Polynomial Spline Collocation Methods for Volterra Integrodifferential Equations with Weakly Singular Kernels

In this paper we investigate the attainable order of (global)convergence of collocation approximations in certain polynomialspline spaces for solution of Volterra integrodifferential equationswith weakly singular kernels. While the use of quasi-uniformmeshes leads, due to the nonsmooth nature of these solutions,to convergence of order less than one, regardless of the degreeof the approximating spline function, collocation on suitablygraded meshes will be shown to yield optimal convergence rates.     

A Spline Collocation Method for Linear Volterra Integro-Differential Equations with Weakly Singular Kernels

The piecewise polynomial collocation method is discussed to solve linear Volterra integro-differential equations with weakly singular or other nonsmooth kernels. Using special graded grids, global convergence estimates are derived. The error analysis is based on certain regularity properties of the solution of the initial value problem.This revised version was published online in October 2005 with corrections to the Cover Date.     

Fractional Multistep Methods for Weakly Singular Volterra Integral Equations of the First Kind with Perturbed Data

ABSTRACT

Fractional multistep methods were introduced by C. Lubich for the quadrature of Abel integral operators and the solution of weakly singular Volterra integral equations of the first kind with exactly given right-hand sides. In the current paper, we consider the regularizing properties of these methods to solve the mentioned integral equations of the first kind for perturbed right-hand sides. Finally, numerical results are presented.     

Generalized Jacobi Spectral-Galerkin Method for Nonlinear Volterra Integral Equations with Weakly Singular Kernels

We propose a generalized Jacobi spectral-Galerkin method for the nonlinear Volterra integral equations (VIEs) with weakly singular kernels. We establish the existence and uniqueness of the numerical solution, and characterize the convergence of the proposed method under reasonable assumptions on the nonlinearity. We also present numerical results which are consistent with the theoretical predictions.     

Spline Collocation for Cordial Volterra Integral Equations

We study the convergence and convergence speed of two versions of spline collocation methods on the uniform grids for linear Volterra integral equations of the second kind with noncompact operators.     

The Numerical Solution of Second-Order Weakly Singular Volterra Integro-Differential Equations

In this paper we investigate the attainable order of convergence of collocation approximations in certain polynomial spline spaces for solutions of a class of second-order volterra integro-differential equations with weakly singular kernels. While the use of quasi-uniform meshes leads, due to the nonsmooth nature of these solutions, to convergence of order less than one, regardless of the degree of the approximating spling function, collocation on suitably graded meshes will be shown to yield optimal convergence rates.     

The Numerical Solution of Two-Dimensional Volterra Integral Equations by Collocation and Iterated Collocation

While the numerical solution of one-dimensional Volterra integralequations of the second kind with regular kernels is now wellunderstood there exist no systematic studies of the approximatesolution of their two-dimensional counterparts. In the presentpaper we analyse the numerical solution of such equations bymethods based on collocation and iterated collocation techniquesin certain polynomial spline spaces. The analysis focuses onthe global convergence and local superconvergence propertiesof the approximating spline functions.     

一类奇性积分方程的多尺度配置快速算法

This paper develops fast multiscale collocation methods for a class of Fredholm integral equations of the second kind with singular kernels. A truncation strategy for the coefficient matrix of the corresponding discrete system is proposed, which forms a basis for fast algorithms. The convergence, stability and computational complexity of these algorithms are analyzed.     

On Spline Collocation for Singular Integral Equations

This paper is devoted to the approximate solution of one-dimensional singular integral equations on a closed curve by spline collocation methods. As the main result we give conditions which are sufficient and in special cases also necessary for the convergence in SOBOLEV norms. The paper is organized as follows. In chapter 1 we indicate some definitions and some facts about projection methods. In chapter 2, we generalize a technique developed in [1] and study the convergence of collocations using splines of odd degree in periodic SOBOLEV spaces. In chapter 3, we apply our method to collocations by splines of even degree and consider the case of systems of equations. And in the last chapter, 4, the results are applied to singular integral equations and compared with known facts about piecewise linear spline collocation for such equations.     

L~1空间中弱奇性积分方程特征值的数值解法探究

在L¹空间中讨论弱奇性积分方程的特征值问题,给出了一种算法,证明所提出算法的合理性,并举出具体算例,通过Matlab编程算出所给算例的近似数值解.     

