Geometric meshes in collocation methods for Volterra integral equations with proportional delays

In this paper we analyse the local superconvergence propertiesof iterated piecewise polynomial collocation solutions for linearsecond-kind Volterra integral equations with (vanishing) proportionaldelays qt (0 < q < 1). It is shown that on suitable geometricmeshes depending on q, collocation at the Gauss points leadsto almost optimal superconvergence at the mesh points. Thiscontrasts with collocation on uniform meshes where the problemregarding the attainable order of local superconvergence remainsopen.     

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.     

Superconvergence of collocation methods for a class of weakly singular Volterra integral equations

We discuss the application of spline collocation methods to a certain class of weakly singular Volterra integral equations. It will be shown that, by a special choice of the collocation parameters, superconvergence properties can be obtained if the exact solution satisfies certain conditions. This is in contrast with the theory of collocation methods for Abel type equations. Several numerical examples are given which illustrate the theoretical results.     

Spline Collocation Approximation to Periodic Solutions of Ordinary Differential Equations

A spline collocation method is proposed to approximate the periodic solution of nonlinear ordinary differential equations. It is proved that the cubic periodic spline collocation solution has the same error bound $O(h^4)$ and superconvergence of the derivative at collocation points as that of the interpolating spline function. Finally a numerical example is given to demonstrate the effectiveness of our algorithm.     

Projection Methods for Integral Equations on the Half-Line

Convergence results are proved for projection methods for integralequations of the form are such that Wiener-Hopf integral equations are included in our analysis. The convergenceresults indicate that the iterated-projection solution may exhibitsuperconvergence. The case of collocation using piecewise-constantbasis functions applied to an integral equation with kernel is discussed in detail, and numerical results are given. For this example superconvergence of the iteratedsolution, and hence also of the collocation solution at thecollocation points, is both proved theoretically and observednumerically.     

Collocation and iterated collocation methods for a class of weakly singular Volterra integral equations

We discuss the convergence properties of spline collocation and iterated collocation methods for a weakly singular Volterra integral equation associated with certain heat conduction problems. This work completes the previous studies of numerical methods for this type of equations with noncompact kernel. In particular, a global convergence result is obtained and it is shown that discrete superconvergence can be achieved with the iterated collocation if the exact solution belongs to some appropriate spaces. Some numerical examples illustrate the theoretical results.     

Spline collocation methods for nonlinear Volterra integral equations with unknown delay

In this paper we introduce and study polynomial spline collocation methods for systems of Volterra integral equations with unknown lower integral limit arising in mathematical economics. Their discretization leads to implicit Runge-Kutta-type methods. The global convergence and local superconvergence properties of these methods are proved, and the theory is illustrated by a numerical example arising in the application of such equations in certain mathematical models of liquidation.     

Superconvergence of functional approximation methods for integral equations

In this work, a functional approximation method for calculating the linear functional of the solution of second-kind Fredholm integral equations is developed. When the method is applied to the collocation method or to the multi-projection method, it generates approximations which exhibit superconvergence.     

The numerical solution of integral-algebraic equations of index 1 by polynomial spline collocation methods

In this paper, we study polynomial spline collocation methods applied to a particular class of integral-algebraic equations of Volterra type. We analyse mixed systems of second and first kind integral equations. Global convergence and local superconvergence results are established.

     

A collocation method for systems of nonlinear ordinary differential equations

A collocation method based on piecewise polynomials is applied to boundary value problems for mth order systems of nonlinear ordinary differential equations. Optimal a priori estimates are obtained for the error of approximation in the maximum norm and superconvergence is verified at particular points.     

Multi-parameter error resolution for the collocation method of Volterra integral equations

In this paper, a multi-parameter error resolution technique is introduced and applied to the collocation method for Volterra integral equations. By using this technique, an approximation of higher accuracy is obtained by using a multi-processor in parallel. Additionally, a correction scheme for approximation of higher accuracy and a global superconvergence result are presented.     

Superconvergence of the iterated collocation methods for Hammerstein equations

In this paper, we analyse the iterated collocation method for Hammerstein equations with smooth and weakly singular kernels. The paper expands the study which began in [16] concerning the superconvergence of the iterated Galerkin method for Hammerstein equations. We obtain in this paper a similar superconvergence result for the iterated collocation method for Hammerstein equations. We also discuss the discrete collocation method for weakly singular Hammerstein equations. Some discrete collocation methods for Hammerstein equations with smooth kernels were given previously in [3, 18].     

