditions satisfy a weak (Agmon–Miranda) discrete maximum principle, and then prove the stability of the Ritz projection with mixed boundary conditions in \(L^\infty \) norm. Such results have a number of applications, but are not available in the literature. Our proof of the maximum-norm stability of the Ritz projection is based on converting the mixed boundary value problem to a pure Neumann problem, which is of independent interest.     

Parallel solution of tridiagonal systems for the Poisson equation

A method is described to solve the systems of tridiagonal linear equations that result from discrete approximations of the Poisson or Helmholtz equation with either periodic, Dirichlet, Neumann, or shear-periodic boundary conditions. The problem is partitioned into a set of smaller Dirichlet problems which can be solved simultaneously on parallel or vector computers leaving a smaller tridiagonal system to be solved on one of the processors.     

A Fast Finite Difference Method for Three-Dimensional Time-Dependent Space-Fractional Diffusion Equations with Fractional Derivative Boundary Conditions

We develop a fast finite difference method for time-dependent variable-coefficient space-fractional diffusion equations with fractional derivative boundary-value conditions in three dimensional spaces. Fractional differential operators appear in both of the equation and the boundary conditions. Because of the nonlocal nature of the fractional Neumann boundary operator, the internal and boundary nodes are strongly coupled together in the coupled linear system. The stability and convergence of the finite difference method are discussed. For the implementation, the development of the fast method is based upon a careful analysis and delicate decomposition of the structure of the coefficient matrix. The fast method has approximately linear computational complexity per Krylov subspace iteration and an optimal-order memory requirement. Numerical results are presented to show the utility of the method.     

Dynamics of stationary structures in a parabolic problem with reflected spatial argument

Properties of stationary structures in a nonlinear optical resonator with an inversion transformer in its two-dimensional feedback are investigated. This system is mathematically described by a scalar parabolic equation with an inversion transformation of its spatial argument and with Neumann conditions on a segment. The evolution of forms of stationary structures and their stability are investigated. It is proved that the number of stable stationary structures increases with lengthening the segment. The central manifold method and Galerkin method are used.     

A high-order compact scheme for the one-dimensional Helmholtz equation with a discontinuous coefficient

In this paper, based on the idea of the immersed interface method, a fourth-order compact finite difference scheme is proposed for solving one-dimensional Helmholtz equation with discontinuous coefficient, jump conditions are given at the interface. The Dirichlet boundary condition and the Neumann boundary condition are considered. The Neumann boundary condition is treated with a fourth-order scheme. Numerical experiments are included to confirm the accuracy and efficiency of the proposed method.     

Scattering of 2-D waves by several obstacles and screens in case of Neumann boundary condition

The Neumann problem for the Helmholtz equation, in a connected plane region bounded by closed and open curves, is studied. The existence of classical solution is proved by potential theory. The problem is reduced to the Fredholm equation of the second kind, which is uniquely solvable. This means that the solution can be computed by standard codes.     

N.N. Krasovskii’s extremal shift method and problems of boundary control

For the boundary-controlled dynamic system obeying a parabolic differential equation with the Neumann boundary condition, the problems of following the reference motion, following the reference control, and guaranteed control (at domination of the controller resource) were solved on the basis of the N.N. Krasovskii method of extremal shift from the theory of positional differential games.     

Exact boundary controllability of the Korteweg–de Vries equation with a piecewise constant main coefficient

This article proposes some interface conditions for a Korteweg–de Vries (KdV) equation with a piecewise constant main coefficient. The well-posedness of the Cauchy problem is proved by using multiplier technique and then an observability inequality for the adjoint problem is obtained. We end up this article with a result of local exact controllability of the discontinuous main coefficient KdV equation with a right Neumann boundary control provided that the length of the domain is small enough and the time of control is sufficiently large.     

