共查询到20条相似文献,搜索用时 15 毫秒
1.
Z.K. Eshkuvatov N.M.A. Nik Long M. Abdulkawi 《Journal of Computational and Applied Mathematics》2010,233(8):1995-4686
The singular integral (SI) with the Cauchy kernel is considered. New quadrature formulas (QFs) based on the modification of discrete vortex method to approximate SI are constructed. Convergence of QFs and error bounds are shown in the classes of functions Hα([−1,1]) and C1([−1,1]). Numerical examples are shown to validate the QFs constructed. 相似文献
2.
3.
A new approach is suggested for constructing “smooth” surfaces which contain discontinuities in functional values or derivatives
at prescribed locations. The approach is based on solving singular integral equations with Cauchy-type kernels. It is applicable
in the interpolation or approximation of scattered data. We investigate, in particular, several two-dimensional biharmonic
and triharmonic problems which have smooth solutions except on given line segments, across which different types of discontinuities
occur. We also discuss some issues concerning the application of the approach in practical problems. 相似文献
4.
Giovanni Capobianco Dajana Conte Ida Del Prete 《Journal of Computational and Applied Mathematics》2009
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. 相似文献
5.
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. 相似文献
6.
We consider the general (composite) Newton-Cotes method for the computation of Cauchy principal value integrals and focus on its pointwise superconvergence phenomenon, which means that the rate of convergence of the Newton-Cotes quadrature rule is higher than what is globally possible when the singular point coincides with some a priori known point. The necessary and sufficient conditions satisfied by the superconvergence point are given. Moreover, the superconvergence estimate is obtained and the properties of the superconvergence points are investigated. Finally, some numerical examples are provided to validate the theoretical results. 相似文献
7.
In this paper, the authors propose a Nyström method to approximate the solutions of Cauchy singular integral equations with constant coefficients having a negative index. They consider the equations in spaces of continuous functions with weighted uniform norm. They prove the stability and the convergence of the method and show some numerical tests that confirm the error estimates. 相似文献
8.
Didenko Victor 《Numerical Functional Analysis & Optimization》2013,34(1-2):107-119
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. 相似文献
9.
Magda Rebelo 《Journal of Computational and Applied Mathematics》2010,234(9):2859-2869
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. 相似文献
10.
Tao Tang 《Numerische Mathematik》1992,61(1):373-382
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. 相似文献
11.
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. 相似文献
12.
A.I. Fedotov 《Nonlinear Analysis: Theory, Methods & Applications》2009,71(12):e303
Here we propose and justify quadrature-difference methods for solving different kinds (linear, nonlinear and multidimensional) of periodic singular integro-differential equations. 相似文献
13.
Midpoint collocation for Cauchy singular integral equations 总被引:1,自引:0,他引:1
G. A. Chandler 《Numerische Mathematik》1992,62(1):483-509
Summary A Cauchy singular integral equation on a smooth closed curve may be solved numerically using continuous piecewise linear functions and collocation at the midpoints of the underlying grid. Even if the grid is non-uniform, suboptimal rates of convergence are proved using a discrete maximum principle for a modified form of the collocation equations. The same techniques prove negative norm estimates when midpoint collocation is used to determine piecewise constant approximations to the solution of first kind equations with the logarithmic potential.This work was supported by the Australian Research Council through the program grant Numerical analysis for integrals, integral equations and boundary value problems 相似文献
14.
J. A. Cuminato 《BIT Numerical Mathematics》1987,27(2):190-202
We establish the uniform convergence of a collocation method for solving a class of singular integral equations. This method uses the Jacobi polynomials {P
n
(, )
} as basis elements and the zeros of a Chebyshev polynomial of the first kind as collocation points. Uniform convergence is shown to hold under the weak assumption that the kernel and the right-hand side are Hölder-continous functions. Convergence rates are also given. 相似文献
15.
Approximations to a solution and its derivatives of a boundary value problem of an nth order linear Fredholm integro-differential equation with weakly singular or other nonsmooth kernels are determined. These approximations are piecewise polynomial functions on special graded grids. For their finding a discrete Galerkin method and an integral equation reformulation of the boundary value problem are used. Optimal global convergence estimates are derived and an improvement of the convergence rate of the method for a special choice of parameters is obtained. To illustrate the theoretical results a collection of numerical results of a test problem is presented. 相似文献
16.
P. P. B. Eggermont 《Numerische Mathematik》1992,62(1):35-48
Summary We analyze the quadrature error associated with operational quadrature methods for convolution equations. The assumptions are that the convolution kernel is inL
1 and that its Laplace transform is analytic and bounded in an obtuse sector of the complex plane. Under these circumstances the Laplace transform has a slow variation property which admits a Fourier analysis of the quadrature error. We provide generalL
p
error estimates assuming suitable smoothness conditions on the function under convolution. 相似文献
17.
In this paper we consider a polynomial collocation method for the numerical solution of Cauchy singular integral equations with fixed singularities over the interval, where the fixed singularities are supposed to be of Mellin convolution type. For the stability and convergence of this method in weightedL
2 spaces, we derive necessary and sufficient conditions. 相似文献
18.
Singular integral equations with Cauchy kernel and piecewise-continuous matrix coefficients on open and closed smooth curves are replaced by integral equations with smooth kernels of the form(t–)[(t–)
2–n
2
(t)
2]–1,0, wheren(t), t , is a continuous field of unit vectors non-tangential to . we give necessary and sufficient conditions under which the approximating equations have unique solutions and these solutions converge to the solution of the original equation. For the scalar case and the spaceL
2() these conditions coincide with the strong ellipticity of the given equation.This work was fulfilled during the first author's visit to the Weierstrass Institute for Applied Analysis and Stochastics, Berlin in October 1993. 相似文献
19.
The problem of the numerical evaluation of Cauchy principal value integrals of oscillatory functions , where −1<τ<1, has been discussed. Based on analytic continuation, if f is analytic in a sufficiently large complex region G containing [−1, 1], the integrals can be transformed into the problems of integrating two integrals on [0,+∞) with the integrand that does not oscillate, and that decays exponentially fast, which can be efficiently computed by using the Gauss-Laguerre quadrature rule. The validity of the method has been demonstrated in the provision of two numerical experiments and their results. 相似文献
20.
Kerli Orav-Puurand Gennadi Vainikko 《Journal of Computational and Applied Mathematics》2010,234(9):2848-2858
Nyström type methods are constructed and justified for a class of Fredholm integral equations of the second kind with kernels which may have weak diagonal and boundary singularities. The proposed approach is based on a suitable smoothing change of variables and product integration techniques. Global convergence estimates are derived and a collection of numerical results is given. 相似文献