求解三维第一类Fredholm积分方程的GMRES法

利用数值求积公式,将三维第一类Fredholm积分方程进行离散,通过引入正则化方法,将离散后的积分方程转化为一离散适定问题,通过广义极小残余算法得到了其数值解．数值模拟结果表明该方法的可行有效性．     

求解第一类Fredholm积分方程的一种新的正则化算法

应用一种新的正则化方法建立了一类新的求解第一类Fredholm积分方程的正则化算法, 并借助Matlab软件给出了数值算例.数值结果与理论分析基本一致,而且表明文中建立的正则化比通常的Tikhonov正则化更精确.     

带复数核的第一类Fredholm积分方程的正则化方法及其应用

本文推广了Ｔｉｋｈｏｎｏｖ正则化方法，导出了带复数核的第一类Ｆｒｅｄｈｏｌｍ积分方程的正则解应满足的正则积分微分方程，并讨论了正则解的收敛性·作为这一方法的应用，数值求解了与二维摇板造波问题相应的一类逆问题，并给出了选择最佳正则参数的一个实用的方法     

第一类弱奇异核Fredholm积分方程的数值求解

第一类弱奇异核Fredholm积分方程由于奇异及本质的不适定性,给求解带来很大难度．本文首先利用克雷斯变换将方程转化,并对转化后的方程进行高斯一勒让德离散,得到一离散不适定的线性方程组,结合正则化方法对该类问题进行数值求解．最后给出了数值模拟,验证了本文方法的可行性及有效性．     

Direct Implementation of Tikhonov Regularization for the First Kind Integral Equation

A common way to handle the Tikhonov regularization method for the first kind Fredholm integral equations, is first to discretize and then to work with the final linear system. This unavoidably inflicts discretization errors which may lead to disastrous results, especially when a quadrature rule is used. We propose to regularize directly the integral equation resulting in a continuous Tikhonov problem. The Tikhonov problem is reduced to a simple least squares problem by applying the Golub-Kahan bidiagonalization (GKB) directly to the integral operator. The regularization parameter and the iteration index are determined by the discrepancy principle approach. Moreover, we study the discrete version of the proposed method resulted from numerical evaluating the needed integrals. Focusing on the nodal values of the solution results in a weighted version of GKB-Tikhonov method for linear systems arisen from the Nyström discretization. Finally, we use numerical experiments on a few test problems to illustrate the performance of our algorithms.     

Fredholm Integro——differential型方程的Legendre小波方法

研究Legendre小波方法求解具有一阶导和二阶导类型的线性Fredholm integro-differential型方程。应用Legendre小波逼近法把这两类方程分别化为代数方程求解．实例说明。Legendre小波在解决这两类方程时的可行性和有效性．     

第二类Fredholm积分方程组的分段泰勒级数展开法

积分方程出现在数学物理的各种问题中,寻求其简单而又有效的解法显得很有必要.提出一种求解第二类线性Fredholm积分方程组的新解法,利用分段泰勒级数展开,通过引入两个参数得到近似解的表达式,并对近似解的收敛性和误差进行分析.通过与已有数值方法的比较,说明此方法的可行性和有效性。     

Complete Hyper-elliptic Integrals of the First Kind and the Chebyshev Property

This paper is devoted to study the following complete hyper-elliptic integral of the first kind $$J(h)=\oint\limits_{\Gamma_h}\frac{\alpha_0+\alpha_1x+\alpha_2x^2+\alpha_3x^3}{y}dx,$$ where $\alpha_i\in\mathbb{R}$, $\Gamma_h$ is an oval contained in the level set $\{H(x,y)=h, h\in(-\frac{5}{36},0)\}$ and $H(x,y)=\frac{1}{2}y^2-\frac{1}{4}x^4+\frac{1}{9}x^9$. We show that the 3-dimensional real vector spaces of these integrals are Chebyshev for $\alpha_0=0$ and Chebyshev with accuracy one for $\alpha_i=0\ (i=1,2,3)$.     

Harmonic wavelet method towards solution of the Fredholm type integral equations of the second kind

Periodic harmonic wavelets (PHW) were applied as basis functions in solution of the Fredholm integral equations of the second kind. Two equations were solved in order to find out advantages and disadvantages of such choice of the basis functions. It is proved that PHW satisfy the properties of the multiresolution analysis.     

