Amplitude equation for the stochastic reaction‐diffusion equations with random Neumann boundary conditions

In this paper, we consider a quite general class of reaction‐diffusion equations with cubic nonlinearities and with random Neumann boundary conditions. We derive rigorously amplitude equations, using the natural separation of time‐scales near a change of stability and investigate whether additive degenerate noise and random boundary conditions can lead to stabilization of the solution of the stochastic partial differential equation or not. The nonlinear heat equation (Ginzburg–Landau equation) is used to illustrate our result. Copyright © 2015 John Wiley & Sons, Ltd.     

Optimal control of backward stochastic heat equation with Neumann boundary control and noise

This paper considers a stochastic control problem in which the dynamic system is a controlled backward stochastic heat equation with Neumann boundary control and boundary noise and the state must coincide with a given random vector at terminal time. Through defining a proper form of the mild solution for the state equation, the existence and uniqueness of the mild solution is given. As a main result, a global maximum principle for our control problem is presented. The main result is also applied to a backward linear-quadratic control problem in which an optimal control is obtained explicitly as a feedback of the solution to a forward–backward stochastic partial differential equation.     

An Impact of Stochastic Dynamic Boundary Conditions on the Evolution of the Cahn-Hilliard System

Abstract

Nonlinear systems are often subject to random influences. Sometimes the noise enters the system through physical boundaries and this leads to stochastic dynamic boundary conditions. A dynamic, as opposed to static, boundary condition involves the time derivative as well as spatial derivatives for the system state variables on the boundary. Although stochastic static (Neumann or Dirichet type) boundary conditions have been applied for stochastic partial differential equations, not much is known about the dynamical impact of stochastic dynamic boundary conditions. The purpose of this article is to study possible impacts of stochastic dynamic boundary conditions on the long term dynamics of the Cahn-Hilliard equation arising in the materials science. We show that the dimension estimation of the random attractor increases as the coefficient for the dynamic term in the stochastic dynamic boundary condition decreases. However, the dimension of the random attractor is not affected by the corresponding stochastic static boundary condition.     

A Legendre–Galerkin Chebyshev collocation method for the Burgers equation with a random perturbation on boundary condition

In the paper, we apply the generalized polynomial chaos expansion and spectral methods to the Burgers equation with a random perturbation on its left boundary condition. Firstly, the stochastic Galerkin method combined with the Legendre–Galerkin Chebyshev collocation scheme is adopted, which means that the original equation is transformed to the deterministic nonlinear equations by the stochastic Galerkin method and the Legendre–Galerkin Chebyshev collocation scheme is used to deal with the resulting nonlinear equations. Secondly, the stochastic Legendre–Galerkin Chebyshev collocation scheme is developed for solving the stochastic Burgers equation; that is, the stochastic Legendre–Galerkin method is used to discrete the random variable meanwhile the nonlinear term is interpolated through the Chebyshev–Gauss points. Then a set of deterministic linear equations can be obtained, which is in contrast to the other existing methods for the stochastic Burgers equation. The mean square convergence of the former method is analyzed. Numerical experiments are performed to show the effectiveness of our two methods. Both methods provide alternative approaches to deal with the stochastic differential equations with nonlinear terms.     

A linear quadratic control problem for the stochastic heat equation driven by Q-Wiener processes

We consider a control problem for the stochastic heat equation with Neumann boundary condition, where controls and noise terms are defined inside the domain as well as on the boundary. The noise terms are given by independent Q-Wiener processes. Under some assumptions, we derive necessary and sufficient optimality conditions stochastic controls have to satisfy. Using these optimality conditions, we establish explicit formulas with the result that stochastic optimal controls are given by feedback controls. This is an important conclusion to ensure that the controls are adapted to a certain filtration. Therefore, the state is an adapted process as well.     

