两参数非线性发展方程的奇摄动尖层解（英文）

本文研究了一类具有非线性发展方程奇摄动问题.引入伸长变量和多重尺度,构造了初始边值问题外部解和尖层、边界层和初始层校正项,得到了问题形式解.利用不动点定理,证明了问题的解的一致有效性.推广了对两参数的奇摄动问题的研究结果.     

两参数非线性反应扩散积分微分方程的内层激波渐近解

研究了一类两参数非线性反应扩散积分微分奇摄动问题.利用奇摄动方法,构造了问题的外部解、内部激波层、边界层及初始层校正项,由此得到了问题解的形式渐近展开式.最后利用积分微分方程的比较定理证明了该问题解的渐近展开式的一致有效性.     

一类双参数奇摄动非线性反应扩散方程

本文研究了一类双参数非线性反应扩散奇摄动问题的模型.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情形下作了讨论.得到了该问题的渐近解,由解的展开式看出本问题的解同时具有初始层和边界层.     

两参数非线性反应-扩散系统的尖层冲击波解

研究了一类两参数反应-扩散系统奇摄动Robin初始-边值问题.首先,利用奇摄动方法,联系到两个小参数构造了问题的外部解.其次,利用伸长变量分别得到了原问题解的的冲击波尖层,边界层和初始层校正项.最后,得到了原问题解的渐近展开式,并利用微分不等式理论证明了渐近解的一致有效性.由本方法求原问题的渐近解,它还可以进一步进行微分,积分等解析运算,从而能了解相应冲击波解的更深层的性态.因此本方法具有良好的应用前景.     

一类广义非线性反应扩散方程奇摄动问题激波解（英文）

本文研究一类广义非线性反应扩散方程奇摄动初始边值问题.首先,构造非线性问题的外部解.其次,利用局部坐标系和伸长变量得到激波层和边界层校正项.最后,利用不动点理论研究了非线性反应扩散方程初始边值问题广义解的渐近性态.     

具有边界摄动弱非线性反应扩散方程的奇摄动

在适当的条件下研究了一类具有边界摄动的非线性反应扩散方程奇摄动初始边值问题．首先,借助正规摄动方法,得到了原问题的外部解．其次,利用伸长变量和幂级数展开理论,构造了解的初始层项．然后,利用微分不等式理论,研究了初始边值问题解的渐近性态．最后,利用一些相关的不等式,讨论了原问题解的存在、唯一性及其一致有效的渐近估计．     

一类含慢变量奇摄动非线性系统初始层现象

研究含有慢变量的一类奇摄动非线性系统初始层现象,通过引进不同量级的伸长变量,构造不同“厚度”的初始层校正项,得到了摄动解关于小参数的N阶近似展开式,揭示了摄动解呈现的“层中层”现象,并利用不动点原理证明了摄动解的存在,给出了解的一致有效的渐近展开式．     

双参数非线性非局部奇摄动抛物型初始-边值问题的广义解

研究了一类广义抛物型方程奇摄动问题.首先在一定的条件下, 提出了一类具有两参数的非线性非局部广义抛物型方程初始 边值问题.其次证明了相应问题解的存在性.然后, 通过Fredholm积分方程得到了初始 边值问题的外部解.再利用泛函分析理论和伸长变量及多重尺度法, 分别构造了初始 边值问题广义解的边界层、初始层项,从而得到了问题的形式渐近展开式.最后利用不动点理论证明了对应的非线性非局部广义抛物型方程的奇异摄动初始 边值问题的广义解的渐近展开式的一致有效性.     

具有奇性方程的非线性奇摄动问题的正解(英文)

本文讨论了一类具有奇性方程的奇摄动初值问题.在适当条件下,利用微分不等式理论,研究了初值问题解的存在性及其渐近性态,并且得到了具有初始层的一致有效解的渐近展开式.     

一类非线性微分-积分时滞反应扩散系统奇摄动问题的广义解

该文研究了一类非线性微分-积分时滞广义反应扩散系统奇摄动问题.在适当的条件下,利用奇摄动方法构造了初始-边值问题广义解的渐近展开式.建立了广义解的微分不等式理论,并证明了相应解的存在性及其解的渐近展开式的一致有效性.     

