数值求解二维抛物型奇异摄动问题的区域分解算法

1.引言奇异摄动问题产生于流体力学、量子力学、声学、光学、化学反应和最优控制等重要领域,其特点是:在讨论的问题中含有小参数。对于这类问题,一般不能求得其精确解,但可以用所谓的渐近方法求其近似解。近三十年来,随着计算机科学与技术的迅猛发展,使得许多科学家参与此类问题的数值求解方法的研究。所采用的主要方法有:指数型拟合差分格式,A.M.Il’in     

微分求积法分析二维亚音速壁板的失稳问题

采用微分求积方法(简称DQ方法)研究了亚音速气流中二维壁板的失稳问题.运用特征值方法分析了壁板在不同边界条件下的失稳特性.结果表明:本方法可有效确定系统的失稳;两端简支和两端固支的壁板出现了发散失稳而未出现颤振失稳;固支—弹性支承的壁板系统出现了颤振失稳,颤振失稳动压与系统参数相关.     

用余弦微分求积法数值求解KdV—Burgers方程

采用余弦微分求积法（CDQM）对（1＋1）维非线性KdV—Burgers方程进行了数值求解．结果表明,所得数值解与方程的精确解相比具有明显的高精度且稳定性高,相对于其他常用方法,且公式简单,使用方便;计算量小,时间复杂性好．     

奇异摄动内层问题的宽度估计及其数值求解

针对奇异摄动内层问题的数值计算,提出了一种内层宽度的估计方法.通过区域分解将原问题拆分为退化问题和内层问题两部分,采用重心形式的有理谱配点法进行离散求解.为了确定内层的宽度,设计了基于一阶导数连续性条件的非线性优化问题.进一步,借助Adomian分解法考虑了一类非线性内层问题的线性化求解.数值实验验证了该方法的可行性和...     

差分进化与有理谱方法求解奇异摄动问题

讨论一种数值求解奇异摄动问题的高精度有理谱配点法。用sinh变换的有理谱配点法使Chebyshev节点在边界层处加密;只需较少的节点即可达到较高的精度。为了获得sinh变换中边界层的宽度;设计了一个以误差最小为目标函数的无约束的非线性优化问题;并给出了求解该优化问题的差分进化算法。数值实验表明;与其他的智能算法和传统的优化算法相比;差分进化算法在sinh变换中的参数优化方面具有明显的优势。     

基于微分求积法的互连线灵敏度分析

提出一种基于微分求积法、对互连线的响应波形进行灵敏度分析的新方法．以传输线时域模型对互连线进行建模,在此基础上导出灵敏度分析模型．由于利用了所有离散点的信息,微分求积法通常只需要较少的采样点就能达到较高的精度．数值算例证明了该方法的有效性．     

层适应网格上求解奇异摄动问题的粒子群算法

针对一类奇异摄动对流扩散问题,将粒子群算法与差分格式相结合,在Bakhvalov-Shishkin网格上进行求解.对于Bakhvalov-Shishkin网格中的网格参数,采用粒子群算法进行优化,构造了求误差范数最小值的目标函数.对两个算例进行了数值计算,实验结果表明,与选择固定的网格参数相比,采用粒子群算法计算能得到...     

奇异摄动系统鲁棒镇定问题

讨论了确定与不确定奇异摄动系统的稳定性问题。首先给出了确定系统稳定及不确定系统鲁棒稳定的条件,同时给出摄动参数的稳定上界,其次,对不稳定系统,给出了状态反馈可镇定的条件及控制器的求解。这些条件均可由MATLAB中的求解器求解。最后,数值实例验证了该方法的可行性和有效性。     

用微分求积法分析输液管道的非线性动力学行为

将微分求积法(Differential Quadrature Method,简称DQM)应用于输液管道的非线性动力学分析,采用此法研究了受非线性约束输液管道的分岔现象和混沌运动问题.从悬臂输液管道模型出发,利用微分求积法形成管道的动力学方程.以分岔图、相平面图、时间历程图和Poincare映射等分析手段考察了系统参数(管内流速)变化对管道振动形态的影响.结果表明,在所研究的系统中存在出现倍周期分岔现象和混沌运动的参数区域,这与前人的研究成果具有一致性.这为一类结构的非线性动力响应问题提供了一种新的研究思路.     

保证闭环性能品质的奇异摄动类降阶方法

1　引言目前,由于H2,H∞和μ等理论所设计的控制器阶数较高,迫切需要对其进行降阶.Zhou等人[1]针对某一类控制系统,提出了保证闭环性能品质的结构均衡截取控制器降阶方法.DoChangOh等人[2]将此方法推广到了奇异摄动降阶方法,它具有低频段误差较小的优点.本文在Zhou等人方法的基础上,利用奇异摄动降阶方法同截取降阶方法的双线性变换关系,用比较简洁的方法证明了DoChangOh等人的结果.并进一步运用这种思想,推广得到了保证闭环性能品质控制器降阶的广义奇异摄动方法.2　保证闭环性能品质的结构均衡截取控制器降阶方法[1]定理1[1].设控制器…     

A new efficient spectral galerkin method for singular perturbation problems

A new spectral Galerkin method is proposed for the convection-dominated convection-diffusion equation. This method employs a new class of trail function spaces. The available error bounds provide a clear theoretical interpretation for the higher accuracy of the new method compared to the conventional spectral methods when applied to problems with thin boundary layers. Efficient solution techniques are developed for the convection-diffusion equations by using appropriate basis functions for the new trial function spaces. The higher accuracy and the effectiveness of the new method for problems with thin boundary layers are confirmed by our numerical experiments.The work of this author is partially supported by NSF grant DMS-9205300.     

