On Volterra partial integral equations in the space of continuous functions of three variables

It is well known that any Volterra integral equation of the second kind with compact operator is uniquely solvable. Partial integral operators are not compact, even in the general case of continuous kernels. Unique solvability conditions for Volterra partial integral equations of the second kind in the space of continuous functions of three variables are considered. Conditions for a Volterra partial integral equation to be equivalent to a three-dimensional Volterra integral equation with compact operator are obtained. Continuum analogues of matrix equations for some problems of scattering theory are reduced to the Volterra partial integral equations under examination. __________ Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 38, Suzdal Conference-2004, Part 3, 2006.     

非线性第二类Volterra积分方程的Chebyshev谱配置法

我们在参考了相关文献的基础上,考察了一类非线性Volterra积分方程的Chebyshev谱配置法.方法中,我们将该类非线性方程转化为两个方程进行数值逼近.我们选择N阶Chebyshev Gauss-Lobatto点作为配置点,对积分项用N阶高斯数值积分公式逼近.收敛性分析结果表明数值误差的收敛阶为N^（1/2）-m,其中m是已知函数最高连续导数的阶数.我们也开展数值实验证实这一理论分析结果.     

Solution of linear integral equations by Gregory's method

Two techniques for using Gregory's method to solve Fredholm integral equations of the second kind are described. Since the kernel function is allowed to be mildly discontinuous, Volterra integral equations of the second kind can be solved in the same manner. Some numerical examples are given.     

Generalized reducible quadrature methods for Volterra integral and integro-differential equations

A general class of convergent methods for the numerical solution of ordinary differential equations is employed to obtain a class of convergent generalized reducible quadrature methods for Volterra integral equations of the second kind and Volterra integro-differential equations.     

Improving the accuracy of solutions of the linear second kind volterra integral equations system by using the taylor expansion method

This paper presents a new and an efficient method for determining solutions of the linear second kind Volterra integral equations system. In this method, the linear Volterra integral equations system using the Taylor series expansion of the unknown functions transformed to a linear system of ordinary differential equations. For determining boundary conditions we use a new method. This method is effective to approximate solutions of integral equations system with a smooth kernel, and a convolution kernel. An error analysis for the proposed method is provided. And illustrative examples are given to represent the efficiency and the accuracy of the proposed method.     

On Discrete Superconvergence Properties of Spline Collocation Methods for Nonlinear Volterra Integral Equations

It is shown that the error corresponding to certain spline collocation approximations for nonlinear Volterra integral equations of the second kind is the solution of a nonlinearly perturbed linear Volterra integral equation. On the basis of this result it is possible to derive general estimates for the order of convergence of the spline solution at the underlying mesh points. Extensions of these techniques to other types of Volterra equations are indicated.     

Volterra积分方程的蒙特卡罗数值求解方法

本文讨论了第一类、第二类以及具有奇异核的Volterra积分方程的数值解问题.利用重要抽样蒙特卡罗随机模拟方法获得积分方程解的近似计算结果.通过对文献中算例的实现表明文中所提方法扩展了Volterra型积分方程的数值求解方法,     

Convergence analysis of discretization methods for nonlinear first kind Volterra integral equations

Summary An existence and uniqueness result is given for nonlinear Volterra integral equations of the first kind. This permits, by means of analogous discrete manipulations, a general convergence analysis for a wide class of discretization methods for nonlinear first kind Volterra integral equations to be presented. A concept of optimal consistency allows twosided error bounds to be derived.     

Block boundary value methods for solving Volterra integral and integro-differential equations

Reducible quadrature rules generated by boundary value methods are considered in block version and applied to solve the second kind Volterra integral equations and Volterra integro-differential equations. These extended block boundary value methods are shown to possess both excellent stability properties and high accuracy for Volterra-type equations. Numerical experiments are presented and the efficiency, accuracy and stability of the schemes are confirmed.     

A numerical method based on hybrid orthonormal Bernstein and improved block-pulse functions for solving Volterra–Fredholm integral equations

This paper deals with the numerical solution of the integral equations of linear second kind Volterra–Fredholm. These integral equations are commonly used in engineering and mathematical physics to solve many of the problems. A hybrid of Bernstein and improved block-pulse functions method is introduced and used where the key point is to transform linear second-type Volterra–Fredholm integral equations into an algebraic equation structure that can be solved using classical methods. Numeric examples are given which demonstrate the related features of the process.     

