奇摄动问题中阶梯状解的缝接法

In view of singularly perturbed problems with complex inner layer phenomenon,including contrast structures(step-step solution and spike-type solution),corner layer behavior and right-hand side discontinuity,we carry out the process with sewing connection.The presented method of sewing connection for singularly perturbed equations is based on the two points singularly perturbed simple boundary problems.By means of sewing orbit smoothness,we get the uniformly valid solution in the whole interval.It is easy to prove the existence of solutions and deal with the high dimensional singularly perturbed problems.     

SINGULAR PERTURBATION OF BOUNDARY VALUE PROBLEM FOR NONLINEAR HIGHER ORDER ORDINARY DIFFERENTIAL EQUATIONS INVOLVING TWO SMALL PARAMETERS

The singularly perturbed boundary value problem for nonlinear higher order ordinary differential equation involving two small parameters has been considered. Under appropriate assumptions, for the three cases:ε/μ2→0(μ→0),μ2/ε→0 (ε→0) andε=μ2, the uniformly valid asymptotic solution is obtained by using the expansion method of two small parameters and the theory of differential inequality.     

The singularly perturbed problem of vector integro-differential equations

The singularly perturbed boundary value problem of scalar integro-differential equations has been studied extensively by the differential inequality method . However, it does not seem possible to carry this method over to a corresponding nonlinear vector integro-differential equation. Therefore , for n-dimensional vector integro-differential equations the problem has not been solved fully. Here, we study this nonlinear vector problem and obtain some results. The approach in this paper is to transform the appropriate integro-differential equations into a canonical or diagonalized system of two first-order equations.     

催化反应中的一类奇摄动边值问题

§1. Introduction We are concered with the singularly perturbed boundary value problemε2y″=y3,(1)y(0)=1, y(1)=2,(2)where ε＞0 is a positive small parameter. This problem arises as models for certain catalytic reactions in chemical engineering. The study of that problem has been paid much attention for the boundary layers of the problem exihibit the behavior of nonexponential decay. There have been some works on this subject ［1］-［4］. In particular, Howes and Chang［1］ gave an accurate…     

一类非线性奇摄动非局部反应扩散系统

Recently the nonlinear singularly perturbed problem has been investigated in theinternational academic circles[1 ,2 ] .Approximation methods have been developed andrefined,including the method of averaging,boundary layer method,matched asymptoticexpanision method and multiple scales method.Many scholars such as O' Malley,Jr.[3] ,Butuzov,Nefedov and Schneider[4] ,Kelley[5] ,Mizoguchi,Yanagida and Life[6] have done agreat deal of work.Using the method of differential inequality and other met…     

非线性奇摄动系统的渐近性态

The asymptotic behavior of solution for a n-dimensional nonlinear singularly perturbed system is studied. Under the appropriate assumptions, the existence of solution for the system is proved and the estimation of the solution is given using the method of differential inequalities.     

The Nonlinear Singularly Perturbed Nonlocal Reaction Diffusion Systems

In this paper the singularly perturbed initial boundary value problems for a nonlocal reaction diffusion system are considered. Using the iteration method and the comparison theorem, the existence and asymptotic behavior of solutions for the problem are studied.     

DISSIPATIVITY AND EXPONENTIAL STABILITY OF θ-METHODS FOR SINGULARLY PERTURBED DELAY DIFFERENTIAL EQUATIONS WITH A BOUNDED LAG

This paper deals with analytic and numerical dissipativity and exponential stability of singularly perturbed delay differential equations with any bounded state-independent lag. Sufficient conditions will be presented to ensure that any solution of the singularly perturbed delay differential equations (DDEs) with a bounded lag is dissipative and exponentially stable uniformly for sufficiently small ε > 0. We will study the numerical solution defined by the linear θ-method and one-leg method and show that they are dissipative and exponentially stable uniformly for sufficiently small ε > 0 if and only if θ = 1.     

THE SINGULARLY PERTURBED NONLINEAR ELLIPTIC SYSTEMS IN UNBOUNDED DOMAINS

Abstract. The singularly perturbed problems for elliptic systems in unbounded domains are considered. Under suitable conditions and by using the comparison theorem the existence and asymptotic behavior of solution for the boundary value problems studied,     

THE INITIAL-BOUNDARY VALUE PROBLEMS OF NONLINEAR AND NONLOCAL SINGULARLY PERTURBED REACTION DIFFUSION SYSTEM

In this paper the initial-boundary value problem for nonlocal singularly perturbed reaction diffusion system are considered.Using the iterative method and the compa- rison theorem,the existence and asymptotic behavior of the solution for the problem are studied.     

Ergodic Control of a Singularly Perturbed Markov Process in Discrete Time with General State and Compact Action Spaces

Ergodic control of singularly perturbed Markov chains with general state and compact action spaces is considered. A new method is given for characterization of the limit of invariant measures, for perturbed chains, when the perturbation parameter goes to zero. It is also demonstrated that the limit control principle is satisfied under natural ergodicity assumptions about controlled Markov chains. These assumptions allow for the presence of transient states, a situation that has not been considered in the literature before in the context of control of singularly perturbed Markov processes with long-run-average cost functionals. Accepted 3 December 1996     

Singularly Perturbed Volterra Integro-differential Equations

Several investigations have been made on singularly perturbed integral equations. This paper aims at presenting an algorithm for the construction of asymptotic solutions and then provide a proof for asymptotic correctness to singularly perturbed systems of Volterra integro-differential equations.     