On spline approximation for a class of integral equations. I: Galerkin and collocation methods with piecewise polynomials

We consider the numerical solution of a class of one-dimensional non-compact integral equations by Galerkin and collocation methods and their iterated variants, using piecewise polynomials as basis functions. In particular, we obtain new results for the stability of the approximation methods, without any restriction on the norm of the integral operators. Furthermore, we extend results of Chandler and Graham^4,6 concerning error estimates and superconvergence to a more general class of operators.     

非线性第二类Volterra积分方程的Chebyshev谱配置法

我们在参考了相关文献的基础上,考察了一类非线性Volterra积分方程的Chebyshev谱配置法.方法中,我们将该类非线性方程转化为两个方程进行数值逼近.我们选择N阶Chebyshev Gauss-Lobatto点作为配置点,对积分项用N阶高斯数值积分公式逼近.收敛性分析结果表明数值误差的收敛阶为N^（1/2）-m,其中m是已知函数最高连续导数的阶数.我们也开展数值实验证实这一理论分析结果.     

Recent advances in the numerical analysis of Volterra functional differential equations with variable delays

The numerical analysis of Volterra functional integro-differential equations with vanishing delays has to overcome a number of challenges that are not encountered when solving ‘classical’ delay differential equations with non-vanishing delays. In this paper I shall describe recent results in the analysis of optimal (global and local) superconvergence orders in collocation methods for such evolutionary problems. Following a brief survey of results for equations containing Volterra integral operators with non-vanishing delays, the discussion will focus on pantograph-type Volterra integro-differential equations with (linear and nonlinear) vanishing delays. The paper concludes with a section on open problems; these include the asymptotic stability of collocation solutions _uh$u_h$ on uniform meshes for pantograph-type functional equations, and the analysis of collocation methods for pantograph-type functional equations with advanced arguments.     

Hammerstein型弱奇异积分方程的β-多项式配置方法

1．引言对VOlterra弱奇异积分方程和积一微分方程之配置方法已有不少文章讨论［1－6］．由于其解在左端点处的非光滑性[3]并要得到m-1次多项式配置解的最优收敛率。，需采用所谓的等级网格．早期M作［1,2］是将等级指数r取为1，a表征核（t—s）ak（t，8）的奇异程度），但...     

多滞量积分方程基于高阶插值的分步配置方法

1．引言考虑多滞量Volterra积分方程其中常数假定已知函数R在定义域内连续，以保证方程（1.1）存在唯一解形如（1．1）的Volterra延滞积分方程常出现在物理问题和生物模型中［2］．由于“滞量”的影响，对其作理论分析和数值研究均比“古典”的Volterra积分方程更为困难．近来人们对Volterra延滞积分方程的数值求解越来越感兴趣[3，4]，但目前的工作基本上只限于单滞量的情形：并采用所谓的“约束”网格（即要求步长人整除一，且假定T是，的整数倍（否则，应在更大的区间上求解），以保证数值解在结点集上具有理想的收敛率．显然，这些限…     

Symmetric collocation for unstructured nonlinear differential-algebraic equations of arbitrary index

Summary. We examine a class of symmetric collocation schemes for the solution of nonlinear boundary value problems for unstructured nonlinear systems of differential-algebraic equations with arbitrary index. We show that these schemes converge with the same orders as one would expect for ordinary differential equations. In particular, we show superconvergence for a special choice of the collocation points. We demonstrate the efficiency of the new approach with some numerical examples.Mathematics Subject Classification (2000): 65L10Revised version received November 21, 2003Supported by DFG research grant Ku964/4.Supported by DFG research grant Me790/11.     

The approximation of generalized turning points by projection methods with superconvergence to the critical parameter

Summary A procedure is given that generates characterizations of singular manifolds for mildly nonlinear mappings between Banach spaces. This characterization is used to develop a method for determining generalized turning points by using projection methods as a discretization. Applications are given to parameter dependent two-point boundary value problems. In particular, collocation at Gauss points is shown to achieve superconvergence in approximating the parameter at simple turning points.     

Chebyshev spectral collocation method for system of nonlinear Volterra integral equations

Numerical Algorithms - We investigate Chebyshev spectral collocation method for system of nonlinear Volterra integral equations. We choose Chebyshev Gauss points as collocation points, and...