Volterra Integral Equation Models for Semiconductor Devices

Van Roosbroeck's bipolar drift diffusion equations cover the qualitative behaviour of many semiconductor devices. The complexity of the model equations however prevents efficient implementations needed in circuit simulations. Under close-to-thermal-equilibrium biasing conditions (zero space charge assumption, low injection limit) the van Roosbroeck system can be replaced by a system of coupled non-linear Volterra integral equations of the second kind. Involving only the macroscopic quantities current, applied voltage and serial resistance this Volterra system can be handled with comparably little effort. Volterra integral equations models are formulated for a large class of semiconductor devices with abrupt pn-junctions. The model equations are made explicit for diodes, transistors and thyristors. A survey on various results concerning Volterra models describing the switching behaviour of pn-diodes is given. The integral equation model allows to recover all relevant properties of the voltage–current characteristics.     

ON SPECTRAL METHODS FOR VOLTERRA INTEGRAL EQUATIONS AND THE CONVERGENCE ANALYSIS

The main purpose of this work is to provide a novel numerical approach for the Volterra integral equations based on a spectral approach. A Legendre-collocation method is proposed to solve the Volterra integral equations of the second kind. We provide a rigorous error analysis for the proposed method, which indicates that the numerical errors decay exponentially provided that the kernel function and the source function are sufficiently smooth. Numerical results confirm the theoretical prediction of the exponential rate of convergence. The result in this work seems to be the first successful spectral approach （with theoretical justification） for the Volterra type equations.     

A Unified Approach to Convergence Analysis of Discretization Methods for Volterra-typeEquations

This paper presents a general convergence analysis of numericalmethods for solving ordinary differential equations and non-linerVolterra integral and integrodifferential euqations. The conceptof analytic and discrete forms is introduced, Prolongation andrestriction operators reduce the problem of comparing the fundamentalforms. Integral inequalities and their discrete analogues proofof a collocation method for solving Volterra integral equationsof the second kind.     

Rationalized Haar functions method for solving Fredholm and Volterra integral equations

This paper presents a computational technique for Fredholm integral equation of the second kind and Volterra integral equation of the second kind. The method is based upon Haar functions approximation. Properties of Rationalized Haar functions are first presented, the operational matrix of integration together with product operational matrix and Newton–Cotes nodes are utilized to reduce the computation of integral equations into some algebraic equations. The method is computationally attractive and applications are demonstrated through illustrative examples.     

Stability of discrete volterra equations of hammerstein type

Stability conditions for Volterra equations with discrete time are obtained using direct Liapunov method, without usual assumption of the summability of the series of the coeffcients. Using such conditions, the stability of some numerical methods for second kind Volterra integral equation is analyzed.     

A survey of recent advances in the numerical treatment of Volterra integral and integro-differential equations

The present survey paper samples recent advances in the numerical analysis of Volterra integral equations of the first and second kind and of integro-differential equations (including equations with weakly singular kernels); except for some important earlier references the discussion focuses on the development which has taken place during the last dozen years. A fairly extensive bibliography (selected to be representative rather than comprehensive) complements the paper.     

On the Divergence of Collocation Solutions in Smooth Piecewise Polynomial Spaces for Volterra Integral Equations

In this paper we present a characterization of those smooth piecewise polynomial collocation spaces that lead to divergent collocation solutions for Volterra integral equations of the second kind. The key to these results is an equivalence result between such collocation solutions and collocation solutions in slightly smoother spaces for initial-value problems for ordinary differential equations. For the latter problems Mülthei (1979/1980) established a complete divergence (and convergence) theory. Our analysis can be extended to furnish divergence results for smooth collocation solutions to Volterra integral equations of the first kind. AMS subject classification (2000) 65R20, 65L20, 65L60.Received May 2004. Accepted September 2004. Communicated by Tom Lyche.Hermann Brunner: This research was supported by the Natural Sciences and Engineering Research Council of Canada (NSERC).     

Stability analysis of methods employing reducible rules for volterra integral equations of the first kind

The concept of (A ₀,S)-stability for Volterra integral equations of the second kind will be extended to that of the first kind equations. We will show that stability polynomials for methods employing reducible quadrature rules, as applied to the first kind equations, can be trivially obtained from the results for the second kind equations.     

利用随机模拟方法求解第二类积分方程

给出一种求解第二类Fredholm和Volterra积分方程的数值算法,算法在数值积分技术的基础上使用Monte Carlo随机模拟方法求积分方程的近似解.通过数值例子证明了该算法是有效的.     

On some iterative numerical methods for a Volterra functional integral equation of the second kind

In this paper, we study an iterative numerical method for approximating solutions of a certain type of Volterra functional integral equations of the second kind (Volterra integral equations where both limits of integration are variables). The method uses the contraction principle and a suitable quadrature formula. Under certain conditions, we prove the existence and uniqueness of the solution and give error estimates for our approximations. We also included a numerical example which illustrates the fast approximations.