Numerical treatment of singularly perturbed two point boundary value problems exhibiting boundary layers

A boundary value method for solving a class of nonlinear singularly perturbed two point boundary value problems with a boundary layer at one end is proposed. Using singular perturbation analysis the method consists of solving two problems; namely, a reduced problem and a boundary layer correction problem. We use Pade’ approximation to obtain the solution of the latter problem and to satisfy the condition at infinity. Numerical examples will be given to illustrate the method.     

一类线性奇异摄动最优控制问题的渐近解

研究了一类线性奇异摄动最优控制问题的空间对照结构,讨论了初始点固定,终端自由的情形.首先根据变分法得到了一阶最优性条件,其次运用退化最优控制问题的解证明了异宿轨道的存在性,从而结合奇异摄动理论证明了原问题空间对照结构解的存在性.进一步根据解的结构,利用边界层函数法构造了奇异摄动最优控制问题一致有效的形式渐近解.最后,通...     

THE ASYMPTOTIC BEHAVIOR OF SOLUTION FOR A CLASS OF STRONGLY NONLINEAR NON-AUTONOMOUS EQUATION

In this paper, using the singularly perturbed theory and the boundary layer corrective method, the asymptotic behavior of solution for a class of strongly nonlinear non-autonomous equations and the infection for asymptotic behavior of the solution with regard to the boundary condition are studied. According to the different regions of the boundary value, the asymptotic expansions of the solution for the original problem are obtained simply and conveniently.     

三阶奇摄动非线性边值问题

利用微分不等式理论,研究了某一类三阶奇摄动非线性边值问题。以二阶边值问题的已知结果为基础,引入Volterra型积分算子,建立了三阶非线性边值问题的上下解方法。在适当条件下,构造出具体的上下解,得出解的存在性和渐进估计。结果表明这种技巧也为三阶奇摄动边值问题的研究提出了崭新的思路。最后举例验证文中定理的正确性。     

一类四阶非线性奇摄动方程的边界层问题

本文研究了一类四阶非线性奇摄动方程的边界层问题,利用在左右边界层的两次匹配,得出了原问题解的一致有效的渐近表达式.这个结果是奇摄动理论在研究高阶微分方程中的一个应用.     

A CLASS OF STRONGLY NONLINEAR SINGULARLY PERTURBED INTERIOR LAYER PROBLEMS

In this paper, a class of strongly nonlinear singularly perturbed interior layer problems are considered by the theory of differential inequalities and the corrective theory of interior layer. The existence of solution is proved and the asymptotic behavior of solution for the boundary value problems are studied. And the satisfying result is obtained.     

一类非线性快慢系统非局部问题的摄动解

主要讨论了一类非线性快慢系统非局部问题的摄动解,在适当的条件下,根据不同边界层利用伸长变量和幂级数展开理论,构造了问题的形式渐近解,并利用微分不等式理论在整个区间上证明了形式渐近解的一致有效性,把奇摄动问题的摄动解推广到快慢系统非局部问题的摄动解.     

一类四阶非线性奇摄动问题的匹配解法

利用匹配条件,讨论了一类四阶非线性奇摄动问题,得出了奇摄动边值问题的渐近展开式.     

Numerical treatment of singularly perturbed two point boundary value problems using initial-value method

In this paper, we describe an initial-value method for linear and nonlinear singularly perturbed boundary value problems in the interval [p,q]. For linear problems, the required approximate solution is obtained by solving the reduced problem and one initial-value problems directly deduced from the given problem. For nonlinear problems the original second-order nonlinear problem is linearized by using quasilinearization method. Then this linear problem is solved as previous method. The present method has been implemented on several linear and non-linear examples which approximate the exact solution. We also present the approximate and exact solutions graphically.     

具有多重解的非线性Robin问题的奇摄动[英文]

本文利用边界层法，研究了具有多重解的非线性Robin问题εx″ f(t,x)x′ g(t,x)=0,0≤t≤1,x′(0,ε）-ax(0,ε)=A,x′(1,ε） bx(1,ε）=B其中ε为正的小参数。在适当的假设下，我们通过给出外部解展开式系数的一般表达式，得到了退化问题的边值为某方程的多重根时的渐近解，推广了有关结果。     

SINGULARLY PERTURBED SEMI-LINEAR BOUNDARY VALUE PROBLEM WITH DISCONTINUOUS FUNCTION

A class of singularly perturbed semi-linear boundary value problems with discontinuous functions is examined in this article.Using the boundary layer function method,the asymptotic solution of such a p...     

