On the Discrete Galerkin Method for Fredholm Integral Equations of the Second Kind

In the present paper, we refine some previous results on thediscrete Galerlcin method and the discrete iterated Galerkinmethod for Fredholm integral equations of the second kind. Byconsidering discrete inner products and discrete projectionson the same node points but with different quadrature rules,we are able to treat more appropriately kernels with discontinuousderivatives. In particular, for Green's function kernels weobtain a Nystr?m-type method which has the same order of convergenceas the corresponding Nystr?m method for infinitely smooth kernels.     

Discrete Galerkin Methods for Fredholm Integral Equations of the Second Kind

The discrete Galerkin and discrete iterated Galerkin methodsarise when the integrals required in the Galerkin and iteratedGalerkin methods are calculated using numerical integration.In this paper, prolongation and restriction operators are usedto give an error analysis for these two discrete Galerkin methods.From this analysis, we can then give conditions on the quadratureerrors, under which the two discrete Galerkin solutions havethe same order of convergence as their exact counterparts.     

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.     

The Minimal Radius of Galerkin Information for the Fredholm Problem of the First Kind

We propose a new scheme of discretization for solving Fredholm integral equations of the first kind and show that for some classes of equations this scheme is order-optimal in the sense of amount of used Galerkin information.     

An Iterative Method for the Solution of Fredholm Integral Equations of the Second Kind

An iterative method for the solution of the linear Fredholmintegral equation is discussed. Various types of quadraturerule are used to replace the integral and the order of convergenceand error estimates are found for each rule. A few examplesare considered, including one with a weakly singular kernel.     

The Galerkin Method for Integral Equations of the First Kind with Logarithmic Kernel: Applications

The aim of this paper is to discuss the numerical performanceof the Galerkin method for the approximate solution of severaltwo-dimensional Fredholm integral equations of the first kindwith logarithmic kernel, and for the approximation of linearfunctionals of the solution. Predicted rates of convergenceare obtained from the theory in Sloan & Spence (1987), andthese are compared with the numerical rates for the case ofpiecewise constant approximation over equal subintervals. Thephenomenon of ‘superconvergence’ is analysed indetail and some examples are given which attain remarkably highrates of convergence.     

The Galerkin Method for Integral Equations of the First Kind with Logarithmic Kernel: Theory

The aim of this paper is to develop a straightforward analysisof the Galerkin method for two-dimensional boundary integralequations of the first kind with logarithmic kernels. A distinctivefeature of the analysis is that no appeal is made to ‘coercivity’,as a result of which some existence questions cannot be answereddirectly. In return, however, the analysis has no special difficultyin handling corners, cusps, or open arcs. Instead of coercivity,the central feature of the analysis is the positive-definiteproperty of the integral operator for small enough contours.Rates of convergence are predicted theoretically and, in particular,certain linear functionals are shown to exhibit ‘superconvergence’.Numerical results supporting the theory are given in the companionpaper Sloan & Spence (1987) for problems on both open andclosed polygonal arcs.     

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.     

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.     

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.     

An Element-Free Galerkin Method for Time-Fractional Diffusion-Wave Equations北大核心CSCD

利用无单元Galerkin法,对Caputo意义下的时间分数阶扩散波方程进行了数值求解和相应误差理论分析。首先用L1逼近公式离散该方程中的时间变量,将时间分数阶扩散波方程转化成与时间无关的整数阶微分方程;然后采用罚函数方法处理Dirichlet边界条件,并利用无单元Galerkin法离散整数阶微分方程;最后推导该方程无单元Galerkin法的误差估计公式。数值算例证明了该方法的精度和效果。     

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.     

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.     

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．     

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.     

An $h$-Adaptive Runge-Kutta Discontinuous Galerkin Method for Hamilton-Jacobi Equations

In [35, 36], we presented an $h$-adaptive Runge-Kutta discontinuous Galerkin method using troubled-cell indicators for solving hyperbolic conservation laws. A tree data structure (binary tree in one dimension and quadtree in two dimensions) is used to aid storage and neighbor finding. Mesh adaptation is achieved by refining the troubled cells and coarsening the untroubled "children". Extensive numerical tests indicate that the proposed $h$-adaptive method is capable of saving the computational cost and enhancing the resolution near the discontinuities. In this paper, we apply this $h$-adaptive method to solve Hamilton-Jacobi equations, with an objective of enhancing the resolution near the discontinuities of the solution derivatives. One- and two-dimensional numerical examples are shown to illustrate the capability of the method.     

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.     

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。     

Further Studies of the Application of Constrained Minimization Methods to Fredholm Integral Equations of the First Kind

Permanent address: Department of Mathematics, University of Queensland, Australia. Following earlier work of Babolian & Delves (J. Inst. MathsApplics (1979) 24, 157–174) the Galerkin equations forintegral equations of the first kind are stablized by imposingasympotic decay rates on the expansion coefficients. Results for the formulation in the l₂ norm are compared withresults of Babolian & Delves where the l₁ norm was used. The importance of the choice of the constants which specifythe decay rates is also considered. Theoretical results andcomputational experiments show that previously used automaticselection of these constants needs to be safeguarded by monitoringthe residuals of the Galerkin equations.