弱奇性Volterra积分不等式解的估计

Medved对弱奇性Gronwall型和Henry型积分不等式解的估计提出一种新方法，本文将他的方法稍加改进用来研究更广的Volterra型弱奇性线笥及非线性积分不等式解的估计，导出解的先验逐点界公式，并举例说明了结果的应用。     

一类新的弱奇性Wendroff型积分不等式解的估计

在参数更广的分布下,讨论了一类新的弱奇性Wendroff型积分不等式解的估计,所得结果推广了已有的相关结果,并将结果应用到研究微分方程解的有界性中.     

一类弱奇性Wendroff型积分不等式

在参数α,β,γ更广的分布下,给出了一类弱奇性Wendroff型积分不等式解的有限形式的界公式,所得结果推广了过去关于弱奇性Wendroff型积分不等式的相关结果,并将结果应用到微分方程中.     

一类非线性弱奇性Wendroff型积分不等式

在参数更广的分布下,讨论了一类含有两项非线性项的弱奇性Wendroff型积分不等式解的估计,所得结果推广了在相应参数不同的分布下已有的相关结果且结果更简洁、更具一般性,并将结果应用到研究微分方程解的有界性中.     

一类两个变量的弱奇性Wendroff积分不等式的推广

建立了一类有两个变量的非线性弱奇性Wendroff型积分不等式解的估计,所得结果推广了过去关于非线性弱奇性Wendroff型积分不等式的相关结果,所给出的解的估计更具有一般性.     

一类非线性时滞Volterra-Fredholm型积分不等式及其应用

在文献马庆华和J.Pecǎri,2008的基础上,建立了一个新的VolterraFredholm型非线性时滞积分不等式.把参考文献中不等式右端被积因子w(u)推广成w_1(u)u和w_1(u)w_2(u)的非线性函数.运用放大技巧、积分微分技巧、变量替换技巧、反函数技巧、常量与变量的辩证关系,给出了不等式中未知函数的估计.推广了文献中相应不等式的结果.最后,用所得结果给出了Volterra-Fredholm积分方程解的估计.     

一类新的非线性离散Bihari型不等式解的估计

本文建立了一类新的具有m项非线性离散Bihaxi型不等式解的估计,所得结果推广了过去关于非线性离散Bihari型不等式的相关结果,并用实例给出了解的估计.     

一类具有高阶奇性解的Hilbert核奇异积分方程

本文利用指数变换研究了一类解具高阶奇性的周期黎曼边值问题,通过转化法研究了一类解具有高阶奇性的Hilber核奇异积分方程,获得了相应的解和可解条件表达式.推广了Hilber核奇异积分方程的结果.     

一类非线性奇异积分方程的离散近似解及其误差估计

当函数f(x,t,u)满足一些[1]中常假定的条件时,我们可借助算子S~(N)和不等式证明非线性奇异积分方程有唯一的离散近似解,这个解可用关于距离的逐次逼近法得到.     

一类二变量积分不等式及其在积分-微分方程中的应用

建立了一类二变量的积分不等式,该不等式包含了一个一重积分和两个二重积分.利用分析技巧,给出了积分不等式中未知函数的估计.这一结果可以作为研究积分-微分方程解的定性性质的工具.     

Hermite-Type Method for Volterra Integral Equation with Certain Weakly Singular Kernel

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…     

Collocation Methods for Weakly Singular Second-kind Volterra Integral Equations with Non-smooth Solution

Collocation type methods are studied for the numerical solutionof the weakly singular Volterra integral equation of the secondkind: where the solution (t) is assumedto have the form f(t) = x(t)+r^?(t), x and being sufficientlysmooth. The solution is approximated near zero by a linear combinationof powers of t?, and away from zero by the usual polynomialrepresentation. Convergence is proved and many numerical experimentsare carried out with examples from the literature. A comparisonis made with a method of Brunner & Norsett (1981), originallydeveloped for (1) with a smooth solution. Special attentionis paid to the numerical approximation of the so-called momentintegrals which emerge in the collocation scheme.     

Generalized Jacobi Spectral-Galerkin Method for Nonlinear Volterra Integral Equations with Weakly Singular Kernels

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.     

第一类弱奇异Volterra积分方程解的渐近展开式

针对核函数和自由项代数且对数奇异的第一类线性Volterra积分方程,通过Laplace变换导出这类方程的解在零点的渐近展开式,对于方程解的奇异性质给出准确刻画.对于核函数仅代数奇异的情形,还得到方程的解在无穷远点的渐近展开式.这些展开式可以分别作为当自变量变小或变大时方程的近似解.最后,给出实例说明展开式的正确性及有...     

Fractional Multistep Methods for Weakly Singular Volterra Integral Equations of the First Kind with Perturbed Data

ABSTRACT

Fractional multistep methods were introduced by C. Lubich for the quadrature of Abel integral operators and the solution of weakly singular Volterra integral equations of the first kind with exactly given right-hand sides. In the current paper, we consider the regularizing properties of these methods to solve the mentioned integral equations of the first kind for perturbed right-hand sides. Finally, numerical results are presented.     

奇摄动Volterra型积分微分方程Robin问题

本文利用上、下解证明了Volterra型积分微分方程解的存在性。然后，应用所获得的微分不等式理论，在适当的假设下，通过构造特殊的上、下解函数，证明Volterra型 奇摄动积分微分方程解的存在性，并给出一致有效的解的渐近估计。     

A Spectral Method for Neutral Volterra Integro-Differential Equation with Weakly Singular Kernel

This paper is concerned with obtaining an approximate solution and an approximate derivative of the solution for neutral Volterra integro-differential equation with a weakly singular kernel. The solution of this equation, even for analytic data, is not smooth on the entire interval of integration. The Jacobi collocation discretization is proposed for the given equation. A rigorous analysis of error bound is also provided which theoretically justifies that both the error of approximate solution and the error of approximate derivative of the solution decay exponentially in $L^∞$ norm and weighted $L^2$ norm. Numerical results are presented to demonstrate the effectiveness of the spectral method.     

The Numerical Solution of Second-Order Weakly Singular Volterra Integro-Differential Equations

In this paper we investigate the attainable order of convergence of collocation approximations in certain polynomial spline spaces for solutions of a class of second-order volterra integro-differential equations with weakly singular kernels. While the use of quasi-uniform meshes leads, due to the nonsmooth nature of these solutions, to convergence of order less than one, regardless of the degree of the approximating spling function, collocation on suitably graded meshes will be shown to yield optimal convergence rates.