矩阵奇异值分解问题重分析的摄动法

本文提出了一般实矩阵奇异值分解问题重分析的摄动法.这是一种简捷、高效的快速重分析方法,对于提高各种需要反复进行矩阵奇异值分解的迭代分析问题的计算效率具有较重要的实用价值.文中导出了奇异值和左、右奇异向量的直到二阶摄动量的渐近估计算式.文末指出了将这种振动分析方法直接推广到一般复矩阵情况的途径.     

关于n阶微分方程边值问题（Ⅲ）——奇摄动的渐近展开

本文研究n阶非线性过值问题(NB)的奇异摄动。在较一般的条件下,应用高阶微分不等式理论证明了摄动解的存在性,并给出了摄动解直到n阶导函数的一致有效渐近展开式,推广和改进了已有的结果。     

一类奇摄动系统奇异极限环的不变环面分支

对一类奇异摄动系统中由奇异极限环产生的不变环面分支进行了研究并利用不变环面的分支理论,讨论了由快系统的二重极限环和三重环分支出的不变环面的存在性.     

指数时程差分与有理谱配点法求解奇异摄动Burgers-Huxley问题

带小参数ε的Burgers-Huxley方程是一类非线性、非定常奇异摄动初边值问题,本文用指数时程差分与有理谱配点法求其数值解.对空间方向的边界层,用带sinh变换的有理谱配点法便Chebyshev节点在边界层处加密,只需取较少节点即可达到较高精度;时间方向采用指数时程差分与4阶Runge-Kutta法相结合的格式,并用围线积分计算矩阵甬数的方法克服了求解奇异摄动问题时遇到的的数值不稳定堆题.数值实验表明,本文提出的方法在求解左、右边界层和内部层的奇异摄动Burgers-Huxley问题都有较高的精度.     

关于奇异摄动问题数值解法的一个猜测的证明

在这篇文章中,我们就二阶常微分方程奇异摄动边值问题证明了Doolan,Miller和Schilders提出的一个猜测,即差分格式的一致收敛阶数不超过退化差分格式对退化问题的收敛阶数。对一般的线性奇异摄动问题,本文的证明方法都适用。     

一类脉冲微分方程的光滑多尺度解

利用奇异摄动方法,提供了一种以光滑的多尺度解的观点来研究一类脉冲微分方程.通过引入适当的奇异摄动项,定义了相应的奇异摄动边值问题,其对应的退化方程即为原脉冲微分方程.利用边界层函数法和缝接法,构造了该奇异摄动边值问题的光滑多尺度解,并有效地刻画原脉冲微分方程的不连续解,同时也证明了多尺度解的存在性及余项估计.最后,通过...     

集中载荷作用下开顶扁球壳的非线性稳定问题的渐近解

本文利用奇异摄动法研究了具有刚性中心的边缘固定的开顶扁球壳在中心集中载荷作用下的非线性稳定问题,得到了几何参数κ值较大时的一致有效的渐近解.     

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...     

一类奇摄动系统奇异极限环的不变环面分支

对一类奇异摄动系统中由奇异极限环产生的不变环面分支进行了研究并利用不变环面的分支理论,讨论了由快系统的二重极限环和三重环分支出的不变环面的存在性．     

轴压加筋圆柱壳Koiter-边界层奇异摄动法

将Koiter理论和奇异摄动理论中的边界层法相结合,处理加筋圆柱壳无因次化非线性边界层型Karman－Donnell方程由分支点和边界层导致的双重奇异性,提出轴压加筋圆柱壳Koiter-边界层奇异摄动法.对AS－2壳分析表明,本方法具有很好的计算效率和计算精度,与数值解相比更能揭示其内在的影响规律.     

Estimation of optimal backward perturbation bounds for the linear least squares problem

In this paper a method of estimating the optimal backward perturbation bound for the linear least squares problem is presented. In contrast with the optimal bound, which requires a singular value decomposition, this method is better suited for practical use on large problems since it requiresO(mn) operations. The method presented involves the computation of a strict lower bound for the spectral norm and a strict upper bound for the Frobenius norm which gives a gap in which the optimal bounds for the spectral and the Frobenius norm must be. Numerical tests are performed showing that this method produces an efficient estimate of the optimal backward perturbation bound.     

