On the stability of the one-step exact collocation methods for the numerical solution of the second kind Volterra integral equation

The purpose of this paper is to analyze the stability properties of one-step collocation methods for the second kind Volterra integral equation through application to the basic test and the convolution test equation.Stability regions are determined when the collocation parameters are symmetric and when they are zeros of ultraspherical polynomials.     

Adaptive boundary element methods for strongly elliptic integral equations

Summary Integral operators are nonlocal operators. The operators defined in boundary integral equations to elliptic boundary value problems, however, are pseudo-differential operators on the boundary and, therefore, provide additional pseudolocal properties. These allow the successful application of adaptive procedures to some boundary element methods. In this paper we analyze these methods for general strongly elliptic integral equations and obtain a-posteriori error estimates for boundary element solutions. We also apply these methods to nodal collocation with odd degree splines. Some numerical examples show that these adaptive procedures are reliable and effective.This work was carried out while Dr. De-hao Yu was an Alexander-von-Humboldt-Stiftung research fellow at the University of Stuttgart in 1987, 1988     

High performance parallel numerical methods for Volterra equations with weakly singular kernels

Non-stationary discrete time waveform relaxation methods for Abel systems of Volterra integral equations using fractional linear multistep formulae are introduced. Fully parallel discrete waveform relaxation methods having an optimal convergence rate are constructed. A significant expression of the error is proved, which allows us to estimate the number of iterations needed to satisfy a prescribed tolerance and allows us to identify the problems where the optimal methods offer the best performance. The numerical experiments confirm the theoretical expectations.     

Second order parallel algorithms for piecewise smooth displacement kernels

Second order parallel algorithms for Fredholm integral equations with piecewise smooth displacement kernels are derived. One is based on a difference scheme of Runge-Kutta type for an unusual partial differential equations for continuous functions of two variables. The other is based on the trapezoidal quadrature rule applied to a modified integral equations. It is found that the Runge-Kutta type algorithm exhibits certain advantages.The work of these authors was supported in part by the NSF Grant DMS-9007030The work of this author was supported in part by a grant from the National Science and Engineering Research Council of Canada     

On the discretization of differential and Volterra integral equations with variable delay

In this paper we analyze the attainable order ofm-stage implicit (collocation-based) Runge-Kutta methods for differential equations and Volterra integral equations of the second kind with variable delay of the formqt (0<q<1). It will be shown that, in contrast to equations without delay, or equations with constant delay, collocation at the Gauss (-Legendre) points will no longer yield the optimal (local) orderO(h ^2m ). This work was supported by the Natural Sciences and Engineering Research Council of Canada (NSERC Research Grant OGP0009406).     

Solution of a nonsymmetric algebraic Riccati equation from a two-dimensional transport model

For the steady-state solution of an integral-differential equation from a two-dimensional model in transport theory, we shall derive and study a nonsymmetric algebraic Riccati equation B^--XF^--F⁺X+XB⁺X=0, where , and with a nonnegative matrix P, positive diagonal matrices D^±, and nonnegative parameters f, and . We prove the existence of the minimal nonnegative solution X^∗ under the physically reasonable assumption , and study its numerical computation by fixed-point iteration, Newton’s method and doubling. We shall also study several special cases; e.g. when and P is low-ranked, then is low-ranked and can be computed using more efficient iterative processes in U and V. Numerical examples will be given to illustrate our theoretical results.     

Approximate solutions of some Mellin equations with conjugation

We prove some stability results for a simple numerical method for Mellin convolution operators with conjugation and investigate the Fredholm property for its local operators.     

The Galerkin scheme for Lavrentiev’sm-times iterated method to solve linear accretive Volterra integral equations of the first kind

In this paper, for the numerical solution of linear accretive Volterra integral equations of the first kind in Hilbert spaces we consider the Galerkin scheme for Lavrentiev’sm-times iterated method, i.e., for each parameter choice for Lavrentiev’sm-times iterated method the arisingm stabilized equations are discretized by the Galerkin scheme. An associated discrepancy principle as parameter choice strategy for this finite-dimensional version of Lavrentiev’sm-times iterated method is proposed, and corresponding convergence results are provided.     

The fast Fourier transform and the numerical solution of one-dimensional boundary integral equations

Summary Here we present a fully discretized projection method with Fourier series which is based on a modification of the fast Fourier transform. The method is applied to systems of integro-differential equations with the Cauchy kernel, boundary integral equations from the boundary element method and, more generally, to certain elliptic pseudodifferential equations on closed smooth curves. We use Gaussian quadratures on families of equidistant partitions combined with the fast Fourier transform. This yields an extremely accurate and fast numerical scheme. We present complete asymptotic error estimates including the quadrature errors. These are quasioptimal and of exponential order for analytic data. Numerical experiments for a scattering problem, the clamped plate and plane estatostatics confirm the theoretical convergence rates and show high accuracy.     

