Multistep Hermite collocation methods for solving Volterra integral equations
type: Item
instance of: scholarly article
title: Multistep Hermite collocation methods for solving Volterra integral equations
zbMATH Open document ID: 1247.65164
author: Somayyeh Fazeli
author: Gholamreza Hojjati
author: Sedaghat Shahmorad
published in: Numerical Algorithms
publication date: 2012-05-08T00:00:00Z
review text: A new method is presented to obtain numerical solutions of a class of nonlinear Volterra integral equations of the form \[ y(t)=g(t)+\int_0^t K(t,s,y(s))ds,\quad t\in I:=[0,T],\tag{\(*\)} \] where \(g:I\to \mathbb R\) is sufficiently smooth and \(K:D\times \mathbb R\to \mathbb R\), with \(D:=\{(t,s):0\leq s\leq t\leq T\}\), is continuous and is Lipschitzian in \(y\). In this paper the authors use (\(*\)) and \[ y'(t)=g'(t)+\int_0^t K_t(t,s,y(s))ds+K(t,t,y(t)),\quad t\in I, \] so that the approximate solution in each subinterval depends on the values of the approximated solution and its first derivative in the fixed number \(r\) of previous time steps and also the values of the approximate solution and its first derivative in the \(m\) collocation points. This technique enables them to find higher order methods with extensive stability region. This new method, which is called multistep Hermite collocation method, has the uniform convergence order \(2m+2r\) for \(r\) steps and \(m\) collocation points. Some numerical examples are presented to show the efficiency of the method.
Mathematics Subject Classification ID: 65R20
Mathematics Subject Classification ID: 45D05
Mathematics Subject Classification ID: 45G10
zbMATH DE Number: 6031711
zbMATH Keywords: multistep collocation methods
zbMATH Keywords: Hermite interpolation
zbMATH Keywords: stability
zbMATH Keywords: convergence
zbMATH Keywords: nonlinear Volterra integral equation
zbMATH Keywords: numerical examples
MaRDI profile type: Publication
full work available at URL: s11075-011-9510-5
OpenAlex ID: W2026610032
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DOI: 10.1007/S11075-011-9510-5
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description: scientific article; zbMATH DE number 6031711