On the Asymptotic and Numerical Analyses of Exponentially III-Conditioned Singularly Perturbed Boundary Value Problems

Asymptotic and numerical methods are used to study several classes of singularly perturbed boundary value problems for which the underlying homogeneous operators have exponentially small eigenvalues. Examples considered include the familiar boundary layer resonance problems and some extensions and certain linearized equations associated with metastable internal layer motion. For the boundary layer resonance problems, a systematic projection method, motivated by the work of De Groen [1], is used to analytically calculate high-order asymptotic solutions. This method justifies and extends some previous results obtained from the variational method of Grasman and Matkowsky [2]. A numerical approach, based on an integral equation formulation, is used to accurately compute boundary layer resonance solutions and their associated exponentially small eigenvalues. For various examples, the numerical results are shown to compare very favorably with two-term asymptotic results. Finally, some Sturm-Liouville operators with exponentially small spectral gap widths are studied. One such problem is applied to analyzing metastable internal layer motion for a certain forced Burgers equation.     

一类拟线性奇摄动流体模型的渐近分析

对一类拟线性流体模型进行研究,借助Linard变换将所研究问题转化为可以用边界函数法处理的问题,进而用边界函数法对该方程组进行分析,构造其(n+1)阶形式渐近解,并证明解的存在唯一性和形式渐近解的一致有效性.     

Singularly Perturbed Markov Chains with Two Small Parameters: A Matched Asymptotic Expansion

This work is concerned with asymptotic properties of solutions to forward equations for singularly perturbed Markov chains with two small parameters. It is motivated by the model of a cost-minimizing firm involving production planning and capacity expansion and a two-level hierarchical decomposition. Our effort focuses on obtaining asymptotic expansions of the solutions to the forward equation. Different from previous work on singularly perturbed Markov chains, the inner expansion terms are constructed by solving certain partial differential equations. The methods of undetermined coefficients are used. The error bound is obtained.     

Asymptotic Solution of a Singularly Perturbed Optimal Problem with Integral Constraint

Journal of Optimization Theory and Applications - Using the so-called direct scheme method, an asymptotic expansion of n-th order to the solution of a class of singularly perturbed linear-quadratic...     

Asymptotic Expansions and a New Numerical Algorithm of the Algebraic Riccati Equation for Multiparameter Singularly Perturbed Systems

In this paper we study a continuous-time multiparameter algebraic Riccati equation (MARE) with an indefinite sign quadratic term. The existence of a unique and bounded solution of the MARE is newly established. We show that the Kleinman algorithm can be used to solve the sign indefinite MARE. The proof of the convergence and the existence of the unique solution of the Kleinman algorithm is done by using the Newton-Kantorovich theorem. Furthermore, we present new algorithms for solving the generalized multiparameter algebraic Lyapunov equation (GMALE) by means of the fixed-point algorithm.     

具有积分边界条件的常微分方程边值问题的数值解

讨论了一类具有积分边界条件的二阶常微分方程非局部边值问题的数值解.对非局部积分边界条件采用了离散的多点边值问题进行逼近,通过常系数情况下解的局部性质,建立了这类边值问题的指数型差分格式,并且给出了格式的误差分析,证明了格式是一致收敛的.     

双参数奇摄动高阶半线性椭圆型方程的渐近解

讨论了一类具有双参数的高阶半线性奇摄动椭圆型方程的边值问题.在适当的假设下,构造了解的形式渐近展开式,并且利用不动点定理,研究了边值问题解的存在性和渐近性态.     

Asymptotic Solution of a Singularly Perturbed Linear-Quadratic Problem in Critical Case with Cheap Control

Using the direct scheme method, we construct an asymptotic expansion for the solution of a singularly perturbed optimal problem in critical case with cheap control and two fixed end-points. The asymptotic solution contains the outer series and two boundary-layer series in the vicinities of the two end-points. The error estimates for state and control variables and the functional are obtained. It is shown that the value of minimized functional does not increase when a higher-order approximation to the optimal control is used. An illustrative example is given.     