Differentiability and its asymptotic analysis for nonlinear singularly perturbed boundary value problem

In this paper, we show differentiability of solutions with respect to the given boundary value data for nonlinear singularly perturbed boundary value problems and its corresponding asymptotic expansion of small parameter. This result fills the gap caused by the solvability condition in Esipova’s result so as to lay a rigorous foundation for the theory of boundary function method on which a guideline is provided as to how to apply this theory to the other forms of singularly perturbed nonlinear boundary value problems and enlarge considerably the scope of applicability and validity of the boundary function method. A third-order singularly perturbed boundary value problem arising in the theory of thin film flows is revisited to illustrate the theory of this paper. Compared to the original result, the imposed potential condition is completely removed by the boundary function method to obtain a better result. Moreover, an improper assumption on the reduced problem has been corrected.     

Multifrequency oscillations of singularly perturbed systems

We consider a system of differential equations that consists of two parts, a regularly perturbed and a singularly perturbed one. We assume that the matrix of the linear part of the regularly perturbed system has pure imaginary eigenvalues, while the matrix of the singularly perturbed part is hyperbolic; i.e., all of its eigenvalues have nonzero real parts.     

Input-to-state stability and sliding mode control of the nonlinear singularly perturbed systems via trajectory-based small-gain theorem

This paper presents the trajectory-based input-to-state stability (ISS) and input-to-output stability (IOS) small-gain theorem, and the finite-time ISS (FTISS) and finite-time IOS (FTIOS) of nonlinear singularly perturbed systems. The contribution of this paper is threefold. Firstly, a novel idea is proposed to analyze the stability of the nonlinear singularly perturbed system, which is regarded as an interconnected system by using two-time-scale decomposition. Secondly, the trajectory-based approach is applied to establish ISS and IOS small-gain theorem for singularly perturbed systems and the FTISS and FTIOS properties are proposed. Thirdly, a novel sliding mode controller is developed for a class of nonlinear singularly perturbed systems. Finally, the effectiveness of proposed method is illustrated by using a numerical example, a DC motor simulation and a multi-agent singularly perturbed system.     

Piecewise shooting reproducing kernel method for linear singularly perturbed boundary value problems

In this letter, a new numerical method is proposed for solving second order linear singularly perturbed boundary value problems with left layers. Firstly a piecewise reproducing kernel method is proposed for second order linear singularly perturbed initial value problems. By combining the method and the shooting method, an effective numerical method is then proposed for solving second order linear singularly perturbed boundary value problems. Two numerical examples are used to show the effectiveness of the present method.     

Cubic spline solution of singularly perturbed boundary value problems with significant first derivatives

We develop a numerical technique for a class of singularly perturbed two-point singular boundary value problems on an uniform mesh using polynomial cubic spline. The scheme derived in this paper is second-order accurate. The resulting linear system of equations has been solved by using a tri-diagonal solver. Numerical results are provided to illustrate the proposed method and to compared with the methods in [R.K. Mohanty, Urvashi Arora, A family of non-uniform mesh tension spline methods for singularly perturbed two-point singular boundary value problems with significant first derivatives, Appl. Math. Comput., 172 (2006) 531–544; M.K. Kadalbajoo, V.K. Aggarwal, Fitted mesh B-spline method for solving a class of singular singularly perturbed boundary value problems, Int. J. Comput. Math. 82 (2005) 67–76].     

Singularly perturbed problems in partial differential equations: a survey

This survey paper contains a surprisingly large amount of material on singularly perturbed partial differential equations and indeed can serve as an introduction to some of the ideas and methods of the singular perturbation theory. Starting from Prandtl's work a large amount of work has been done in the area of singular perturbations. This paper limits its coverage to some standard singular perturbation models considered by various workers and the methods developed by numerous researchers after 1980–2000. In this review we have covered singularly perturbed partial differential equations. About ODEs the survey has already been done by us [see M.K. Kadalbajoo, K.C. Patidar, Appl. Math. Comput. 130 (2002) 457–510].     

A new fitted operator finite difference method to solve systems of evolutionary reaction-diffusion equations

Abstract

In recent years, fitted operator finite difference methods (FOFDMs) have been developed for numerous types of singularly perturbed ordinary differential equations. The construction of most of these methods differed though the final outcome remained similar. The most crucial aspect was how the difference operator was designed to approximate the differential operator in question. Very often the approaches for constructing these operators had limited scope in the sense that it was difficult to extend them to solve even simple one-dimensional singularly perturbed partial differential equations. However, in some of our most recent work, we have successfully designed a class of FOFDMs and extended them to solve singularly perturbed time-dependent partial differential equations. In this paper, we design and analyze a robust FOFDM to solve a system of coupled singularly perturbed parabolic reaction-diffusion equations. We use the backward Euler method for the semi-discretization in time. An FOFDM is then developed to solve the resulting set of boundary value problems. The proposed method is analyzed for convergence. Our method is uniformly convergent with order one and two, respectively, in time and space, with respect to the perturbation parameters. Some numerical experiments supporting the theoretical investigations are also presented.