A CLASS OF SINGULARLY PERTURBED NONLINEAR BOUNDARY VALUE PROBLEM

The singularly perturbed boundary value problem for the nonlinear boundary conditions is considered. Under suitable conditions,the asymptotic behavior of solution for the original problems is studied by using theory of differential inequalities.     

奇摄动非线性边值问题

本文讨论了一类奇摄动非线性边值问题 .利用伸长变量和边界层校正法 ,得到了问题解的形式渐近展开式 .再用微分不等式理论 ,证明了解的一致有效性     

The nonlinear nonlocal singularly perturbed problems for reaction diffusion equations with a boundary perturbation

The nonlinear nonlocal singularly perturbed initial boundary value problems for reaction diffusion equations with a boundary perturbation is considered. Under suitable conditions, the outer solution of the original problem is obtained. Using the stretched variable, the composing expansion method and the expanding theory of power series the initial layer is constructed. And then using the theory of differential inequalities the asymptotic behavior of solution for the initial boundary value problems is studied. Finally the existence and uniqueness of solution for the original problem and the uniformly valid asymptotic estimation are discussed.     

具有边界摄动的非线性非局部反应扩散方程的奇摄动问题

A class of nonlinear nonlocal singularly perturbed Robin initial boundary value problems for reaction diffusion equations with boundary perturbation is considered. Under suitable conditions, firstly, the outer solution of the original problem is obtained; secondly, by using the stretched variable, the composing expansion method and the expanding theory of power series, the initial layer is constructed; and finally, by using the theory of differential inequalities the asymptotic behavior of solutions for initial boundary value problems is studied, and including some relational inequalities the existence and uniqueness of solutions for the original problem and the uniformly valid asymptotic estimation are discussed.     

一类具有边界摄动的非线性泛函椭圆型方程奇摄动问题

研究了具有边界摄动的非线性泛函椭圆型方程奇摄动边值问题．在适当的条件下．利用伸长变量、微分不等式理论，讨论了问题解的渐近性态和原问题解的存在唯一性。     

THE SINGULARLY PERTURBED BOUNDARY VALUE PROBLEM OF NONLINEAR INTEGRO-DIFFERENTIAL SYSTEM

In this paper, using the differential inequality and singularly perturbed theory, the singularly perturbed boundary value problems for nonlinear integro-differential system has been studied. Under some appropriate assumptions, the existence of solution has been proved and the uniform validness of the asymptotic expansions for arbitrary nth-order have been obtained simply and conveniently.     

An asymptotic initial value method for second order singular perturbation problems of convection-diffusion type with a discontinuous source term

In this paper a numerical method is presented to solve singularly perturbed two points boundary value problems for second order ordinary differential equations consisting a discontinuous source term. First, in this method, an asymptotic expansion approximation of the solution of the boundary value problem is constructed using the basic ideas of a well known perturbation method WKB. Then some initial value problems and terminal value problems are constructed such that their solutions are the terms of this asymptotic expansion. These initial value problems are happened to be singularly perturbed problems and therefore fitted mesh method (Shishkin mesh) are used to solve these problems. Necessary error estimates are derived and examples provided to illustrate the method.     

Singularly perturbed solution for third order nonlinear equations with two parameters

A class of singularly perturbed boundary value problems for nonlinear equation of the third order with two parameters is considered. Under suitable conditions, using the theory of differential inequalities the existence and asymptotic behavior of the solution for boundary value problem are studied.     

A novel method for solving a class of singularly perturbed boundary value problems based on reproducing kernel method

In this paper, a novel method is presented for solving a class of singularly perturbed boundary value problems. Firstly the original problem is reformulated as a new boundary value problem whose solution does not change rapidly via a proper transformation; then the reproducing kernel method is employed to solve the boundary value new problem. Numerical results show that the present method can provide very accurate analytical approximate solutions.