Hermite-Type Method for Volterra Integral Equation with Certain Weakly Singular Kernel

1.IntroductionThispaPerconsidersthenumericalsolutionofthesecondkindVolterraintegralequationy(t)+(Ky)(t)=g(t),(1.1)wherey(t)istheunknownsolution,g(t)isagivenfUnctionandKistheintegraJoperatorforsomegivenkernelfunctionK,(Ky)(t)=l'K(f)y(8)ids.(1.2)Suchequationsarisefromcertaindiffusionproblems.BecauseKisnotcompact,sothestandaxdstabilityproofSfornumericaJmethodsdonotfit.ManypeoplehaveworkedonHermite-typecollocationmethodsforsecond-kindVolterraintegralequationswithsmoothkernels[3,4,5'6],butver…     

On Discrete Superconvergence Properties of Spline Collocation Methods for Nonlinear Volterra Integral Equations

It is shown that the error corresponding to certain spline collocation approximations for nonlinear Volterra integral equations of the second kind is the solution of a nonlinearly perturbed linear Volterra integral equation. On the basis of this result it is possible to derive general estimates for the order of convergence of the spline solution at the underlying mesh points. Extensions of these techniques to other types of Volterra equations are indicated.     

Jacobi Spectral Methods for Volterra-Urysohn Integral Equations of Second Kind with Weakly Singular Kernels

Abstract

In this article, we discuss Jacobi spectral Galerkin and iterated Jacobi spectral Galerkin methods for Volterra-Urysohn integral equations with weakly singular kernels and obtain the convergence results in both the infinity and weighted L²-norm. We show that the order of convergence in iterated Jacobi spectral Galerkin method improves over Jacobi spectral Galerkin method. We obtain the convergence results in two cases when the exact solution is sufficiently smooth and non-smooth. For finding the improved convergence results, we also discuss Jacobi spectral multi-Galerkin and iterated Jacobi spectral multi-Galerkin method and obtain the convergence results in weighted L²-norm. In fact, we prove that the iterated Jacobi spectral multi-Galerkin method improves over iterated Jacobi spectral Galerkin method. We provide numerical results to verify the theoretical results.     

On Optimization of Direct Methods of Solving Weakly Singular Integral Equations

We find the exact order of optimal accuracy of adaptive direct methods for the approximate solution of integral equations with potential-type kernels and for Peierls integral equations arising in transport theory. Moreover, for these equations we indicate the adaptive direct method of optimal order.     

Collocation Methods and Their Modifications for Cauchy Singular Integral Equations on the Interval

In this paper we present polynomial collocation methods and their modi.cations for the numerical solution of Cauchy singular integral equations over the interval [-1, 1]. More precisely, the operators of the integral equations have the form with piecewise continuous coefficients a and b, and with a Jacobi weight . Using the splitting property of the singular values of the collocation methods, we obtain enough stable approximate methods to .nd the least square solution of our integral equation. Moreover, the modifications of the collocation methods enable us to compute kernel and cokernel dimensions of operators from a C*-algebra, which is generated by operators of the Cauchy singular integral equations.     

一类新的弱奇性Volterra 积分不等式解的估计

收稿研究了一类新的含有多个非线性项的弱奇性Volterra积分不等式解的估计,所得结果推广了过去关于弱奇性Volterra型积分不等式的相关结果,并用实例给出了解的估计.     

Collocation Method for Singular Integral Equations on Holder Spaces

The classical polynomial collocation method is considered for a class of Cauchy singular integral equations with variable coefficients on a bounded interval. This method is naturally extended to the case of a non-zero index of the underlying Fredholm operator. This is done by using the structure of the kernel and a complement of the image of this operator. For the extended method we directly obtain error bounds in the norm of the weighted Holder spaces. As an illustration, some numerical results are given.     

L~1空间中直接离散法求解弱奇异积分方程特征值的探讨

在L~1空间中讨论弱奇异积分方程的特征值问题.利用弱奇异积分算子为紧算子,便可以对其直接进行离散化算法求出特征值.并将直接离散的方法与以往的迭代后离散的方法用实例通过Matlab作图进行对比,说明直接进行离散的方法更佳.