Cn中双球相交域上的奇异积分方程

Cn中双球相交域上具有全纯核的奇异积分的Sokhotsky-Plemelj公式具有一种特殊的形式,它在边界上是分片连续的.利用这个Sokhotsky-Plemelj公式,在适当条件下得到了一个特殊的合成公式,并得到了常系数奇异积分方程和方程组的特征方程一个直接解,并把常系数奇异积分方程和方程组化为一类与之等价的Fred...     

一类对偶积分方程组正则化为第一类含对数核的Fredholm奇异积分方程组解法

引入辅助未知函数及辅助未知函数的积分关系式,表示原未知函数,将对偶积分方程组退耦.应用Sonine第一有限积分公式,实现化为Abel型积分方程组,应用Abel反演变换并化简,正则化为含对数核的第一类Fredholm奇异积分方程组.由此给出奇异积分方程组的一般性解,进而获得对偶积分方程组的解析解,同时严格地证明了,对偶积分方程组和由它化成的含对数核的奇异积分方程组的等价性,以及对偶积分方程组解的存在性和唯一性.     

复Finsler流形上的Koppelman-Leray公式

利用不变积分核(Berndtsson核)、复Finsler度量和联系于Chern-Finsler联络的非线性联络来研究复Finsler流形上的积分表示理论,得到了Koppelman公式和Koppelman-Leray公式,并给出了∂-方程的解.     

Clifford 分析中无界域上正则函数带Haseman位移的边值问题

该文在引入修正的Cauchy核的基础上,讨论了Clifford 分析中无界域上正则函数带 Haseman 位移的边值问题. 首先给出了无界域上Cauchy 型积分的Plemelj公式,再利用积分方程方法和压缩不动点定理证明了问题解的存在唯一性.     

超空间中次正则函数的Cauchy积分公式*

在本文中, 首先给出了超空间中次正则函数(sandwich方程 D_xfD_x=0的解)的一些性质, 然后证明了超空间中的Cauchy-Pompeiu公式, 最后得到了超空间中的Cauchy积分公式和Cauchy积分定理.     

一类有限部积分方程数值解的误差分析

本文研究了一类有限部积分方程数值解的误差.利用离散的极值原理和复合中点公式的超收敛性,获得了积分方程配置格式的误差分析理论,改进了有限部积分方程数值解的相关研究成果.并给出数值实验验证了理论分析的正确性.     

弱奇性Volterra积分不等式解的估计

Medved对弱奇性Gronwall型和Henry型积分不等式解的估计提出一种新方法，本文将他的方法稍加改进用来研究更广的Volterra型弱奇性线笥及非线性积分不等式解的估计，导出解的先验逐点界公式，并举例说明了结果的应用。     

非均匀调和函数的两类边值问题与积分表达式

本文给出非均匀指数函数的定义及性质,并且进一步引入了非均匀三角函数、非均匀双曲函数和非均匀对数函数.最后利用非均匀指数函数表达形式和非均匀解析函数的Cauchy积分理论,建立了非均匀泊松积分公式和非均匀施瓦茨积分公式,获得了非均匀调和函数在两类特殊边界上的狄利克雷问题和诺伊曼问题解的显示表达式.     

随机Volterra积分方程相容解的稳定性

本文考虑了随机Volterra积分方程相容解的稳定性.应用Lyapunov第二方法,并以推广的Ito公式为工具,给出了随机Volterra积分方程相容解的几乎确定指数稳定和矩指数稳定的充分性原则.     

用正交函数求超奇异积分的近似值及其误差估计

基于 Hadamard有限部分积分定义, 当密度函数是多项式、正弦函数和余弦函数时, 本文推导出了计算超奇异积分准确值的公式, 进而利用这些公式给出了密度函数为一般连续函数的超奇异积分近似值的计算方法. 本文还对近似值进行了误差分析, 据此可以在事先给定的误差下来计算超奇异积分的近似值. 最后将前面的理论应用到超奇异积分方程求近似解的问题. 数值算例表明该方法的可行性和有效性.     

Legendre小波求解超奇异积分

超奇异积分的数值算法一直是近些年来研究的重要课题. 基于超奇异积分的 Hadamard 有限部分积分定义, 本文给出了利用 Legendre 小波计算超奇异积分的方法. 当奇异点位于区间内时, 由于 Legendre 小波具有很好的正交性、显式表达式以及小波函数的可计算性, 将区间内的奇异点变换到区间端点处, 再利用区间端点处 Hadamard 有限部分积分的定义,进而可以计算 p+1(p∈N⁺) 阶超奇异积分. 文中最后给出的算例表明了该方法的可行性和有效性.     

HIGH ORDER NUMERICAL METHODS TO TWO DIMENSIONAL HEAVISIDE FUNCTION INTEGRALS

In this paper we design and analyze a class of high order numerical methods to two dimensional Heaviside function integrals. Inspired by our high order numerical methods to two dimensional delta function integrals [19], the methods comprise approximating the mesh cell restrictions of the Heaviside function integral. In each mesh cell the two dimensional Heaviside function integral can be rewritten as a one dimensional ordinary integral with the integrand being a one dimensional Heaviside function integral which is smooth on several subsets of the integral interval. Thus the two dimensional Heaviside function integral is approximated by applying standard one dimensional high order numerical quadratures and high order numerical methods to one dimensional Heaviside function integrals. We establish error estimates for the method which show that the method can achieve any desired accuracy by assigning the corresponding accuracy to the sub-algorithms. Numerical examples are presented showing that the second- to fourth-order methods implemented in this paper achieve or exceed the expected accuracy.     

