A perspective on the numerical treatment of Volterra equations
type: Item
instance of: scholarly article
title: A perspective on the numerical treatment of Volterra equations
zbMATH Open document ID: 0976.65121
DOI: 10.1016/S0377-0427(00)00470-2
author: Christopher T. H. Baker
published in: Journal of Computational and Applied Mathematics
publication date: 2001-02-18T00:00:00Z
review text: Following a short introduction to the basic theory of Volterra integral and integro-differential equations, the author surveys the principle classes of of numerial methods for such functional equations: these include methods based on elementary quadrature formulas; Runge-Kutta type methods; and (exact and discretized) collocation methods. The subsequent discussion of the analysis of these numerical methods focuses on their stability properties. An extensive list of references complements this survey.
Mathematics Subject Classification ID: 65R20
Mathematics Subject Classification ID: 45G10
Mathematics Subject Classification ID: 45J05
Mathematics Subject Classification ID: 45E10
Mathematics Subject Classification ID: 45D05
zbMATH DE Number: 1566014
zbMATH Keywords: Volterra and Abel integral equations
zbMATH Keywords: quadrature methods
zbMATH Keywords: survey paper
zbMATH Keywords: integro-differential equations
zbMATH Keywords: Runge-Kutta type methods
zbMATH Keywords: collocation methods
zbMATH Keywords: stability
reviewed by: H. Brunner
describes a project that uses: Algorithm 689
describes a project that uses: Algorithm 627
MaRDI profile type: Publication
cites work: Q4358706
cites work: FFT techniques in the numerical solution of convolution equations
cites work: Q3755937
cites work: Q3798129
cites work: Stability analysis of continuous implicit Runge-Kutta methods for Volterra integro-differential systems with unbounded delays
cites work: Q3964484
cites work: Stability Regions in the Numerical Treatment of Volterra Integral Equations
cites work: Regions of Stability in the Numerical Treatment of Volterra Integro-Differential Equations
cites work: Stability analysis of Runge–Kutta methods applied to a basic Volterra integral equation
cites work: On some variation of parameter methods for integrodifferential, integral, and quasilinear partial integrodifferential equations
cites work: Natural Continuous Extensions of Runge-Kutta Methods for Volterra Integral Equations of the Second Kind and Their Applications
cites work: A stability analysis of the trapezoidal method for Volterra integral equations with completely positive kernels
cites work: A nonlinear variation of constants method for integro differential and integral equations
cites work: The Numerical Solution of Nonlinear Volterra Integral Equations of the Second Kind by Collocation and Iterated Collocation Methods
cites work: Stability of collocation for weakly singular Volterra equations
cites work: Stability results for collocation methods for Volterra integral equations
cites work: Algorithm 689: Discretized collocation and iterated collocation for nonlinear Volterra integral equations of the second kind
cites work: Theory and performance of a subroutine for solving Volterra integral equations
cites work: Algorithm 627
cites work: A nonlinear variation of constants formula for volterra equations
cites work: Q3880258
cites work: Q3942318
cites work: A survey of recent advances in the numerical treatment of Volterra integral and integro-differential equations
cites work: Nonpolynomial Spline Collocation for Volterra Equations with Weakly Singular Kernels
cites work: The Numerical Solution of Weakly Singular Volterra Integral Equations by Collocation on Graded Meshes
cites work: Q3677107
cites work: Q3221350
cites work: High-order methods for the numerical solution of Volterra integro- differential equations
cites work: Polynomial Spline Collocation Methods for Volterra Integrodifferential Equations with Weakly Singular Kernels
cites work: Implicit Runge-Kutta-Nyström methods for general second-order Volterra integro differential equations
cites work: The approximate solution of initial-value problems for general Volterra integro-differential equations
cites work: The numerical treatment of Volterra integro-differential equations with unbounded delay
cites work: Collocation Methods for Nonlinear Volterra Integro-Differential Equations with Infinite Delay
cites work: Q3991268
cites work: Implicitly linear collocation methods for nonlinear Volterra equations
cites work: Q4022164
cites work: Open problems in the discretization of volterra integral equations
cites work: 1896--1996: One hundred years of Volterra integral equations of the first kind
cites work: Runge-Kutta Theory for Volterra Integral Equations of the Second Kind
cites work: Geometric meshes in collocation methods for Volterra integral equations with proportional delays
cites work: Higher accuracy methods for second-kind Volterra integral equations based on asymptotic expansions of iterated Galerkin methods
cites work: On mixed collocation methods for Volterra integral equations with periodic solution
cites work: The piecewise polynomial collocation method for nonlinear weakly singular Volterra equations
cites work: The iterative correction method for Volterra integral equations
cites work: On global superconvergence of iterated collocation solutions to linear second-kind Volterra integral equations
cites work: Q3750978
cites work: Stability of numerical methods for Volterra integro-differential equations
cites work: Superconvergence of collocation methods for Volterra and Abel integral equations of the second kind
cites work: Volterra integral and differential equations
cites work: Numerical solution of non-linear Volterra integral equations
cites work: On the stability of Volterra integral equations with a lagging argument
cites work: Q3213489
cites work: Equations with unbounded delay: a survey
cites work: Q4933708

