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].     

Iterated Galerkin versus Iterated Collocation for Integral Equations of the Second Kind

We consider the numerical solution of one-dimensional Fredholmintegral equations of the second kind by the Galerkin and collocationmethods and their iterated variants, using spline bases. Inparticular, we state and prove new superconvergence resultsfor the iterated solutions, under general smoothness requirementson the kernel and solution. We find that the smoothness requirementsfor the iterated collocation method are more stringent thanthose for the iterated Galerkin method, and show by examplethat these more stringent smoothness conditions are in a certainsense necessary. In the light of these results the Galerkinand collocation schernes are compared.     

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.     

The iterated Galerkin method for linear integro-differential equations

The iterated Galerkin method of I.H. Sloan, which yields superconvergence for the approximate solution of integral equations, is generalized to get similar results for the larger class of integro differential equations. Two approaches are investigated to obtain superconvergence in different norms. Moreover, some additional aspects are discussed.     

On global superconvergence of iterated collocation solutions to linear second-kind Volterra integral equations

It is well known that, in contrast to Fredholm integral equations, iterated collocation solutions (based on collocation at the Gauss points) to Volterra integral equations of the second kind exhibit optimal discrete superconvergence only at the mesh points. Here, we show that some degree of global superconvergence is possible on the entire interval.     

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.     

A fast multiscale Kantorovich method for weakly singular integral equations

In this paper, we use the idea of Kantorovich regularization to develop the fast multiscale Kantorovich method and the fast iterated multiscale Kantorovich method. For some kinds of weakly singular integral equations with nonsmooth inhomogeneous terms, we show that our two proposed methods can still obtain the optimal order of convergence and superconvergence order, respectively. Numerical examples are given to demonstrate the efficiency of the methods.     

Mixed problem for pseudoparabolic integro-differential equation with degenerate kernel

A mixed problem for a certain nonlinear third-order intregro-differential equation of the pseudoparabolic type with a degenerate kernel is considered. The method of degenerate kernel is essentially used and developed and the Fourier method of variable separation is employed for this equation. A system of countable systems of algebraic equations is first obtained; after it is solved, a countable system of nonlinear integral equations is derived. The method of sequential approximations is used to prove the theorem on the unique solvability of the mixed problem.     

Error Analysis of Discrete Legendre Multi-Projection Methods for Nonlinear Fredholm Integral Equations

In this paper, polynomially-based discrete M-Galerkin and M-collocation methods are proposed to solve nonlinear Fredholm integral equation with a smooth kernel. Using su?ciently accurate numerical quadrature rule, we establish superconvergence results for the approximate and iterated approximate solutions of discrete Legendre M-Galerkin and M-collocation methods in both infinity and L²-norm. Numerical examples are presented to illustrate the theoretical results.     

Degenerate kernel method for multi-variable Hammerstein equations

In this paper we extend the degenerate kernel method for single-variable Hammerstein equations of a previous paper to include multi-variable Hammerstein equations.     

Numerical Solutions for second-kind Volterra integral equations by Galerkin methods

In this paper, we study the global convergence for the numerical solutions of nonlinear Volterra integral equations of the second kind by means of Galerkin finite element methods. Global superconvergence properties are discussed by iterated finite element methods and interpolated finite element methods. Local superconvergence and iterative correction schemes are also considered by iterated finite element methods. We improve the corresponding results obtained by collocation methods in the recent papers [6] and [9] by H. Brunner, Q. Lin and N. Yan. Moreover, using an interpolation post-processing technique, we obtain a global superconvergence of the O(h ^2r )-convergence rate in the piecewise-polynomial space of degree not exceeding (r–1). As a by-product of our results, all these higher order numerical methods can also provide an a posteriori error estimator, which gives critical and useful information in the code development.     

Extrapolation for the Approximations to the Solution of a Boundary Integral Equation on Polygonal Domains

In this paper, we consider a boundary integral equation of second kind rising from potential theory. The equation may be solved numerically by Galerkin's method using piecewise constant functions. Because of the singularities produced by the corners, we have to grade the mesh near the corner. In general, Chandler obtained the order 2 superconvergence of the iterated Galerkin solution in the uniform norm. It is proved in this paper that the Richardson extrapolation increases the accuracy from order 2 to order 4.     