PDE工具箱实现偏微分方程的有限元求解

偏微分方程在科学和工程上有着广泛的应用。有限元法是一种重要的偏微分方程数值解法。编程实现从偏微分方程到有限元求解全过程需要很好的理论基础和编程技巧,难度较高。该文介绍了偏微分方程有限元求解的基本理论和一般Neumann条件下椭圆型方程的有限元求解具体过程,并通过两个实例,电机磁场问题和热传导问题,介绍了使用PDE工具箱实现偏微分方程的有限元解法。实验结果表明这一方法具有操作简单明了,运算速度快,计算误差可控制等优点。     

von Neumann stability analysis of first-order accurate discretization schemes for one-dimensional (1D) and two-dimensional (2D) fluid flow equations

In literature, the von Neumann stability analysis of simplified model equations, such as the wave equation, is typically used to determine stability conditions for the non-linear partial differential fluid flow equations (Navier–Stokes and Euler). However, practical experience suggests that such simplistic stability conditions are grossly inadequate for computations involving the system of coupled flow equations. The goal of this paper is to determine stability conditions for the full system of fluid flow equations – the Euler equations are examined, as any conditions derived for the Euler equations will apply to the Navier–Stokes (NS) equations in the limit of convection-dominated flows. A von Neumann stability analysis is conducted for the one-dimensional (1D) and two-dimensional (2D) Euler equations. The system of equations is discretized on a staggered grid using finite-difference discretization techniques; the use of a staggered grid allows equivalence to finite-volume discretization. By combining the different discretization techniques, ten solution schemes are formulated – eight solution schemes are considered for the 1D Euler equations, and two schemes for the 2D Euler equations. For each scheme, error amplification matrices are determined from the stability analysis, stable and unstable regimes are identified, and practical stability limits are predicted in terms of the maximum-allowable CFL (Courant–Friedrichs–Lewy) number as a function of Mach number. The predictions are verified for selected schemes using the Riemann problem at incompressible and compressible Mach numbers. Very good agreement is obtained between the analytically predicted and the “experimentally” observed CFL values. The successfully tested stability limits are presented in graphical form, which offer a viable alternative to complicated mathematical expressions often reported in published literature, and should benefit everyday CFD (Computational Fluid Dynamics) users. The stability regions are used to discuss the effect of time integration (explicit vs. implicit), density bias in continuity equation and momentum convection term linearization on stability. A comparison of the predicted stability limits for 1D and 2D Euler equations with commonly-used stability conditions arising from the wave equation shows that the stability thresholds for the Euler equations lie well below those predicted by the wave equation analysis; in addition, the 2D Euler stability limits are more restrictive as compared to 1D Euler limits. Since the present analysis accounts for the full system of fluid flow (Euler) equations, the derived stability conditions can be used by CFD practitioners to estimate a timestep or CFL number to guide the stability of their computations.     

Optimal control of a stochastic delay heat equation with boundary-noise and boundary-control

In this paper, a controlled stochastic delay heat equation with Neumann boundary-noise and boundary-control is considered. The existence and uniqueness of the mild solution for the associated Hamilton–Jacobi–Bellman equations are obtained by means of the backward stochastic differential equations, which is applied to the optimal control problem.     

基于连续体模型的人群疏散系统的边界控制

研究一类存在扰动的一维人群疏散系统的边界控制问题.以走廊中的人群动态为例,基于数量守恒定律建立人群动态模型;由非线性偏微分方程描述系统模型,并直接在分布参数的范畴内,设计Robin、Neumann、Dirichlet三种边界控制律,用于控制行人在疏散过程中的移动方式,避免拥堵的产生;利用李雅普诺夫方法对边界控制律作用下的人群疏散系统稳定性给出详细证明,并通过一个仿真实例验证边界控制律的有效性.研究成果可以应用到生活中单入口单出口场所的人群动态管理.     

Schwarz iterations for the efficient solution of screen problems with boundary elements

This paper investigates two domain decomposition algorithms for the numerical solution of boundary integral equations of the first kind. The schemes are based on theh-type boundary element Galerkin method to which the multiplicative and the additive Schwarz methods are applied. As for twodimensional problems, the rates of convergence of both methods are shown to be independent of the number of unknowns. Numerical results for standard model problems arising from Laplaces' equation with Dirichlet or Neumann boundary conditions in both two and three dimensions are discussed. A multidomain decomposition strategy is indicated by means of a screen problem in three dimensions, so as to obtain satisfactory experimental convergence rates.