An iterative method based on equation decomposition for the fourth‐order singular perturbation problem

In this article, we propose an iterative method based on the equation decomposition technique ( 1 ) for the numerical solution of a singular perturbation problem of fourth‐order elliptic equation. At each step of the given method, we only need to solve a boundary value problem of second‐order elliptic equation and a second‐order singular perturbation problem. We prove that our approximate solution converges to the exact solution when the domain is a disc. Our numerical examples show the efficiency and accuracy of our method. Our iterative method works very well for singular perturbation problems, that is, the case of 0 < ε ? 1, and the convergence rate is very fast. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013     

A uniformly convergent difference scheme for a semilinear singular perturbation problem

Summary We present a difference scheme for solving a semilinear singular perturbation problem with any number of turning points of arbitrary orders. It is shown that a solution of the scheme converges, uniformly in a perturbation parameter, to that of the continuous problem.     

双曲-抛物偏微分方程奇异摄动问题的差分解法

本文对双曲-抛物偏微分方程奇异摄动问题构造了一个指数型拟合差分格式.我们不仅在方程中加了一个拟合因子,而且在逼近第二个初始条件时也加了拟合因子.我们利用问题的渐近解证明了差分格式关于小参数的一致收敛性.     

Semilinear Finite Element Method

In the Ritz-Galerkin method the linear subspace of the trial solution is extended to a closed subset. Some results, such as orthogonalization and minimum property of the error function, are obtained. A second order scheme is developed for solving a linear singular perturbation elliptic problem and error estimates are given for a uniform mesh size. Numerical results for linear and semilinear singular perturbation problems are included.     

Perturbation singulière d'un problème de frottement sec non monotone

In this Note we deal with a singularly perturbed system constituted by a differential inclusion which has a unique solution for each value of the perturbation parameter. The associated degenerated problem, that corresponds to a dynamic dry friction problem, has many solutions. We show that perturbed problem solutions converge to a particular solution of the degenerated problem when the perturbation parameter goes to zero. The singular perturbation approach allows an analysis of a criterion used to select a solution of the degenerated problem, and suggests a method to study more elaborated dry friction problems.     

Perturbation Identities for Regularized Tikhonov Inverses and Weighted Pseudoinverses

We consider the perturbation analysis of two important problems for solving ill-conditioned or rank-deficient linear least squares problems. The Tikhonov regularized problem is a linear least squares problem with a regularization term balancing the size of the residual against the size of the weighted solution. The weight matrix can be a non-square matrix (usually with fewer rows than columns). The minimum-norm problem is the minimization of the size of the weighted solutions given by the set of solutions to the, possibly rank-deficient, linear least squares problem.It is well known that the solution of the Tikhonov problem tends to the minimum-norm solution as the regularization parameter of the Tikhonov problem tends to zero. Using this fact and the generalized singular value decomposition enable us to make a perturbation analysis of the minimum-norm problem with perturbation results for the Tikhonov problem. From the analysis we attain perturbation identities for Tikhonov inverses and weighted pseudoinverses.     

Stability of a singularly perturbed problem

We prove the stability of the mixed problem for a system of telegraph equations under a perturbation of one of the boundary conditions by a sum of a singular perturbation (a small parameter multiplying the highest derivative) and a small regular perturbation. The solution of the problem consists of the current and voltage in a segment of a telegraph line. One of its ends is short-circuited, and a capacitor of small capacity, together with a nonlinear resistance whose volt-ampere characteristic is perturbed by a small term, is connected to the other end. We prove the convergence of the solution of the problem to the unique continuous piecewise continuously differentiable solution of the unperturbed problem bifurcating at some instant of time from its unique classical solution.     

Piecewise homotopy perturbation method for solving linear and nonlinear weakly singular VIE of second kind

The purpose of this paper is to obtain the approximation solution of linear and strong nonlinear weakly singular Volterra integral equation of the second kind, especially for such a situation that the equation is of nonsmooth solution and the situation that the problem is a strong nonlinear problem. For this purpose, we firstly make a transform to the equation such that the solution of the new equation is as smooth as we like. Through modifying homotopy perturbation method, an algorithm is successfully established to solve the linear and nonlinear weakly singular Volterra integral equation of the second kind. And the convergence of the algorithm is proved strictly. Comparisons are made between our method and other methods, and the results reveal that the modified homotopy perturbation is effective.     

On Models for an Epidemic

We consider deterministic and stochastic models for the progressof malaria. A simple model takes the form of a set of delaydifferential equations in which a small parameter multipliesthe highest derivatives and the delays. A solution is soughtas a series in powers of the small parameter, using the specialmethods appropriate to this type of singular perturbation problem.We then consider the threshold problem for the primary host. The stochastic model also presents a singular perturbation problem,and the asymptotic method is extended appropriately. This enablesus to examine the distribution of the duration of a simple epidemic,and the stochastic equivalent of the threshold problem.