声波方程系数逆演问题的非线性积分方程及时卷特征迭代法(TCC)

本文将一维、二维和三维声波方程系数逆演问题转化为非线性积分方程。在一维情形,非线性积分方程是第二类的,在二维和三维情形,非线性积分方程是有趣的积分几何问题,并考虑求解上述积分方程的迭代法。本文所讨论的非线性积分方程及相应的迭代解法,对于声波方程系数逆演问题的理论及数值方法研究及在科学与工程中的应用有重要作用。     

声学外问题的边界积分方程方法

本文考虑了二维Ｈｅｌｍｈｏｌｔｚ方程的Ｄｉｒｉｃｈｌｅｔ问题．首先利用单层位势将问题归结为第一类边界积分方程，然后利用有限元方法求解该边界积分方程的近似解，给出近似解的收敛性、一致收敛性以及最优收敛速度定理，并进行解的稳定性分析．     

2.5维介质Born近似波速反演唯一性

考虑脉冲源引起的２．５维弱不均匀介质波速反演问题，利用线性化方法得到了波速的二维小扰动满足的积分方程，这是一个积分几何的问题，进而由Ｆｏｕｒｉｅｒ变换和脉冲函数的性质将此二维积分方程化为单变量的积分方程，最后用压缩映象理论证明了积分方程解的唯一性。本文给出了二给波速反演的一种新算法。同时，唯一性结果证明了已有的迭代算法的合理性。     

某些带变换的边值问题和奇异积分方程

本文主要考虑某些带变换的奇异积分方程。尽管用经典的正则化方法可以求解，我们还是给出一种新的分区跳跃方法，使它们化为较简单的可以系统地求解并可迅即讨论的Ｆｒｅｄｈｏｌｍ积分方程。     

含多重积分的Volterra型积分方程与不等式

本文考虑一定有序局部紧空间上含多重积分的Ｖｏｌｔｅｒｒａ型积分方程，给出了此类方程可解的充分条件，建立了相应的积分不等式．本文结果包含许多已知结论作为特殊情况．     

积分双半群与抽象动力方程的可解性

本文从一类抽象积分微分方程的可解性，引人积分双半群的概念，并讨论了积分双半群的一些性质．利用积分双半群，说明了一类抽象动力方程的可解性，特别是抽象动力方程边值问题的可解性．     

偏微分方程边值反问题的数值方法研究

本文研究了奇异积分方程在反边值问题中的应用问题.利用圆周上的自然积分方程及其反演公式,把Laplace方程的边值反问题转化为一对超奇异积分方程和弱奇异积分方程的组合,通过选取三角插值近似奇异积分的计算并构造相应的配置格式,并使用Tikhonov正则化方法求解所得到的线性方程组.数值实验表明了该方法的有效性.     

一类积分方程的稳定性与吸收域

本文通过建立非线性积分不等式，了一类具有时滞积分方程的渐近稳定性，获得了简捷而实用的充分准则，并给出了确定吸引域的方法。     

滞时Volterra积分方程数值方法的数值稳定性分析

本文给出数值方法解Volterra积分方程的稳定性分析,我们判定可约积分方法的数值稳定性基于如下试验方程其中τ是正常数,p和q是复值的。在上述试验方程的情况下,我们研究θ-方法及可约积分方法的稳定性。     

复合结构与流体耦合运动方程的时域分析方法

本文就复合结构与流体耦合的时域运动方程，利用核函数矩阵的特点，将二阶微分积分方程变形为Volterra型积分方程，然后引入积分变换，得到一组一阶常微分方程组，该微分方程组的形式与现代控制论中的状态方程类似。     

G可积函数的Lebesgue可测性

Botsko在连续和可导的知识基础上推广了Riemann积分,得到了一种新的积分,称为G积分．G积分既不同于Riemann积分也不同于Lebesgue积分．本文通过对G积分的研究,得到了G可积函数一定Lebesgue可测,从而有界G可积函数一定Lebesgue可积;同时我们还证明了这两个积分值相等．     

Integral representation of the four-parametric generalized Mittag-Leffler function

In this paper, we give a new integral representation of the four-parametric generalized Mittag-Leffler function introduced and studied by Djrbashian (Dzherbashian). The representation, obtained in this paper, contains an iterated integral, wherein the internal integral is a Cauchy-type integral, and the external one is a simple improper integral along the so-called Hankel path. The representation also contains the values of the special Wright functions.     

