非Fourier温度场分布的奇摄动解

应用非Fourier热传导定律构建了单层材料中温度场模型，即一类在无界域上带小参数的奇摄动双曲方程，通过奇摄动展开方法，得到了该问题的渐近解.首先应用奇摄动方法得到了该问题的外解和边界层矫正项，通过对内解和外解的最大模估计和关于时间导数的最大模估计以及线性抛物方程理论，得到了内外解的存在唯一性，从而得到了解的形式渐近展开式.通过余项估计，给出了渐近解的L2估计，得到了渐近解的一致有效性，从而得到了无界域上温度场的分布.通过奇摄动分析，给出了非Fourier 温度场与Fourier 温度场的关系，描述了非Fourier温度场的具体形态.     

Volteera型积分微分方程组边值问题的奇摄动

该文研究二阶积分微分方程组边值问题奇摄动，在适当的条件下，利用渐近分析方法和对角化技巧，还得解的存在性和给出解的渐近展开式与相应的余项估计．然后，应用这些结果到三阶常微分方程组边值问题的奇摄动，最后也得到解的一致有效的渐近展开式．     

一类非线性奇摄动问题的匹配解法

利用匹配渐近展开法,讨论了一类非线性奇摄动问题的解,得出了奇摄动边值问题的零次渐近展开式.     

高维非线性系统边值问题的奇摄动

利用渐近方法和对角化技巧研究了伴有边界摄动的高维非线性系统边值问题的奇摄动，在适当的假设下，证得摄动问题解的存在并导出其解关于ε的高阶近似．     

非线性超抛物型方程广义渐近解的研究

讨论了一类超抛物型方程的非线性奇摄动问题.首先引入了相应问题的比较定理,然后利用奇摄动方法构造了问题的形式渐近解,最后利用比较定理,证明了问题广义解的存在性及其渐近性态.     

一类含有小迟滞的奇摄动方程组的渐近解

本文研究了一类含有小迟滞的奇摄动方程组的渐近解.利用原问题的退化形式和伸长变量,依据边界层特有的性质,获得了边界层的渐近解.推广了奇摄动方程组初值问题和小迟滞问题的研究结果.     

二阶非线性奇摄动微分方程的渐近解

研究了二阶非线性奇摄动微分方程的边值问题.利用匹配原则和微分不等式原理,得到一阶非线性问题的渐近解,进而得到二阶奇摄动问题的解的渐近估计.     

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.     

积分方程的奇摄动问题

本文利用边界层函数法，构造并证明了奇摄动的非线性积分方程组的渐近解．     

一类高阶椭圆型方程奇摄动边值问题

本文讨论了一类２ｍ阶奇摄动椭圆型方程边值问题的渐近性态，得到了问题解的一致有效的渐近展开式。     

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.     

A subdivision collocation method for solving two point boundary value problems of order three

In this paper, we propose a method for the numerical solution of self adjoint singularly perturbed third order boundary value problems in which the highest order derivative is multiplied by a small parameter $\varepsilon$. In this method, first we introduce the derivatives of two scale relations satisfied by the subdivision schemes. After that we use these derivatives to construct the subdivision collocation method for the numerical solution of singularly perturbed boundary value problems. Convergence of the subdivision collocation method is also discussed. Numerical examples are presented to illustrate the proposed method.     

奇摄动燃烧反应扩散问题

本文讨论了一类奇摄动燃烧反应扩用Ｒｏｂｉｎ问题,利用微分不等式理论,证明了问题解的存在性等式并得到了解的斩近估计。     

Wavelet optimized finite difference method using interpolating wavelets for self-adjoint singularly perturbed problems

We design a wavelet optimized finite difference (WOFD) scheme for solving self-adjoint singularly perturbed boundary value problems. The method is based on an interpolating wavelet transform using polynomial interpolation on dyadic grids. Small dissipation of the solution is captured significantly using an adaptive grid. The adaptive feature is performed automatically by thresholding the wavelet coefficients. Numerical examples have been solved and compared with non-standard finite difference schemes in [J.M.S. Lubuma, K.C. Patidar, Uniformly convergent non-standard finite difference methods for self-adjoint singular perturbation problems, J. Comput. Appl. Math. 191 (2006) 228–238]. The proposed method outperforms the non-standard finite difference for studying singular perturbation problems for small dissipations (very small ) and effective grid generation. Therefore, the proposed method is better for studying the more challenging cases of singularly perturbed problems.     

On some difference schemes for singular singularly-perturbed boundary value problems

Summary Singularly perturbed boundary value ordinary differential problems are considered, where the problem defining the reduced solution is singular. For numerical approximation, families of symmetric difference schemes, which are equivalent to certain collocation schemes based on Gauss and Lobatto points, are used. Convergence results, previously obtained for the regular singularly perturbed case, are extended. While Gauss schemes are extended with no change, Lobatto schemes require a small modification in the mesh selection procedure. With meshes as prescribed in the text, highly accurate solutions can be obtained with these schemes for singular singularly perturbed problems at a very reasonable cost. This is demonstrated by examples.This research was completed while the author was visiting the Department of Applied Mathematics, Weizmann Inst., Rehovot, Israel. The author was supported in part under NSERC grant A4306     

在部分区域上的奇摄动反应扩散方程初始边值问题解的渐近性态

讨论一类在部分区域上的奇摄动反应扩散方程初始边值问题,利用算子理论,得到了相应问题解的渐近性态。     

Admissibility of simplified equations in the dynamics of gyroscopic systems

Based on the asymptotic approach of /1/, rigorous mathematical methods are used to single out some known simplified models from the theory of gyroscopic systems and to prove that they may legitimately be employed to solve problems in dynamics (including stability problems). The initial system is of the singularly perturbed type /2/. The use of methods from stability theory /3, 4/ yields conditions under which transition to a simplified (computational) model is permissible. Several papers have been devoted to the solution of such problems for singularly perturbed equations /5/ by methods of Lyapunov theory.     

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 nonlocal singularly perturbed problems for nonlinear hyperbolic differential equation

The author discussed a class of singularly perturbed problems for differential equation fiee {1--7]). Now we consider the non1ocal singu1arly perturbed problem as follows:where E is a positive small parameter anHere x = (xl, x2,' ) x.) E n, fl denotes a bounded region in R", 0fl signilies a boundary offl for class Cl cr (cr 6 (0, 1) is H5lder exponent), T0 is a positive constant, L1 is a uniformlyelliptic operator, L2 is a first order differential operator, T is an integral operator, K(x…     

一类反应扩散方程奇摄动Robin问题的广义解

讨论了一类奇摄动反应扩散方程R ob in问题.在适当的条件下,研究了问题广义解的渐近性态.