On the Numerical Solution of Fredholm Integral Equations of the First Kind

Two previously given methods for the numerical solution of Fredholmintegral equations of the first kind are investigated by theuse of polynomial spline functions. As a byproduct a new methodis presented for obtaining the eigensolutions of the kernelfunction. From numerical experiments zero order regularizationappears to give more accurate results than higher orders. Acomparison is made as to the relative accuracy and speed ofthe two methods. A scheme is presented to enable a choice tobe made, from the set of truncated solutions, as to which memberis closest to the solution of the integral equation.     

First Kind Cordial Volterra Integral Equations 1

For (pure) cordial Volterra integral equations of the first kind, we establish stability estimates of the solution in the scales of Banach spaces of C ^m -smooth functions on (0, T] and on [0, T]. This enables to estimate the error of polynomial or generalized polynomial approximate solutions which are easy to be determined.     

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.     

On One Class of Nonclassical Linear Volterra Integral Equations of the First Kind

Ukrainian Mathematical Journal - On the basis of a new approach, we prove the uniqueness theorem and construct Lavrent’ev’s regularizing operators for the solution of nonclassical...     

On the Solution of Fredholm Integral Equations of the First Kind in L2

A method of solving numerically an integral equation of thefirst kind in L² is given and precise conditions are given forthe validity of the method. The method should be useful in physicalproblems where the kernel is calculated numerically and is notgiven as a known function.     

Iterative Method for Solving Integral Equations of the First Kind

A new iterative method is proposed for solving integral equations of the first kind. The efficiency of the method is demonstrated using examples of typical integral kernels.     

Fredholm Integral Equations of the First Kind and Topological Information Theory

The Fredholm integral equations of the first kind are a classical example of ill-posed problem in the sense of Hadamard. If the integral operator is self-adjoint and admits a set of eigenfunctions, then a formal solution can be written in terms of eigenfunction expansions. One of the possible methods of regularization consists in truncating this formal expansion after restricting the class of admissible solutions through a-priori global bounds. In this paper we reconsider various possible methods of truncation from the viewpoint of the ${\varepsilon}$ -coverings of compact sets.     

A New Analysis of Volterra-Fredholm Boundary Integral Equations of Second Kind

In this paper, we study the existence, uniqueness and regularity of the solutions to Volterra-Fredholm boundary integral equations of second kind in a kind of boundary function spaces.     

Fractional Linear Multistep Methods for Abel-Volterra Integral Equations of the First Kind

Fractional powers of linear multistep methods are suggestedfor the numerical solution of weakly singular Volterra integralequations of the first kind. The proposed methods are convergentof the order of the underlying multistep method. The stabilityproperties are directly related to those of the multistep method.     

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.     

Nonlinear Smoothing and the EM Algorithm for Positive Integral Equations of the First Kind

We study a modification of the EMS algorithm in which each step of the EMS algorithm is preceded by a nonlinear smoothing step of the form , where S is the smoothing operator of the EMS algorithm. In the context of positive integral equations (à la positron emission tomography) the resulting algorithm is related to a convex minimization problem which always admits a unique smooth solution, in contrast to the unmodified maximum likelihood setup. The new algorithm has slightly stronger monotonicity properties than the original EM algorithm. This suggests that the modified EMS algorithm is actually an EM algorithm for the modified problem. The existence of a smooth solution to the modified maximum likelihood problem and the monotonicity together imply the strong convergence of the new algorithm. We also present some simulation results for the integral equation of stereology, which suggests that the new algorithm behaves roughly like the EMS algorithm. Accepted 1 April 1997     

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.     

On the Second-order Equations of Mixed Type of First Kind

In this paper the generalized Tricomi problem for the second-order equation of mixed type of first kind is considered. The uniqueness or solutions is proved under very weak conditions oil the coefficients or equation and the boundary curve of domain. The existence of H¹ strong solutions is proved for the Tricomi problem.     

Convergence of Collocation Method with Delta Functions for Integral Equations of First Kind

Integral equations of first kind with periodic kernels arising in solving partial differential equations by interior source methods are considered. Existence and uniqueness of solution in appropriate spaces of linear analytic functionals is proved. Rate of convergence of collocation method with Dirac’s delta-functions as the trial functions is obtained in case of uniform meshes. In case of an analytic kernel the convergence rate is exponential.     

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.     

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.     

On the Uniqueness of Solution for a Kind of Singular Integral Equations

In this article, from the discussion of some properties of entire functions, the uniqueness of solution for a kind of singular integral equations with multiple complex translations on the real axis is proved.     

Wavelet Collocation Methods for a First Kind Boundary Integral Equation in Acoustic Scattering

In this paper we consider a wavelet algorithm for the piecewise constant collocation method applied to the boundary element solution of a first kind integral equation arising in acoustic scattering. The conventional stiffness matrix is transformed into the corresponding matrix with respect to wavelet bases, and it is approximated by a compressed matrix. Finally, the stiffness matrix is multiplied by diagonal preconditioners such that the resulting matrix of the system of linear equations is well conditioned and sparse. Using this matrix, the boundary integral equation can be solved effectively.     

A Method for Solving Fredholm Integral Equations of the First Kind Based on Chebyshev Wavelets

In this paper, we suggest a method for solving Fredholm integral equation of the first kind based on wavelet basis. The continuous Legendre and Chebyshev wavelets of the first, second, third and fourth kind on [0,1] are used and are utilized as a basis in Galerkin method to approximate the solution of integral equations. Then, in some examples the mentioned wavelets are compared with each other.