A perspective on the numerical treatment of Volterra equations (cites work)
cites work: Stability of difference Volterra equations: Direct Lyapunov method and numerical procedure

A perspective on the numerical treatment of Volterra equations (cites work)
cites work: Boundedness of discrete Volterra equations

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cites work: Stability of continuous and discrete Volterra integro-differential equations by Liapunov approach

A perspective on the numerical treatment of Volterra equations (cites work)
cites work: Stability of collocation methods for Volterra integro-differential equations

A perspective on the numerical treatment of Volterra equations (cites work)
cites work: Q4009295

A perspective on the numerical treatment of Volterra equations (cites work)
cites work: Stability Results for One-Step Discretized Collocation Methods in the Numerical Treatment of Volterra Integral Equations

A perspective on the numerical treatment of Volterra equations (cites work)
cites work: Convergence analysis of discretization methods for nonlinear first kind Volterra integral equations

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

A perspective on the numerical treatment of Volterra equations (cites work)
cites work: Improving the accuracy of collocation solutions of Volterra integral equations of the first kind by local interpolation

A perspective on the numerical treatment of Volterra equations (cites work)
cites work: On monotone Abel-Volterra integral equations on the half line

A perspective on the numerical treatment of Volterra equations (cites work)
cites work: Uniform Error Estimates of Galerkin Methods for Monotone Abel-Volterra Integral Equations on the Half-Line

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cites work: Qualitative behaviour and stability of solutions of discretised nonlinear Volterra integral equations of convolution type

A perspective on the numerical treatment of Volterra equations (cites work)
cites work: Q4955280

A perspective on the numerical treatment of Volterra equations (cites work)
cites work: Nonlinear Volterra integro-differential equations -- stability and numerical stability of \(\theta\)-methods

A perspective on the numerical treatment of Volterra equations (cites work)
cites work: Fractional Calculus: Integral and Differential Equations of Fractional Order

A perspective on the numerical treatment of Volterra equations (cites work)
cites work: Abel integral equations. Analysis and applications

A perspective on the numerical treatment of Volterra equations (cites work)
cites work: Q4692887

A perspective on the numerical treatment of Volterra equations (cites work)
cites work: On the Stability of Volterra–Runge–Kutta Methods

A perspective on the numerical treatment of Volterra equations (cites work)
cites work: Order of Convergence of One-Step Methods for Volterra Integral Equations of the Second Kind

A perspective on the numerical treatment of Volterra equations (cites work)
cites work: Fast numerical solution of weakly singular Volterra integral equations

A perspective on the numerical treatment of Volterra equations (cites work)
cites work: Fast Numerical Solution of Nonlinear Volterra Convolution Equations

A perspective on the numerical treatment of Volterra equations (cites work)
cites work: Convergence and stability results in Runge-Kutta type methods for Volterra integral equations of the second kind

A perspective on the numerical treatment of Volterra equations (cites work)
cites work: Backward differentiation type formulas for Volterra integral equations of the second kind

A perspective on the numerical treatment of Volterra equations (cites work)
cites work: On the stability of multistep formulas for Volterra integral equations of the second kind