Applications of integral equations in particle-size statistics

We discuss the application of integral equations techniques to two broad areas of particle statistics, namely, stereology and packing. Problems in stereology lead to the inversion of Abel-type integral equations; and we present a brief survey of existing methods, analytical and numerical, for doing this. Packing problems lead to Volterra equations which, in simple cases, can be solved exactly and, in other cases, need to be solved numerically. Methods for doing this are presented along with some numerical results.     

On the limits of generalization of the Kolmogorov integral

The generalization of the Kolmogorov integral to functions with values in a Banach space is considered. It is proved that the resulting integral turns out to be essentially more general than the Bochner integral and is exactly equivalent to an integral of McShane type, whose definition requires that the scaling function be measurable.Translated from Matematicheskie Zametki, vol. 77, no. 2, 2005, pp. 258–272.Original Russian Text Copyright © 2005 by A. P. Solodov.This revised version was published online in April 2005 with a corrected issue number.     

An accurate application of the integral method applied to the diffusion of oxygen in absorbing tissue

Accurate integral methods are applied to a one dimensional moving boundary problem describing the diffusion of oxygen in absorbing tissue. These methods have been well studied for classic Stefan problems but this situation is unusual because there is no condition which contains the velocity of the moving boundary explicitly. This paper begins by giving a short time solution and then discusses some of the previous integral methods found in the literature. The main drawbacks of these solutions are that they cannot be solved from $t=0$ and also cannot determine the end behaviour. This is due to the non-uniform initial profile which integral methods typically fail to capture. The use of a novel transformation removes this non-uniformity and, on applying optimal integral methods to the resulting system, leads to simple and yet very accurate approximate solutions that overcome the deficiencies of previous methods.     

On approximation and numerical solution of Fredholm integral equations of second kind using quasi-interpolation

In this article a method is presented, which can be used for the numerical treatment of integral equations. Considered is the Fredholm integral equation of second kind with continuous kernel, since this type of integral equation appears in many applications, for example when treating potential problems with integral equation methods.The method is based on the approximation of the integral operator by quasi-interpolating the density function using Gaussian kernels. We show that the approximation of the integral equation, gained with this method, for an appropriate choice of a certain parameter leads to the same numerical results as Nyström’s method with the trapezoidal rule. For this, a convergence analysis is carried out.     

Properties of estimators of baseline hazard functions in a semiparametric cure model

We consider a semiparametric cure model combining the Cox model with the logistic model. There are the two distinct methods for estimating the nonparametric baseline hazard function of the model; one is based on a pseudo partial likelihood and the other is to use an EM algorithm. In this paper, we discuss the consistency and the asymptotic normality of the estimators from the two methods. Then, we show that the estimator from the pseudo partial likelihood can be characterized by the (forward) Volterra integral equation, and the estimator from the EM algorithm by the Fredholm integral equation. These characterizations reveal differences in the properties between the estimators from the two methods. In addition, a simulation study is performed to numerically confirm the results in several finite samples.     

求解数值积分的两类新的Monte-Carlo方法

基于对积分区间的多区间划分及子区间上的随机取点,提出了两类新的计算定积分的Monte-Carlo方法——"类矩形"及"类梯形"方法,并给出了其误差阶.最后,通过数值实验比较且验证了方法的高效性.     

A new stabilizing technique for boundary integral methods for water waves

Boundary integral methods to compute interfacial flows are very sensitive to numerical instabilities. A previous stability analysis by Beale, Hou and Lowengrub reveals that a very delicate balance among terms with singular integrals and derivatives must be preserved at the discrete level in order to maintain numerical stability. Such balance can be preserved by applying suitable numerical filtering at certain places of the discretization. While this filtering technique is effective for two-dimensional (2-D) periodic fluid interfaces, it does not apply to nonperiodic fluid interfaces. Moreover, using the filtering technique alone does not seem to be sufficient to stabilize 3-D fluid interfaces.

Here we introduce a new stabilizing technique for boundary integral methods for water waves which applies to nonperiodic and 3-D interfaces. A stabilizing term is added to the boundary integral method which exactly cancels the destabilizing term produced by the point vortex method approximation to the leading order. This modified boundary integral method still has the same order of accuracy as the point vortex method. A detailed stability analysis is presented for the point vortex method for 2-D water waves. The effect of various stabilizing terms is illustrated through careful numerical experiments.

     

对数三角函数的定积分

定义了三种积分表示的两元函数.这些两元函数有伽马函数表示,可以展开为幂级数.在积分符号内展开被积函数,先积分,再求和,也得到级数展开.对比展开系数,就得到一些对数三角函数定积分的值.选取合适的围道,得到其他两类对数三角函数定积分的值.