不定积分的代数解法

基于积分运算是微分运算的逆运算,利用线性代数方法,提出了一种求解几类特殊函数不定积分的新方法,并结合实例验证了方法的有效性.     

解大系统稳定性的积分方程法

本文通过分解积分方程组，并建立积分方程法比较原理，化大系统为低维系统，进而讨论了带时滞的时变大系统的稳定性，给出了新的结果，这一方法也可以用来讨论其它类型大系统的稳定性问题．     

A Domain Integral Equation for the Bergman Kernel

Certain integral operators involving the Szegö, the Bergman and the Cauchy kernels are known to have the reproducing property. Both the Szegö and the Bergman kernels have series representations in terms of an orthonormal basis. In this paper we derive the Cauchy kernel by means of biorthogonality. The ideas involved are then applied to construct a non-Hermitian kernel admitting a reproducing property for a space associated with the Bergman kernel. The construction leads to a domain integral equation for the Bergman kernel.^{1 2}     

用积分方程法解板的振动问题*

本文把带有集中质量、弹性支承和弹簧支撑着的质量块(振子)的薄板的振动微分方程化成为积分方程的特征值问题。然后利用广义函数理论和积分方程理论,得到了用一无穷阶矩阵的标准特征值形式给出的频率方程,从而方便地得到了固有频率和振型。并讨论了这种方法的收敛性。     

Application of Two-Parameter Stabilizing Functions in Solving a Convolution-Type Integral Equation by Regularization Method

This paper examines the solution of convolution-type integral equations of the first kind by applying the Tikhonov regularization method with two-parameter stabilizing functions. The class of stabilizing functions is expanded in order to improve the accuracy of the resulting solution. The features of the problem formulation for identification and adaptive signal correction are described. A method for choosing regularization parameters in problems of identification and adaptive signal correction is suggested.     

Boundary Integral Operator for the Fractional Laplace Equation in a Bounded Lipschitz Domain

We study the boundary integral operator induced from fractional Laplace equation in a bounded Lipschitz domain. As an application, we study the boundary value problem of a fractional Laplace equation.     

二维手性介质层电磁散射问题的积分方程方法

该文考虑了时谐的电磁平面波入射到外面包裹一层手性介质的良导体障碍物的散射问题.建立了一个二维散射模型并用积分方程方法讨论了解的存在性和唯一性.     

周期手性结构电磁散射问题的积分方程方法

In this paper, we consider the electromagnetic scattering by a periodic chiral structure. The media is homogeneous and the structure is periodic in one direction and invariant in another direction. The electromagnetic fields inside the chiral medium are governed by Maxwell equations together with the Drude-BornFedorov equations. We simplify the problem to a two-dimensional scattering problem and discuss the existence and the uniqueness of solutions by an integral equation approach. We show that for all but possibly a discrete set of wave numbers, the integral equation has a unique solution.     

新的辅助方程法构造KdV方程的行波解

应用一种新的辅助方程法成功地获得了(1+1)维KdV方程的多个含有参数的精确行波解,所得的解涵盖了已有结果．与其它方法相比,所给出的方法具有简单高效、计算量小、速度快、易于求解等特点．另外,所给的方法还可以用来求解其它的一大类非线性发展方程的精确行波解．     

Partial Decomposition of a Domain Containing Thin Tubes for Solving the Heat Equation

An initial–boundary value problem for the heat equation in a three-dimensional domain containing thin cylindrical tubes is considered. The Neumann condition is set on the lateral boundaries of the tubes. The original three-dimensional problem is reduced to a hybrid-dimensional one in which the heat equation in the tubes is replaced by the one-dimensional heat equation in shorter cylinders (subtubes), and the three- and one-dimensional equations are matched on the bases of the subtubes. The difference between the solutions of the original and hybrid-dimensional problems is estimated using two geometric characteristics: the distance between the bases of the tubes and subtubes and the reciprocals of the minimal positive eigenvalues of the Neumann problem for the Laplace operator in the tube cross sections.     

The Qualocation Method for Symm's Integral Equation on a Polygon

This paper discusses the convergence of the qualocation method for Symm's integral equation on closed polygonal boundaries in IR². Qualocation is a Petrov-Galerkin method in which the outer integrals are performed numerically by special quadrature rules. Before discretisation a nonlinear parametrisation of the polygon is introduced which varies more slowly than arc-length near each corner and leads to a transformed integral equation with a regular solution. We prove that the qualocation method using smoothest splines of any order k on a uniform mesh (with respect to the new parameter) converges with optimal order O (h^k). Furthermore, the method is shown to produce superconvergent approximations to linear functionals, retaining the same high convergence rates as in the case of a smooth curve.     

Learning the Algebraic Method of Solving Problems

This study of students' attempts to formulate and solve algebra word problems shows that the logic underlying algebraic problem solving methods is little understood. Students' prior experiences with solving problems in arithmetic gives them a compulsion to calculate which is manifested in the meaning they give to “the unknown” and how they use letters, their interpretation of what an equation is, and the methods they choose to solve equations. At every stage of the process of solving problems by algebra, students were deflected from the algebraic path by reverting to thinking grounded in arithmetic problem solving methods.     

An Iterative Variant of the Degenerate Kernel Method for Solving Fredholm Integral Equations

An iterative variant of the classical degenerate kernel method for solving Fredholm integral equations of the second kind is presented and its convergence properties are studied.     

Orr-Sommerfeld方程的Green函数和积分方程解法

本文提出了一种求解线性稳定性理论Orr-Sommerfeld方程的方法.我们首先定义了该方程的Green函数,并将它表达成矩阵形式;然后证明了Green函数的互易性;最后导出了等价于原方程的线性积分方程.该方法适用于两固壁间任意Reynolds数下各种主流速度分布的情况.     

An Approximate Method for Solving an Inverse Coefficient Problem for the Heat Equation

Journal of Applied and Industrial Mathematics - We construct a numerical method for recovering a variable coefficient in the Cauchy problem and also in the initial boundary value problem for the...     

An Integral Geometry Problem in a Nonconvex Domain

We consider the problem of recovering the solenoidal part of a symmetric tensor field f on a compact Riemannian manifold (M,g) with boundary from the integrals of f over all geodesics joining boundary points. All previous results on the problem are obtained under the assumption that the boundary M is convex. This assumption is related to the fact that the family of maximal geodesics has the structure of a smooth manifold if M is convex and there is no geodesic of infinite length in M. This implies that the ray transform of a smooth field is a smooth function and so we may use analytic techniques. Instead of convexity of M we assume that M is a smooth domain in a larger Riemannian manifold with convex boundary and the problem under consideration admits a stability estimate. We then prove uniqueness of a solution to the problem for     

关于矩阵方程求解的另一点注记

对任意给定的矩阵A∈Pm×n,B∈Pm×s(s≤n),探讨了矩阵方程AX=B有列满秩解,同时BY=A有行满秩解的充分必要条件,并且给出了基于矩阵的等价、齐次方程组的同解、向量组的等价及线性空间语言的推广.     

Domain Approximation Method for Solving Multifacility Location Problems on a Sphere

This paper considers the problem of optimally locating new facilities on a sphere among existing facilities so that the sum of all weighted geodesic distance pairs is minimized. A method involving an iterative solution is presented. The procedure involves the approximation of the domain of objective function, which in the limit approaches to that of the original objective function. Computational experience with the procedure is described.