An initial-value technique to solve third-order reaction–diffusion singularly perturbed boundary-value problems

The aim of this paper is to build an efficient initial-value technique for solving a third-order reaction–diffusion singularly perturbed boundary-value problem. Using this technique, a third-order reaction–diffusion singularly perturbed boundary-value problem is reduced to three approximate unperturbed initial-value problems and then Runge–Kutta fourth-order method is used to solve these unperturbed problems numerically.     

Numerical solutions of the Burgers–Huxley equation by the IDQ method

In this paper, the Burgers–Huxley equation has been solved by a generalized version of the Iterative Differential Quadrature (IDQ) method for the first time. The IDQ method is a method based on the quadrature rules. It has been proposed by the author applying to a certain class of non-linear problems. Stability and error analysis are performed, showing the efficiency of the method. Besides, an error bound is tried. In the discussion about the numerical examples, the generalized Burgers–Huxley equation is involved too.     

An efficient method for solving nonlinear singular initial value problems

In [M.M. Hosseini, Modified Adomain decomposition method for specific second order ordinary differential equations, Appl. Math. Comput. 186 (2007), pp. 117–123] an efficient modification of Adomian decomposition method has been proposed for solving some cases of ordinary differential equations. In this paper, this method is generalized to more cases. The proposed method can be applied to linear, nonlinear, singular and nonsingular problems. Here, it is focused on nonlinear singular initial value problems of ordinary differential equations. The scheme is tested for some examples and the obtained results demonstrate reliability and efficiency of the proposed method.     

一类非线性系统的时标分解和反馈线性化

在过去的十几年,已经证明基于微分几何理论的精确线性化是一种有效的非线性系统分析与设计方法。但是当这种方法应用在高阶非线性系统时,会出现Ｋｒｏｎｅｅｋｅｒ指数较大。本文提出一种建立在奇异摄动和反馈线性化相结合的方法可较好地应用于具有双时标性质的高阶非线性系统,并将其成功地应用在鱼雷纵平面非线性运动方程。     

A novel numerical method for a class of problems with the transition layer and Burgers’ equation

A numerical method for solving a class of quasi-linear singular two-point boundary value problems with a transition layer is presented in this paper. For the problem ? u _xx +a(u+f(x))u _x +b(x, u)=0, we develop a multiple scales method. First, this method solves the location of the transition layer, then it approximates the singular problem with reduced problems in the non-layer domain and pluses a layer corrected problem which nearly has an effect in the layer domain. Both problems are transformed into first-order problems which can be solved easily. For the problem ? u _xx +b(x, u)=0, we establish a similar method which approximate the problem with reduced problems and a two-point boundary value problem. Unsteady problems are also considered in our paper. We extend our method to solve Burgers’ equation problems by catching the transition layer with the formula of shock wave velocity and approximating it by a similar process.     

高阶系统的奇异摄动模型的平衡降阶

随着控制理论和电子计算机技术的迅速发展,尽管为研究与设计高阶复杂系统提供了方便的条件,但是对复杂高阶系统进行模型简化仍然是控制系统设计与仿真工作者的一个重要研究课题.针对奇异摄动模型的平衡降阶方法,提出了采用时矩输出拟合来改善降阶系统输出响应逼近程度的改进方法,通过给出的实例仿真证实了这一点.     

Solving singular nonlinear boundary value problems by combining the homotopy perturbation method and reproducing kernel Hilbert space method

This paper investigates singular nonlinear boundary value problems (BVPs). The numerical solutions are developed by combining He's homotopy perturbation method (HPM) and reproducing kernel Hilbert space method (RKHSM). He's HPM is based on the use of traditional perturbation method and homotopy technique. The HPM can reduce a nonlinear problem to a sequence of linear problems and generate a rapid convergent series solution in most cases. RKHSM is also an analytical technique, which can solve powerfully singular linear BVPs. Therefore, we solve singular nonlinear BVPs using advantages of these two methods. Three numerical examples are presented to illustrate the strength of the method.     

Application of homotopy perturbation method to multidimensional partial differential equations

In this paper, an application of homotopy perturbation method (HPM) is applied to solve the kindly of multidimensional partial differential equation such as Helmholtz equation. Comparisons are made between the Adomians decomposition method and HPM. The results reveal that the HPM is very effective and simple and gives the exact solution.