共查询到20条相似文献,搜索用时 359 毫秒
1.
Wilhelm Hock 《Numerische Mathematik》1981,38(2):155-178
Summary In [6] it has been shown that the midpoint rule applied to second kind volterra integral equations possesses an asymptotic expansion in even powers of the stepsizeh. In this paper we describe an extrapolation method based on the midpoint rule, together with a mechanism of step size control. 相似文献
2.
Ian H. Sloan 《Numerische Mathematik》1988,54(1):41-56
Summary The collocation method is a popular method for the approximate solution of boundary integral equations, but typically does not achieve the high order of convergence reached by the Galerkin method in appropriate negative norms. In this paper a quadrature-based method for improving upon the collocation method is proposed, and developed in detail for a particular case. That case involves operators with even symbol (such as the logarithmic potential) operating on 1-periodic functions; a smooth-spline trial space of odd degree, with constant mesh spacingh=1/n; and a quadrature rule with 2n points (where ann-point quadrature rule would be equivalent to the collocation method). In this setting it is shown that a special quadrature rule (which depends on the degree of the splines and the order of the operator) can yield a maximum order of convergence two powers ofh higher than the collocation method. 相似文献
3.
Wilhelm Hock 《Numerische Mathematik》1979,33(1):77-100
Summary This paper deals with linear multistep methods applied to nonlinear, nonsingular Volterra integral equations of the second kind. Analogously to the theory of W.B. Gragg, the existence of asymptotic expansions in the stepsizeh is proved. Under certain conditions only even powers ofh occur. As a special case, the midpoint rule is treated, a short numerical example for the applicability to extrapolation techniques is given. 相似文献
4.
J. N. Lyness 《Numerische Mathematik》1985,46(4):611-622
Summary The classical Euler Maclaurin Summation Formula expresses the difference between a definite integral over [0, 1] and its approximation using the trapezoidal rule with step lengthh=1/m as an asymptotic expansion in powers ofh together with a remainder term. Many variants of this exist some of which form the basis of extrapolation methods such as Romberg Integration. in this paper a variant in which the integral is a Cauchy Principal Value integral is derived. The corresponding variant of the Fourier Coefficient Asymptotic Expansion is also derived. The possible role of the former in numerical quadrature is discussed.This work was supported by the Applied Mathematical Sciences subprogram of the Office of Energy Research, U.S. Department of Energy, under contract W-31-109-Eng-38 相似文献
5.
Summary Numerical integration formulas are discussed which are obtained by differentiation of the Volterra integral equation and by applying backward differentiation formulas to the resulting integro-differential equation. In particular, the stability of the method is investigated for a class of convolution kernels. The accuracy and stability behaviour of the method proposed in this paper is compared with that of (i) a block-implicit Runge-Kutta scheme, and (ii) the scheme obtained by applying directly a quadrature rule which is reducible to the backward differentiation formulas. The present method is particularly advantageous in the case of stiff Volterra integral equations. 相似文献
6.
This paper deals with a semi-discrete version of the Galerkin method for the single-layer equation in a plane, in which the
outer integral is approximated by a quadrature rule. A feature of the analysis is that it does not require high precision
quadrature or exceptional smoothness of the data. Instead, the assumptions on the quadrature rule are that constant functions
are integrated exactly, that the rule is based on sufficiently many quadrature points to resolve the approximation space,
and that the Peano constant of the rule is sufficiently small. It is then shown that the semi-discrete Galerkin approximation is well posed. For the
trial and test spaces we consider quite general piecewise polynomials on quasi-uniform meshes, ranging from discontinuous
piecewise polynomials to smoothest splines. Since it is not assumed that the quadrature rule integrates products of basis
functions exactly, one might expect degradation in the rate of convergence. To the contrary, it is shown that the semi-discrete
Galerkin approximation will converge at the same rate as the corresponding Galerkin approximation in the and norms.
Received March 15, 1996 / Revised version received June 2, 1997 相似文献
7.
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. 相似文献
8.
Summary The trapezoidal rule is applied to the numerical calculation of a known integral representation of the complementary incomplete gamma function (a,x) in the regiona<–1 andx>0. Since this application of the rule is not standard, a careful investigation of the remainder terms using the Euler-Maclaurin formula is carried out. The outcome is a simple numerical procedure for obtaining values of incomplete gamma functions with surprising accuracy in the stated region.This work has been supported by the Ministero della Pubblica Istruzione and the Consiglio Nazionale delle Ricerche 相似文献
9.
Alberto Durán 《International Journal of Mathematical Education in Science & Technology》2013,44(6):907-910
Very often in practice one has to evaluate a definite integral of a function that has no explicit anti‐derivative or whose anti‐derivative has values that are not easily obtained. One way of handling this it to use a numerical technique, such as the trapezoidal rule or Simpson's rule, to approximate the value of the integral. This paper describes how the value of a definite integral could be approximated using a Monte Carlo technique and a computer. Another application of this technique is the estimation of the value of pi. 相似文献
10.
关于一些数值求积公式的渐近性 总被引:19,自引:0,他引:19
刘彬清 《应用数学与计算数学学报》2000,14(2):83-87
该文给出了一些数值求积公式的渐近性质,这些公式包括求积分的矩形法则、梯形法则和抛物线法则,包含于余项中的中介点的位置当积分区间的长度趋于零时被确定,对应于该法则的校正公式被得到,它们具有较高的代数精度,我们也进行了一些数值试验,得到较满意的数值结果。 相似文献
11.