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cites work: Convergence and Stability Analysis for Modified Runge—Kutta Methods in the Numerical Treatment of Second-Kind Volterra Integral Equations

A perspective on the numerical treatment of Volterra equations (cites work)
cites work: Superconvergence of Numerical Solutions to Volterra Integral Equations with Singularities

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cites work: Variable Step Size Predictor-Corrector Schemes for Second Kind Volterra Integral Equations

A perspective on the numerical treatment of Volterra equations (cites work)
cites work: Implicit Runge-Kutta methods for some integrodifferential-algebraic equations

A perspective on the numerical treatment of Volterra equations (cites work)
cites work: Implicit Runge-Kutta methods for singularly perturbed integro-differential systems

A perspective on the numerical treatment of Volterra equations (cites work)
cites work: A survey of singularly perturbed Volterra equations

A perspective on the numerical treatment of Volterra equations (cites work)
cites work: Continuous collocation approximations to solutions of first kind Volterra equations

A perspective on the numerical treatment of Volterra equations (cites work)
cites work: Q3314929

A perspective on the numerical treatment of Volterra equations (cites work)
cites work: Method of Variation of Parameters for Dynamic Systems

A perspective on the numerical treatment of Volterra equations (cites work)
cites work: Q4872463

A perspective on the numerical treatment of Volterra equations (cites work)
cites work: Q3681889

A perspective on the numerical treatment of Volterra equations (cites work)
cites work: Fractional Linear Multistep Methods for Abel-Volterra Integral Equations of the First Kind

A perspective on the numerical treatment of Volterra equations (cites work)
cites work: A Stability Analysis of Convolution Quadraturea for Abel-Volterra Integral Equations

A perspective on the numerical treatment of Volterra equations (cites work)
cites work: Runge-Kutta theory for Volterra integrodifferential equations

A perspective on the numerical treatment of Volterra equations (cites work)
cites work: Runge-Kutta Theory for Volterra and Abel Integral Equations of the Second Kind

A perspective on the numerical treatment of Volterra equations (cites work)
cites work: Fractional Linear Multistep Methods for Abel-Volterra Integral Equations of the Second Kind

A perspective on the numerical treatment of Volterra equations (cites work)
cites work: On the Stability of Linear Multistep Methods for Volterra Convolution Equations

A perspective on the numerical treatment of Volterra equations (cites work)
cites work: Q3968435

A perspective on the numerical treatment of Volterra equations (cites work)
cites work: Nonsmooth data error estimates for approximations of an evolution equation with a positive-type memory term

A perspective on the numerical treatment of Volterra equations (cites work)
cites work: Generalised Discrete Gronwall Lemmas

A perspective on the numerical treatment of Volterra equations (cites work)
cites work: Q3811663

A perspective on the numerical treatment of Volterra equations (cites work)
cites work: Q3038572

A perspective on the numerical treatment of Volterra equations (cites work)
cites work: The repetition factor and numerical stability of Volterra integral equations

A perspective on the numerical treatment of Volterra equations (cites work)
cites work: Admissibility and Nonlinear Volterra Integral Equations

A perspective on the numerical treatment of Volterra equations (cites work)
cites work: Q4127960

A perspective on the numerical treatment of Volterra equations (cites work)
cites work: Q3968436

A perspective on the numerical treatment of Volterra equations (cites work)
cites work: Analysis of explosion for nonlinear Volterra equations

A perspective on the numerical treatment of Volterra equations (cites work)
cites work: Solving Volterra Integral Equations with ODE Codes

A perspective on the numerical treatment of Volterra equations (cites work)
cites work: Applications and numerical analysis of partial differential Volterra equations: A brief survey

A perspective on the numerical treatment of Volterra equations (cites work)
cites work: Volterra integral equations

A perspective on the numerical treatment of Volterra equations (cites work)
cites work: Q4241037

A perspective on the numerical treatment of Volterra equations (cites work)
cites work: On the Numerical Stability of Reducible Quadrature Methods for Second Kind Volterra Integral Equations

A perspective on the numerical treatment of Volterra equations (cites work)
cites work: The Construction of Reducible Quadrature Rules for Volterra Integral and Integro-differential Equations

A perspective on the numerical treatment of Volterra equations (cites work)
cites work: Q3897985