Integral equation methods for the Yukawa-Beltrami equation on the sphere

An integral equation method for solving the Yukawa-Beltrami equation on a multiply-connected sub-manifold of the unit sphere is presented. A fundamental solution for the Yukawa-Beltrami operator is constructed. This fundamental solution can be represented by conical functions. Using a suitable representation formula, a Fredholm equation of the second kind with a compact integral operator needs to be solved. The discretization of this integral equation leads to a linear system whose condition number is bounded independent of the size of the system. Several numerical examples exploring the properties of this integral equation are presented.     

On the Chebyshev type inequality for seminormed fuzzy integral

The Chebyshev type inequality for seminormed fuzzy integral is discussed. The main results of this paper generalize some previous results obtained by the authors. We also investigate the properties of semiconormed fuzzy integral, and a related inequality for this type of integral is obtained.     

The superconvergence of the composite midpoint rule for the finite-part integral

The composite midpoint rule is probably the simplest one among the Newton-Cotes rules for Riemann integral. However, this rule is divergent in general for Hadamard finite-part integral. In this paper, we turn this rule to a useful one and, apply it to evaluate Hadamard finite-part integral as well as to solve the relevant integral equation. The key point is based on the investigation of its pointwise superconvergence phenomenon, i.e., when the singular point coincides with some a priori known point, the convergence rate of the midpoint rule is higher than what is globally possible. We show that the superconvergence rate of the composite midpoint rule occurs at the midpoint of each subinterval and obtain the corresponding superconvergence error estimate. By applying the midpoint rule to approximate the finite-part integral and by choosing the superconvergence points as the collocation points, we obtain a collocation scheme for solving the finite-part integral equation. More interesting is that the inverse of the coefficient matrix of the resulting linear system has an explicit expression, by which an optimal error estimate is established. Some numerical examples are provided to validate the theoretical analysis.     

关于 C-Pettis 积分的一个控制收敛定理

In this paper, we give the Riemann-type extensions of Dunford integral and Pettis integral, C-Dunford integral and C-Pettis integral. We discuss the relationship between the C-Pettis integral and Pettis integral, and prove a controlled convergence theorem for the C-Pettis integral.     

关于亚音速绕流问题的强奇性基本解法

本文对求解亚音速流的偶极子基本解法作了新的处理,导出了一个关于求解偶极子强度的强奇性积分方程,给出了强奇性积分有效主值的定义及计算公式.由此可以导出多种整体连续分布的数值基本解法.适用于亚音速气动力计算.     

圆内平面弹性问题的边界积分公式

根据双解析函数可以得到单位圆内平面弹性问题应力函数的边界积分公式,但式中包含强奇异积分,不能用于直接计算．将边界上的应力函数展开为Fourier级数,再利用广义函数论中的几个公式进行卷积计算,可以得到不含强奇异积分核的边界积分公式,通过边界的应力函数值和法向导数的积分,直接得到圆内应力函数值,并给出几个算例,表明该结果用于求解单位圆内平面弹性问题十分方便．     

探讨Abel积分零点的一个定理

本文研究了具有中心环域的可积多项式系统,探讨此系统在用多项式进行微扰的情况下,其所对应的Abel积分的零点个数的计算方法,给出了一个较为实用的定理,并举出了若干个应用的例子．     

Numerical solution of Fredholm integral equations of the second kind by using integral mean value theorem II. High dimensional problems

In this work, we generalize the numerical method discussed in [Z. Avazzadeh, M. Heydari, G.B. Loghmani, Numerical solution of Fredholm integral equations of the second kind by using integral mean value theorem, Appl. math. modelling, 35 (2011) 2374–2383] for solving linear and nonlinear Fredholm integral and integro-differential equations of the second kind. The presented method can be used for solving integral equations in high dimensions. In this work, we describe the integral mean value method (IMVM) as the technical algorithm for solving high dimensional integral equations. The main idea in this method is applying the integral mean value theorem. However the mean value theorem is valid for multiple integrals, we apply one dimensional integral mean value theorem directly to fulfill required linearly independent equations. We solve some examples to investigate the applicability and simplicity of the method. The numerical results confirm that the method is efficient and simple.