利用随机模拟方法求解第二类积分方程

给出一种求解第二类Fredholm和Volterra积分方程的数值算法,算法在数值积分技术的基础上使用Monte Carlo随机模拟方法求积分方程的近似解.通过数值例子证明了该算法是有效的.     

First Kind Cordial Volterra Integral Equations 2

We reduce a cordial Volterra integral equation of the first kind to a cordial integral equation of the second kind and establish a unique solvability of those in the scales of Banach spaces of C ^m -smooth functions on [0, T] and (0, T]. Stability estimates w.r.t. perturbation of data are derived. Polynomial and spline collocation methods are discussed for the solving the equation.     

基于Legrndre小波的第一类Fredholm积分方程的数值解法研究

提出利用Legendre小波函数去获得第一类Fredholm积分方程的数值解,函数定义在区间[0,1)上,然后结合Garlerkin方法将原问题转化为线性代数方程组.而且还对算法的收敛性和误差进行了分析,最后通过两个数值算例验证了所提算法的可行性及有效性.     

L~1空间中第二类Fredholm积分方程数值解法探究

在L¹空间中对第二类Fredholm积分方程提出了一种求其数值解的算法,证明了算法的收敛性,并给出相应的误差估计.数值算例进一步验证了算法的合理性.     

含第一类临界参数抽象动力方程边值问题之积分算子求解法

该文讨论了一类抽象Volterra型积分算子，利用此获得了含第一类临界参数的抽象动力方程边值问题的解。这种新的求解法我们称为积分算子求解法。     

用第二类Fredholm积分方程求解弹性半空间上弹性板的垂直振动

根据混合边值条件,建立均布简谐荷载作用下弹性半空间上弹性板振动的对偶积分方程.用Abel变换化对偶积分方程为第二类Fredholm积分方程,并进行了数值计算.     

Using Sinc-collocation method for solving weakly singular Fredholm integral equations of the first kind

In this paper, Sinc-collocation method is used to approximate the solution of weakly singular nonlinear Fredholm integral equations of the first kind. Some of the important advantages of this method are rate of convergence of an approximate solution and simplicity for performing even in the presence of singularities. The convergence analysis of the proposed method is proved by preparing the theorems which show the errors decay exponentially and guarantee the applicability of that. Finally, several numerical examples are considered to show the capabilities, validity, and accuracy of the numerical scheme.     

一类对偶积分方程组正则化为Cauchy奇异积分方程组解法

本文基于Mellin变换法求解复杂更一般形式的对偶积分方程组．通过积分变换，由实数域化成复数域上的方程组，引入未知函数的积分变换，移动积分路径，应用Cauchy积分定理，实现退耦正则化为Cauchy奇异积分方程组，由此给出一般性解，并严格证明了对偶积分方程组退耦正则化为Cauchy奇异积分方程组与原对偶积分方程组等价性，以及对偶积分方程组解的存在性和唯一性．给出的解法和理论解，作为求解复杂对偶积分方程组一种有效解法，可供求解复杂的数学、物理、力学中的混合边值问题应用．     

A Modified Convergence Analysis For Solving Fredholm Integral Equations of The First Kind

In this paper, we consider a modified convergence analysis for solving Fredholm integral equations of the first kind in Hilbert space setting using Tikhonov regularization. We follow a general approach which not only includes, as special case, the results of Groetsch [2] but also obtain the same with weaker assumptions.     

球面上第二类Fredholm积分方程配置方法

球面上第二类 Fredholm积分方程经球坐标变换可化为矩形域 H0 上的问题求解 .用有限元法构造H0 上的插值函数 ,它必须满足在 H0 的左、右两边连续 ,然后用配置方程求方程的近似解     

An algorithm based on the regularization and integral mean value methods for the Fredholm integral equations of the first kind

In this paper, an algorithm based on the regularization and integral mean value methods, to handle the ill-posed multi-dimensional Fredholm equations, is introduced. The application of this algorithm is based on the transforming the first kind equation to a second kind equation by the regularization method. Then, by converting the first kind to a second kind, the integral mean value method is employed to handle the resulting Fredholm integral equations of the second kind. The efficiency of the approach will be shown by applying the procedure on some examples.