On the σ - reciprocal system for model order reduction

In this paper a definition of the oσ-reciprocal system is introduced. This allows us to show that the generalized singular perturbation approximation method for model reduction is related to the direct truncation of the σ -reciprocal system. On the basis of this definition, two new model order reduction algorithms are presented. The suitability of the proposed reduction methods is illustrated by means of two numerical examples.     

B-Spline Collocation Method for Nonlinear Singularly-Perturbed Two-Point Boundary-Value Problems

A B-spline collocation method is presented for nonlinear singularly-perturbed boundary-value problems with mixed boundary conditions. The quasilinearization technique is used to linearize the original nonlinear singular perturbation problem into a sequence of linear singular perturbation problems. The B-spline collocation method on piecewise uniform mesh is derived for the linear case and is used to solve each linear singular perturbation problem obtained through quasilinearization. The fitted mesh technique is employed to generate a piecewise uniform mesh, condensed in the neighborhood of the boundary layers. The convergence analysis is given and the method is shown to have second-order uniform convergence. The stability of the B-spline collocation system is discussed. Numerical experiments are conducted to demonstrate the efficiency of the method.     

Method for solving nonlinear initial value problems by combining homotopy perturbation and reproducing kernel Hilbert space methods

The nonlinear singular initial value problems including generalized Lane–Emden-type equations are investigated by combining homotopy perturbation method (HPM) and reproducing kernel Hilbert space method (RKHSM). He’s HPM is based on the use of traditional perturbation method and homotopy technique and can reduce a nonlinear problem to some linear problems and generate a rapid convergent series solution in most cases. RKHSM is also an analytical technique, which can overcome the difficulty at the singular point of non-homogeneous, linear singular initial value problems; especially when the singularity appears on the right-hand side of this type of equations, so it can solve powerfully linear singular initial value problems. Therefore, using advantages of these two methods, more general nonlinear singular initial value problems can be solved powerfully. Some numerical examples are presented to illustrate the strength of the method.     

Coercive singular perturbations II: Reduction and convergence

An algebra of pseudodifferential singular perturbations is introduced. It provides a constructive machinery in order to reduce an elliptic singularly perturbed operator (in $\text{R}$ⁿ or on a smooth manifold without border) to a regular perturbation. The technique developed is applied to some singularly perturbed boundary value problems as well. Special attention is given to a singular perturbation appearing in the linear theory of thin elastic plates. A Wiener-Hopf-type operator containing the small parameter reduces this singular perturbation to a regular one. It also gives rise to a natural recurrence process for the construction of high-order asymptotic formulae for the solution of the perturbed problem. The method presented can be extended to the general coercive singular perturbations.     

奇摄动线性代数方程组及其对病态方程的应用

本文首先从一个曲柄导杆机构的优化问题提出了含小参数线性代数方程组的奇摄动问题。然后利用摄动方法证明了这个问题解的存在唯一性,同时给出了解的渐近展开和误差估计。最后讨论了所得结果对求解病态方程的应用。     

Stability theorems for singular perturbation of eigenvalues

Two stability, i.e., convergence in multiplicity, theorems for eigenvalues of non-self-adjoint singular perturbation problems are proved in this paper. Use is made of quadratic interpolation between Hubert spaces to obtain an extension of the notion of a holomorphic family of type (B). Applications to singular perturbation problems for non-self-adjoint elliptic partial differential operators, and to singular potential perturbation problems with complex coupling constants, are presented.This research was partially supported by NSF Grant 02 MCS-7902663     

A numerical method for singularly perturbed weakly coupled system of two second order ordinary differential equations with discontinuous source term

In this paper, a numerical method based on finite difference scheme and Shishkin mesh for singularly perturbed two second order weakly coupled system of ordinary differential equations with discontinuous source term is presented. An error estimate is derived to show that the method is uniformly convergent with respect to the singular perturbation parameter. Numerical results are presented to illustrate the theoretical results.     

A class of singularly perturbed semilinear differential equations with interior layers

In this paper singularly perturbed semilinear differential equations with a discontinuous source term are examined. A numerical method is constructed for these problems which involves an appropriate piecewise-uniform mesh. The method is shown to be uniformly convergent with respect to the singular perturbation parameter. Numerical results are presented that validate the theoretical results.