Asymptotic Expansion of Eigenvalues of the Laplace Operator in Domains with Singularly Perturbed Boundary

In this paper, we consider eigenvalue problems for the Laplace operator in three-dimensional domains with singularly perturbed boundary. Perturbations are generated by a complementary Dirichlet boundary condition on a small nonclosed surface inside the domain. The convergence and the asymptotic behavior of simple eigenvalues of the problem are considered.__________Translated from Matematicheskie Zametki, vol. 78, no. 2, 2005, pp. 299–307.Original Russian Text Copyright © 2005 by M. I. Cherdantsev.     

Asymptotic Behavior of Solutions of Linear Singularly Perturbed General Separated Boundary-Value Problems with Initial Jump

We obtain asymptotic estimates for solutions of singularly perturbed boundary-value problems with initial jumps.     

一类两参数半线性奇摄动问题解的渐近性态

本文研究了一类两参数半线性奇摄动问题的基本模型.利用奇摄动方法,对该问题解的结构在两个小参数相互关联的三种不同情形下作了讨论.得到了该问题在三种不同情形下的渐近解并证明了在三种情形下解的结构与渐近性态.     

Numerical Analysis of a Nonlinear Singularly Perturbed Delay Volterra Integro-Differential Equation on an Adaptive Grid

In this paper, we study a nonlinear first-order singularly perturbed Volterra integro-differential equation with delay. This equation is discretized by the backward Euler for differential part and the composite numerical quadrature formula for integral part for which both an a priori and an a posteriori error analysis in the maximum norm are derived. Based on the a priori error bound and mesh equidistribution principle, we prove that there exists a mesh gives optimal first order convergence which is robust with respect to the perturbation parameter. The a posteriori error bound is used to choose a suitable monitor function and design a corresponding adaptive grid generation algorithm. Furthermore, we extend our presented adaptive grid algorithm to a class of second-order nonlinear singularly perturbed delay differential equations. Numerical results are provided to demonstrate the effectiveness of our presented monitor function. Meanwhile, it is shown that the standard arc-length monitor function is unsuitable for this type of singularly perturbed delay differential equations with a turning point.     

Singularly Perturbed Markov Chains: Convergence and Aggregation

Asymptotic properties of singularly perturbed Markov chains having measurable and/or continuous generators are developed in this work. The Markov chain under consideration has a finite-state space and is allowed to be nonstationary. Its generator consists of a rapidly varying part and a slowly changing part. The primary concerns are on the properties of the probability vectors and an aggregated process that depend on the characteristics of the fast varying part of the generators. The fast changing part of the generators can either consist of l recurrent classes, or include also transient states in addition to the recurrent classes. The case of inclusion of transient states is examined in detail. Convergence of the probability vectors under the weak topology of L² is obtained first. Then under slightly stronger conditions, it is shown that the convergence also takes place pointwise. Moreover, convergence under the norm topology of L² is derived. Furthermore, a process with aggregated states is obtained which converges to a Markov chain in distribution.     

具有初始跳跃的双曲型方程奇异摄动问题的数值解法

本文考虑具有初始跳跃的二阶双曲型方程初边值问题.首先给出解的导数估计.然后在一非均匀网格上构造了一个差分格式,最后在能量范数意义下证明了差分格式解的一致收敛性.     

Uniform Convergence Analysis for Singularly Perturbed Elliptic Problems with Parabolic Layers

In this paper,using Lin's integral identity technique,we prove the optimal uniform convergence θ(N_x~(-2)In~2N_x N_y~(-2)In~2N_y) in the L~2-norm for singularly per- turbed problems with parabolic layers.The error estimate is achieved by bilinear fi- nite elements on a Shishkin type mesh.Here N_x and N_y are the number of elements in the x- and y-directions,respectively.Numerical results are provided supporting our theoretical analysis.     

薄层流中的一类三阶奇摄动边值问题的渐近分析

本文研究薄层流中出现的一类三阶奇摄动数学模型．本文不采用研究其渐近等价的二阶奇摄动微分方程的方法,而利用边界层函数法,直接讨论该数学模型的渐近解,并严格地证明了解的存在唯一性和其渐近解的一致有效性．本文的结果不仅去掉了以往方法所必须的位势条件,纠正了某个不适定的假设,而且推广了以往的结果．