闭光滑流形上的奇异积分方程

<正> §1.前言 自Giraud G.以来,已有不少关于闭光滑流形上的奇异积分和奇异积分方程的研究(见[10]),但利用多复变函数的Cauchy型积分作为工具者,至今不多(如[3—8]),而在一维奇异积分方程论中,复变函数的Cauchy型积分起着基本的作用.本文试以定理在多复变函数论中的拓广为基础,讨论闭光滑流形上奇异积分的合成和奇异积分方程的求解,其方法和结论,都是与Giraud G.等人的工作全然不同的.     

Plemelj公式及其应用

本文介绍了Ｃ￣ｎ空间中函数经Ｂｏｃｈｎｅｒ－Ｍａｒｔｉｎｅｌｌｉ变换后的Ｐｌｅｍｅｌｊ公式和它在Ｓｔｅｉｎ流形上的拓广，同时还介绍了Ｃ￣ｎ空间和Ｓｔｅｉｎ流形上微分形式在Ｂｏｃｈｎｅｒ－Ｍａｒｔｉｎｅｌｌｉ变换下的跳跃公式以及这些公式分别在全纯开拓，闭开拓，方程和线性奇异积分方程上的应用．     

闭光滑流形上的高阶线性微-积分方程

利用积分变换技巧,作者给出了C~n中闭光滑可定向流形上一个新的Bochner-Martinelli型积分的高阶偏导数的奇异积分的Hadamard主值,获得了高阶奇异积分的Plemelj公式和合成公式,还讨论了相应的变系数线性微分积分方程的正则化,证明其可转化为一类等价的Fredholm方程。并且指出其特征方程当给出一组适当的边值条件时,在L~*中存在唯一解。     

关于多复变数的积分变换公式及其应用

利用整体分析方法,给出了一个多复变数的整体积分变换公式,获得了C^n中一闭逐块光滑可定向流形上的Bochner-Martinelli型积分高阶偏导具有Hadamard主值的Plemelj公式和相应奇异积分的合成公式,拓广的Poincaré-Bertrand公式.作为应用,我们还讨论了一类高阶Cauchy边值问题和一类多复变数线性高阶奇异微积分方程的正则化问题.     

闭逐块光滑流形上奇异积分的Poincaré-Bertrand公式

本文把复变函数论中著名的Poincare-Bertrand公式拓广到闭逐块C（1）光滑流形上Bochner-Martinelli型奇异积分的含有点ζ的立体角系数α（ζ）的更一般的置换公式.     

关于外微分式的奇异积分方程

钟同德、孙继广讨论了光滑流形上含 B—M 核的奇异积分方程,本文是把这些结果推广到密度函数为(n,n-1)型外微分式的情形。     

数学年刊第28卷B辑第2期(2007)

目次和提要一类奇异摄动系统的不变环面和次调和解分支叶志勇林茂安对非自治奇异摄动系统中由快系统的极限环产生的次调和解和不变环面的分支进行了讨论.利用Pofncar6映射，blow一up变换以及积分流形理论，得到了不变环面存在的条件，研究了次调和解的鞍结点分支.线性权合映射格点同步分析的一种断途径卢文联陈天平对一种新的分析线性祸合映射格点同步问题方法，引入了系统轨道在同步流形上的(非正交)投影.利用这个投影，同步间题归结成轨道与相应的投影间的距离趋于零.由此，给出了局部或全局同步的判据.在讨论同步流形的稳定性中，对应于…     

Stein流形上一个带有算子映射函数的积分公式

本文得到Ｓｔｅｉｎ流形上一个带有算子映射函数Ｓ（ｚ，ζ）和实参数的积分公式，由这个公式不但可以推出Ｓｔｅｉｎ流形上全纯函数和光滑函数的一些已有积分公式和它们相应的拓广式，而且还可以得到一些新的积分公式。     

实轴上的特征奇异积分方程

该文首先得到了实轴上的特征奇异积分方程的可解性理论.由此,得到了实轴上解具一阶奇异性的特征奇异积分方程的可解性理论,纠正了文献[8]中的错误.     

线性流形上矩阵方程A^TXA=B的中心对称解

利用矩阵对的商奇异值分解，给出了线性流形上矩阵方程A^TXA=B存在中心对称解的充要条件及其通解的表达式.另外，导出了线性流形上矩阵方程的解集合中与给定矩阵的最佳逼近解的表达式.     