Ensemble Averaging for Dynamical Systems Under Fast Oscillating Random Boundary Conditions

This article is devoted to providing a theoretical underpinning for ensemble forecasting with rapid fluctuations in body forcing and in boundary conditions. Ensemble averaging principles are proved under suitable “mixing” conditions on random boundary conditions and on random body forcing. The ensemble averaged model is a nonlinear stochastic partial differential equation, with the deviation process (i.e., the approximation error process) quantified as the solution of a linear stochastic partial differential equation.     

Finite difference schemes for the fourth-order parabolic equations with different boundary value conditions

In this paper, the fourth-order parabolic equations with different boundary value conditions are studied. Six kinds of boundary value conditions are proposed. Several numerical differential formulae for the fourth-order derivative are established by the quartic interpolation polynomials and their truncation errors are given with the aid of the Taylor expansion with the integral remainders. Effective difference schemes are presented for the third Dirichlet boundary value problem, the first Neumann boundary value problem and the third Neumann boundary value problem, respectively. Some new embedding inequalities on the discrete function spaces are presented and proved. With the method of energy analysis, the unique solvability, unconditional stability and unconditional convergence of the difference schemes are proved. The convergence orders of derived difference schemes are all O(τ² + h²) in appropriate norms. Finally, some numerical examples are provided to confirm the theoretical results.     

基于Monte-Carlo随机有限元方法的随机边界条件下自然对流换热不确定性研究

为分析边界条件不确定性对方腔内自然对流换热的影响,发展了一种求解随机边界条件下自然对流换热不确定性传播的Monte-Carlo随机有限元方法.通过对输入参数场随机边界条件进行Karhunen-Loeve展开及基于Latin(拉丁)抽样法生成边界条件随机样本,数值计算了不同边界条件随机样本下方腔内自然对流换热流场与温度场,并用采样统计方法计算了随机输出场的平均值与标准偏差.根据计算框架编写了求解随机边界条件下方腔内自然对流换热不确定性的MATLAB随机有限元程序,分析了随机边界条件相关长度与方差对自然对流不确定性的影响.结果表明:平均温度场及流场与确定性温度场及流场分布基本相同;随机边界条件下Nu数概率分布基本呈现正态分布,平均Nu数随着相关长度和方差增加而增大;方差对自然对流换热的影响强于相关长度的影响.     

ON THE SINGULAR ONE DIMENSIONAL P-LAPLACIAN-LIKE EQUATION WITH NEUMANN BOUNDARY CONDITIONS

1IntroductionInthispaper,weconsiderthesolvabilityofsingularonedimensionalpLaplacian-likeequationwithNeumannboundaryconditionswhereA(()ispositivefor(>0,h(t)EL'([0,1]),gisacontinuousfunctiondefinedon(--co,0)U(0, co)suchthatg(f)-0asf'l-cojg(()- coas(-0 ,g(()---coas(-0--,andg(')(>0for'/0.Afunctionu(t)issaidtobeasolutionof(1.1),ifthefollowingconditionsaresatisfied:Itiseasytoseethatthenecessaryconditionfortheexistenceofthesolutionof(1.1)isInfact,fromtheassumptionsong,weknowthatu(t)/0foralltE(0,1)…     

Random attractor of the stochastic strongly damped wave equation

In this paper we study the asymptotic dynamics of the stochastic strongly damped wave equation with homogeneous Neumann boundary condition. We investigate the existence of a random attractor for the random dynamical system associated with the equation.     

Solvability of discrete Neumann boundary value problems

In this article we gain solvability to a nonlinear, second-order difference equation with discrete Neumann boundary conditions. Our methods involve new inequalities on the right-hand side of the difference equation and Schaefer's Theorem in the finite-dimensional space setting.     

Analysis and discretization of the volume penalized Laplace operator with Neumann boundary conditions

We study the properties of an approximation of the Laplace operator with Neumann boundary conditions using volume penalization. For the one-dimensional Poisson equation we compute explicitly the exact solution of the penalized equation and quantify the penalization error. Numerical simulations using finite differences allow then to assess the discretization and penalization errors. The eigenvalue problem of the penalized Laplace operator with Neumann boundary conditions is also studied. As examples in two space dimensions, we consider a Poisson equation with Neumann boundary conditions in rectangular and circular domains.     