本文研究了圆周上带希尔伯特核的柯西奇异积分的复合梯型公式.利用连续的分片线性函数逼近被积函数,得到积分公式的误差估计;然后用积分公式构造求解对应奇异积分方程的两种格式;最后给出数值实验验证理论分析结果. 相似文献
12.
A Nyström method for the discretization of thermal layer potentials is proposed and analyzed. The method is based on considering the potentials as generalized Abel integral operators in time, where the kernel is a time dependent surface integral operator. The time discretization is the trapezoidal rule with a corrected weight at the endpoint to compensate for singularities of the integrand. The spatial discretization is a standard quadrature rule for surface integrals of smooth functions. We will discuss stability and convergence results of this discretization scheme for second-kind boundary integral equations of the heat equation. The method is explicit, does not require the computation of influence coefficients, and can be combined easily with recently developed fast heat solvers. 相似文献
13.
Summary. In this article, we study the incompressible Navier–Stokes equations in ℝ3. The non linear integral equation satisfied by the Fourier transform of the Laplacian of the velocity field can be interpreted
in terms of a branching process and a composition rule along the associated tree. We derive from this representation new classes
where global existence and uniqueness can be proven.
Received: 27 November 1996 / In revised form: 30 May 1997 相似文献
14.
We develop a fourth-order piecewise quartic spline rule for Hadamard
integral. The quadrature formula of Hadamard integral is obtained by replacing
the integrand function with the piecewise quartic spline interpolation function. We
establish corresponding error estimates and analyze the numerical stability. The
rule can achieve fourth-order convergence at any point in the interval, even when
the singular point coincides with the grid point. Since the derivative information of
the integrand is not required, the rule can be easily applied to solve many practical
problems. Finally, the quadrature formula is applied to solve the electromagnetic
scattering from cavities with different wave numbers, which improves the whole
accuracy of the solution. Numerical experiments are presented to show the efficiency
and accuracy of the theoretical analysis. 相似文献
15.
N. I. Ioakimidis 《BIT Numerical Mathematics》1983,23(1):92-104
The direct quadrature method for the numerical solution of singular integral equations with Hilbert kernel is investigated and a very accurate natural interpolation formula for the approximation of the unknown function is proposed. It is further proved that this formula coincides with Nyström's natural interpolation formula for the Fredholm integral equation of the second kind equivalent to the original integral equation if the same quadrature rule is used in both cases. 相似文献
16.
Summary.
The cruciform crack problem of
elasticity gives rise to an integral equation of the second
kind on [0,1] whose
kernel has a fixed singularity at (0,0).
We introduce a transformation of
[0,1] onto itself such that an arbitrary number
of derivatives vanish at the
end points 0 and 1. If the transformed kernel
is dominated near the origin by
a Mellin kernel then we have given conditions
under which the use of a
modified Euler-Maclaurin quadrature rule and the
Nystr?m method gives an approximate solution
which converges to the exact solution of the
original equation. The method is
illustrated with a numerical example.
Received May 10, 1994 相似文献
17.
Summary. In this paper we present a new quadrature method for computing Galerkin stiffness matrices arising from the discretisation
of 3D boundary integral equations using continuous piecewise linear boundary elements. This rule takes as points some subset
of the nodes of the mesh and can be used for computing non-singular Galerkin integrals corresponding to pairs of basis functions
with non-intersecting supports. When this new rule is combined with standard methods for the singular Galerkin integrals we
obtain a “hybrid” Galerkin method which has the same stability and asymptotic convergence properties as the true Galerkin
method but a complexity more akin to that of a collocation or Nystr?m method. The method can be applied to a wide range of
singular and weakly-singular first- and second-kind equations, including many for which the classical Nystr?m method is not
even defined. The results apply to equations on piecewise-smooth Lipschitz boundaries, and to non-quasiuniform (but shape-regular)
meshes. A by-product of the analysis is a stability theory for quadrature rules of precision 1 and 2 based on arbitrary points
in the plane. Numerical experiments demonstrate that the new method realises the performance expected from the theory.
Received January 22, 1998 / Revised version received May 26, 1999 / Published online April 20, 2000 –? Springer-Verlag 2000 相似文献
18.
二阶线性常微分方程的两点边值问题的泰勒展开式解法 总被引:2,自引:0,他引:2
本文用泰勒展开公式求解二阶线性常微分方程的两点边值问题.首先将两点边值问题化为一个F redho lm积分方程,进一步通过泰勒展开公式化F redho lm积分方程为线性方程组,利用G ramm er法则可求得问题的近似解. 相似文献
19.
Frank MüllerWerner Varnhorn 《Applied mathematics and computation》2011,217(13):6409-6416
In this article a method is presented, which can be used for the numerical treatment of integral equations. Considered is the Fredholm integral equation of second kind with continuous kernel, since this type of integral equation appears in many applications, for example when treating potential problems with integral equation methods.The method is based on the approximation of the integral operator by quasi-interpolating the density function using Gaussian kernels. We show that the approximation of the integral equation, gained with this method, for an appropriate choice of a certain parameter leads to the same numerical results as Nyström’s method with the trapezoidal rule. For this, a convergence analysis is carried out. 相似文献
20.
We define the Skorohod integral of an operator-valued process with respect to a cylindrical Hilbertian Wiener process. We study the resulting process, and establish a generalized Itô formula. We define also a Stratonovitch integral, and establish the corresponding chain rule. Our work is inspired by the finite-dimensional results in [10]. 相似文献