The solvability of some kinds of singular integral equations of convolution type with variable integral limits

In this paper, we discuss several classes of convolution type singular integral equations with variable integral limits in class $ H^*_1 $. By means of the theory of complex analysis, Fourier analysis and integral transforms, we can transform singular integral equations with variable integral limits into the Riemann boundary value problems with discontinuous coefficients. Under the solvability conditions, the existence and uniqueness of the general solutions can be obtained. Further, we analyze the asymptotic properties of the solutions at the nodes. Our work improves the Noether theory of singular integral equations and boundary value problems, and develops the knowledge architecture of complex analysis.     

Some new explicit bounds for weakly singular integral inequalities with applications to fractional differential and integral equations

Some new weakly singular integral inequalities of Gronwall-Bellman type are established, which generalized some known weakly singular inequalities and can be used in the analysis of various problems in the theory of certain classes of differential equations, integral equations and evolution equations. Some applications to fractional differential and integral equations are also indicated.     

Optimal and Approximate Solutions of Singular Integral Equations by Means of Reproducing Kernels

A reproducing kernel method is proposed to obtain the optimal and approximate solutions of Carleman singular integral equations. Therefore, we will be mostly interested in singular integral equations with a Cauchy type kernel and whose coefficients are real or complex valued functions. The new method and corresponding concepts allow the analysis of associated discrete singular integral equations and corresponding inverse source problems in appropriate frameworks.     

Regularization for Higher Order Singular Integral Equations

By means of the definition of Hadamard principal value permutation formula and composition formula for higher order singular integrals developing in [5], the regularization problems for higher order singular integral equations with variable coefficients are discussed, and the higher order singular integral equations with constant coefficient are solved. It is the first time to treat the high order singular integral equations of arbitrary degree in the theory of singular integral equations.      

Existence and uniqueness for non-linear singular integral equations used in fluid mechanics

Non-linear singular integral equations are investigated in connection with some basic applications in two-dimensional fluid mechanics. A general existence and uniqueness analysis is proposed for non-linear singular integral equations defined on a Banach space. Therefore, the non-linear equations are defined over a finite set of contours and the existence of solutions is investigated for two different kinds of equations, the first and the second kind. Moreover, the existence of solutions is further studied for non-linear singular integral equations over a finite number of arbitrarily ordered arcs. An application to fluid mechanics theory is finally given for the determination of the form of the profiles of a turbomachine in two-dimensional flow of an incompressible fluid.     

The piecewise polynomial collocation method for nonlinear weakly singular Volterra equations

Second-kind Volterra integral equations with weakly singular kernels typically have solutions which are nonsmooth near the initial point of the interval of integration. Using an adaptation of the analysis originally developed for nonlinear weakly singular Fredholm integral equations, we present a complete discussion of the optimal (global and local) order of convergence of piecewise polynomial collocation methods on graded grids for nonlinear Volterra integral equations with algebraic or logarithmic singularities in their kernels.

     

Volume and surface integral equations for electromagnetic scattering by a dielectric body

We derive and analyze two equivalent integral formulations for the time-harmonic electromagnetic scattering by a dielectric object. One is a volume integral equation (VIE) with a strongly singular kernel and the other one is a coupled surface-volume system of integral equations with weakly singular kernels. The analysis of the coupled system is based on standard Fredholm integral equations, and it is used to derive properties of the volume integral equation.     

热载荷作用下平面裂纹问题的奇异积分方程与边界无法

从边界积分方程出发,导出了二维裂纹体热传导问题及热弹性问题的积分方程组,继而使用奇异积分方程与边界元相结合的方法,为其建立了相应的数值求解方法。此外,利用奇异积分方程的主部分析法,严格地证明了裂纹尖端温度梯度场的1/√r 奇异性,并且给出了奇性温度梯度场的精确解。最后。对一些典型例子,做了数值计算。     

3D Fredholm integral equations for scattering by dielectric structures

We consider 3D singular integral equations that describe problems of interaction of an electromagnetic wave with 3D dielectric structures. By using the theory of singular integral equations, we reduce these equations to Fredholm integral equations of the second kind.     

A schauder decomposition and weakly singular integral equations in Lp

In this paper, a Schauder decomposition in L_p is used to obtain numerical solutions for the Fredholm integral equations of the second kind. Considerations are also given to weakly singular integral equations and two dimensional weakly singular integral equations.