Improved trial methods for a class of generalized Bernoulli problems

The aim of this article is to develop improved trial methods for the solution of a generalized exterior Bernoulli free boundary problem. At the free boundary, we prescribe the Neumann boundary condition and update the free boundary with the help of the remaining Dirichlet boundary condition. Appropriate update rules are obtained by expanding the state's Dirichlet data at the actual boundary via a Taylor expansion of first and second order. The resulting trial methods converge linearly for both cases, although the trial method based on the second order Taylor expansion is much more robust. Nevertheless, via results of shape sensitivity analysis, we are able to modify the update rules such that their convergence is improved. The feasibility of the proposed trial methods and their performance is demonstrated by numerical results.     

Diffusion approximation for nonlinear evolutionary equations with large interaction and fast boundary fluctuation

Approximations are derived for both nonlinear heat equations and singularly perturbed nonlinear wave equations with highly oscillating random force on boundary and strong interaction. By a diffusion approximation method, if the interaction is large and the singular perturbation is small enough, the approximation of the nonlinear wave equation is an one dimensional stochastic ordinary differential equation with white noise from the boundary which is exactly the same as that of the nonlinear heat equation.     

The boundary element method for stochastic potential problems

Stochastic Dirichlet and Neumann boundary value problems and stochastic mixed problems have been formulated. As a result the stochastic singular integral equations have been obtained. A way of solving these equations by means of discretization of a boundary using stochastic boundary elements has been presented, resulting in a set of random algebraic equations. It has been proved that for Dirichlet and Neumann problems probabilistic characteristics (i.e. moments: expected value and correlation function) fulfilled deterministic singular integral equations. A numerical method of evaluation of moments on a boundary and inside a domain has been presented.     

Carleman estimates for stochastic parabolic equations with Neumann boundary conditions and applications

By a dual method, two Carleman estimates for forward and backward stochastic parabolic equations with Neumann boundary conditions are established. Then they are used to study a null controllability problem and a state observation problem for some stochastic forward parabolic equations with Neumann boundary conditions.     

非线性发展方程的Taylor展开方法

提出了积分非线性发展方程的新方法,即Taylor展开方法．标准的Galerkin方法可以看作0-阶Taylor展开方法,而非线性Galerkin方法可以看作1-阶修正Taylor展开方法A·D2此外,证明了数值解的存在性及其收敛性．结果表明,在关于严格解的一些正则性假设下,较高阶的Taylor展开方法具有较高阶的收敛速度．最后,给出了用Taylor展开方法求解二维具有非滑移边界条件Navier-Stokes方程的具体例子．     

Stochastic Cahn-Hilliard equations driven by Poisson random measures

We study a stochastic Cahn-Hilliard equation driven by a Poisson random measure with Neumann boundary conditions. The global weak solution is established for the equation. Moreover, the existence of a Lyapunov function for the equation and an invariant measure associated with the transition semigroup are proved.     

A nonlocal nonlinear diffusion equation with blowing up boundary conditions

We deal with boundary value problems (prescribing Dirichlet or Neumann boundary conditions) for a nonlocal nonlinear diffusion operator which is analogous to the porous medium equation. First, we prove existence, uniqueness and the validity of a comparison principle for these problems. Next, we impose boundary data that blow up in finite time and study the behavior of the solutions.     

Small-amplitude solutions of the sine-Gordon equation on an interval under dirichlet or Neumann boundary conditions

Summary We give a complete classification of the small-amplitude finite-gap solutions of the sine-Gordon (SG) equation on an interval under Dirichlet or Neumann boundary conditions. Our classification is based on an analysis of the finite-gap solutions of the boundary problems for the SG equation by means of the Schottky uniformization approach.On leave from IPPI, Moscow, Russia