Finite sections for singular integral operators by weighted Chebyshev polynomials

A finite section method for the approximate solution of singular integral equations with piecewise continuous coefficients on intervals is considered. The problem is transformed in such a way that results which were previously obtained for singular integral equations on the unit circle using localization methods in Banach algebras are applicable to it. Thus, necessary and sufficient conditions for the stability of the approximation method can be proved.     

On parameters influencing the stability of quadrature methods for singular integral equations with conjugation

In this paper, we study an approximation method for solving singular integral equations with conjugation on an open arc. The stability of the method depends on the invertibility of certain operators which belong to well-known algebras. We investigate properties of these operators and show how to choose the parameters of the approximation method so that the Fredholm indices of the operators mentioned become equal to zero.     

Quadrature-difference methods for solving linear and nonlinear singular integro-differential equations

Here we propose and justify quadrature-difference methods for solving different kinds (linear, nonlinear and multidimensional) of periodic singular integro-differential equations.     

A hybrid collocation method for a nonlinear Volterra integral equation with weakly singular kernel

This work is concerned with the numerical solution of a nonlinear weakly singular Volterra integral equation. Owing to the singular behavior of the solution near the origin, the global convergence order of product integration and collocation methods is not optimal. In order to recover the optimal orders a hybrid collocation method is used which combines a non-polynomial approximation on the first subinterval followed by piecewise polynomial collocation on a graded mesh. Some numerical examples are presented which illustrate the theoretical results and the performance of the method. A comparison is made with the standard graded collocation method.     

Superconvergence of numerical solutions to weakly singular Volterra integro-differential equations

Summary We discuss the application of a class of spline collocation methods to first-order Volterra integro-differential equations (VIDEs) which contain a weakly singular kernel (t–s)^– with 0<<1. It will be shown that superconvergence properties may be obtained by using appropriate collocation parameters and graded meshes. The grading exponents of graded meshes used are not greater thanm (the polynomial degree) which is independent of . This is in contrast to the theories of spline collocation methods for Volterra (or Fredholm) integral equation of the second kind. Numerical examples are given to illustrate the theoretical results.     

On a method for solving a two-dimensional nonlinear integral equation of the second kind

In this article, the existence of at least one solution of a nonlinear integral equation of the second kind is proved. The degenerate method is used to obtain a nonlinear algebraic system, where the existence of at least one solution of this system is discussed. Finally, computational results with error estimates are obtained using Maple software.     

Toeplitz matrix method and nonlinear integral equation of Hammerstein type

The existence and uniqueness solution of the nonlinear integral equation of Hammerstein type with discontinuous kernel are discussed. The normality and continuity of the integral operator are proved. Toeplitz matrix method is used, as a numerical method, to obtain a nonlinear system of algebraic equations. Also, many important theorems related to the existence and uniqueness of the produced algebraic system are derived. Finally, numerical examples, when the kernel takes a logarithmic and Carleman forms, are discussed and the estimate error, in each case, is calculated.     

Error bounds and parameter choice strategies for simplified regularization in Hilbert scales

Simplified regularization in the setting of Hilbert scales has been considered for obtaining stable approximate solutions for ill-posed operator equations. The derived error estimates using an a posteriori as well as an a priori parameter choice strategy are shown to be of optimal order with respect to certain natural assumptions on the ill-posedness of the equation.The work of M. Thamban Nair is partially supported by IC&SR, I.I.T., Madras     

Optimal order results for a class of regularization methods using unbounded operators

A class of regularization methods using unbounded regularizing operators is considered for obtaining stable approximate solutions for ill-posed operator equations. With an a posteriori as well as an a priori parameter choice strategy, it is shown that the method yields the optimal order. Error estimates have also been obtained under stronger assumptions on the generalized solution. The results of the paper unify and simplify many of the results available in the literature. For example, the optimal results of the paper include, as particular cases for Tikhonov regularization, the main result of Mair (1994) with an a priori parameter choice, and a result of Nair (1999) with an a posteriori parameter choice. Thus the observations of Mair (1994) on Tikhonov regularization of ill-posed problems involving finitely and infinitely smoothing operators is applicable to various other regularization procedures as well. Subsequent results on error estimates include, as special cases, an optimal result of Vainikko (1987) and also some recent results of Tautenhahn (1996) in the setting of Hilbert scales.     

Blowing up behavior for a class of nonlinear VIEs connected with parabolic PDEs

We study the blowing-up behavior of solutions of a class of nonlinear integral equations of Volterra type that is connected with parabolic partial differential equations with concentrated nonlinearities. We present some analytic results and, in the case of the kernel of Abel-kind with power nonlinearity and fixed initial data, we give a numerical approximation by using one-point collocation methods.