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1.
The piecewise polynomial collocation method is discussed to solve linear Volterra integro-differential equations with weakly singular or other nonsmooth kernels. Using special graded grids, global convergence estimates are derived. The error analysis is based on certain regularity properties of the solution of the initial value problem.This revised version was published online in October 2005 with corrections to the Cover Date. 相似文献
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The $z$-transform is introduced to analyze a full discretization
method for a partial integro-differential equation (PIDE) with a
weakly singular kernel. In this method, spectral collocation is
used for the spatial discretization, and, for the time stepping,
the finite difference method combined with the convolution
quadrature rule is considered. The global stability and
convergence properties of complete discretization are derived and
numerical experiments are reported. 相似文献
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The main purpose of this work is to provide a numerical method for the solution of Volterra functional integro-differential equations of neutral type based on a spectral approach. We analyze the convergence properties of the spectral method to approximate smooth solutions of Volterra functional integro-differential equations of neutral type. It is shown that for the neutral integro-differential equations, the spectral methods yield an exponential order of convergence. 相似文献
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Generalized Jacobi Spectral-Galerkin Method for Nonlinear Volterra Integral Equations with Weakly Singular Kernels
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We propose a generalized Jacobi spectral-Galerkin method for the nonlinear
Volterra integral equations (VIEs) with weakly singular kernels. We establish the
existence and uniqueness of the numerical solution, and characterize the convergence
of the proposed method under reasonable assumptions on the nonlinearity. We also
present numerical results which are consistent with the theoretical predictions. 相似文献
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This paper is concerned with a compact difference scheme with the truncation error of order 3/2 for time and order 4 for space to an evolution equation with a weakly singular kernel. The integral term is treated by means of the second order convolution quadrature suggested by Lubich. The stability and convergence are proved by the energy method. A numerical experiment is reported to verify the theoretical predictions. 相似文献
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针对带有弱奇异核的第二类Fredholm积分方程数值解法问题,介绍了两种方法.一种方法是直接用L~1空间中的离散化方法求其数值解;另一种方法是将弱奇异核通过迭代变为连续核,再用L~1空间中的离散化方法求其数值解,且通过对具体算例作图分析,从而得出直接用L~1空间中离散化方法更好. 相似文献
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In this paper, the convergence analysis of the Volterra integral equation of second kind with weakly singular kernel and pantograph delays is provided. We use some function transformations and variable transformations to change the equation into a new Volterra integral equation with pantograph delays defined on the interval [-1, 1], so that the Jacobi orthogonal polynomial theory can be applied conveniently. We provide a rigorous error analysis for the proposed method in the L ∞ -norm and the weighted L 2-norm. Numerical examples are presented to complement the theoretical convergence results. 相似文献
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讨论了带弱奇异核的非线性抛物积分微分方程的Hermite型各向异性矩形元逼近.在各向异性网格下导出了关于Riesz投影的L~2和H~1模的误差估计.在半离散和向后欧拉全离散格式下,基于Riesz投影的性质并利用平均值技巧,分别得到了L~2模意义下的最优误差估计. 相似文献
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Yunxia WEI Yanping CHEN Xiulian SHI Yuanyuan ZHANG 《Frontiers of Mathematics in China》2019,14(2):435-448
This paper is concerned with obtaining an approximate solution for a linear multidimensional Volterra integral equation with a regular kernel. We choose the Gauss points associated with the multidimensional Jacobi weight function ω(x)=∏di=1(1-xi)^α(1+xi)^β,-1<α,β<1/d-1/2 (d denotes the space dimensions) as the collocation points. We demonstrate that the errors of approxima te solution decay exponentially. Numerical results are presen ted to demonstrate the effectiveness of the Jacobi spectral collocation method. 相似文献
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本文给出了数值求解一类偏积分微分方程的空间半离散Legendre-Galerkin谱方法;证明了解的稳定性及收敛性。 相似文献
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在L1空间中,研究带弱奇异核的第二类Fredholm积分方程.将弱奇异核转换成连续核,给出了一种数值求解的算法,并举出具体算例. 相似文献
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The numerical solution of linear Volterra integral equations of the second kind is discussed. The kernel of the integral equation may have weak diagonal and boundary singularities. Using suitable smoothing techniques and polynomial splines on mildly graded or uniform grids, the convergence behavior of the proposed algorithms is studied and a collection of numerical results is given. 相似文献
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In this paper, we study a nonlinear first-order singularly perturbed Volterra integro-differential equation with delay. This equation is discretized by the backward Euler for
differential part and the composite numerical quadrature formula for integral part for which
both an a priori and an a posteriori error analysis in the maximum norm are derived. Based
on the a priori error bound and mesh equidistribution principle, we prove that there exists
a mesh gives optimal first order convergence which is robust with respect to the perturbation parameter. The a posteriori error bound is used to choose a suitable monitor function
and design a corresponding adaptive grid generation algorithm. Furthermore, we extend
our presented adaptive grid algorithm to a class of second-order nonlinear singularly perturbed delay differential equations. Numerical results are provided to demonstrate the
effectiveness of our presented monitor function. Meanwhile, it is shown that the standard arc-length monitor function is unsuitable for this type of singularly perturbed delay
differential equations with a turning point. 相似文献
17.
A class of numerical methods is developed for second order Volterra integrodifferential equations by using a Legendre spectral approach.We provide a rigorous error analysis for the proposed methods,which shows that the numerical errors decay exponentially in the L∞-norm and L2-norm.Numerical examples illustrate the convergence and effectiveness of the numerical methods. 相似文献
19.
G. Q. Han & L. Q. Zhang 《计算数学(英文版)》1995,13(4):306-314
1.IntroductionThispaPerconsidersthenumericalsolutionofthesecondkindVolterraintegralequationy(t)+(Ky)(t)=g(t),(1.1)wherey(t)istheunknownsolution,g(t)isagivenfUnctionandKistheintegraJoperatorforsomegivenkernelfunctionK,(Ky)(t)=l'K(f)y(8)ids.(1.2)Suchequationsarisefromcertaindiffusionproblems.BecauseKisnotcompact,sothestandaxdstabilityproofSfornumericaJmethodsdonotfit.ManypeoplehaveworkedonHermite-typecollocationmethodsforsecond-kindVolterraintegralequationswithsmoothkernels[3,4,5'6],butver… 相似文献
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研究了带弱奇异核的抛物型积分微分方程的非协调有限元方法,在不需要Ritz-Volterra投影的情况下,在半离散和全离散的格式下分别得到了与协调有限元方法相同的误差估计. 相似文献