Representation of a multiple integral of special form by a series

Multiple integrals generalizing the iterated kernels of linear integral equations are expressed by a series each of whose terms is proportional to the product of two orthogonal functions in the case of a similar representation of the kernel. Besides integral equations, these integrals have applications in the theory of Markov processes. The results obtained are illustrated by several examples.     

第二类端点奇异Fredholm积分方程的分数阶退化核方法

本文针对第二类端点奇异Fredholm积分方程构造基于分数阶Taylor展开的退化核方法，设计了两种计算格式，一是在全区间上使用分数阶Taylor展开式近似核函数，二是在包含奇点的小区间上采用分数阶插值，在剩余区间上采用分段二次多项式插值逼近核函数.讨论了两种退化核方法收敛的条件，并给出了混合插值法的收敛阶估计.数值算例表明对于非光滑核函数分数阶退化核方法有着良好的计算效果，且混合二次插值法比全区间上的分数阶退化核方法有着更广泛的适用范围.     

Superconvergent Nyström and degenerate kernel methods for eigenvalue problems

The Nyström and degenerate kernel methods, based on projections at Gauss points onto the space of (discontinuous) piecewise polynomials of degree ?r-1, for the approximate solution of eigenvalue problems for an integral operator with a smooth kernel, exhibit order 2r. We propose new superconvergent Nyström and degenerate kernel methods that improve this convergence order to 4r for eigenvalue approximation and to 3r for spectral subspace approximation in the case where the kernel is sufficiently smooth. Moreover for a simple eigenvalue, we show that by using an iteration technique, an eigenvector approximation of order 4r can be obtained. The methods introduced here are similar to that studied by Kulkarni in [10] and exhibit the same convergence orders, so a comparison with these methods is worked out in detail. Also, the error terms are analyzed and the obtained methods are numerically tested. Finally, these methods are extended to the case of discontinuous kernel along the diagonal and superconvergence results are also obtained.     

Generalized Sobolev's phi function for the resolvent of a Milne-type integral equation with a degenerate kernel

It is well known in the field of radiative transfer that Sobolev was the first to introduce the resolvent into Milne's integral equation with a displacement kernel. Thereafter it was shown that the resolvent plays an important role in the theory of formation of spectral lines. In the theory of line-transfer problems, the kernel representation in Milne's integral equation has been used to provide an approximate solution in a manner similar to that given by the discrete ordinales method.In this paper, by means of invariant imbedding we show how to determine an exact solution of a Milne-type integral equation with a degenerate kernel, whose form is more general than the Pincherle-Gourast kernel. A Cauchy system for the resolvent is expressed in terms of generalized Sobolev's Φ- and Ψ-functions, which are computed by solving a system of differential equations for auxiliary functions. Furthermore, these functions are expressed in terms of components of the kernel representation.     

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.     

An Iterative Variant of the Degenerate Kernel Method for Solving Fredholm Integral Equations

An iterative variant of the classical degenerate kernel method for solving Fredholm integral equations of the second kind is presented and its convergence properties are studied.     

Superconvergence analysis for Maxwell's equations in dispersive media

In this paper, we consider the time dependent Maxwell's equations in dispersive media on a bounded three-dimensional domain. Global superconvergence is obtained for semi-discrete mixed finite element methods for three most popular dispersive media models: the isotropic cold plasma, the one-pole Debye medium, and the two-pole Lorentz medium. Global superconvergence for a standard finite element method is also presented. To our best knowledge, this is the first superconvergence analysis obtained for Maxwell's equations when dispersive media are involved.

     

Boundary-Value problems for systems of integro-differential equations with Degenerate Kernel

By using methods of the theory of generalized inverse matrices, we establish a criterion of solvability and study the structure of the set of solutions of a general linear Noether boundary-value problem for systems of integro-differential equations of Fredholm type with degenerate kernel.