A survey of singularly perturbed Volterra equations
type: Item
instance of: scholarly article
title: A survey of singularly perturbed Volterra equations
zbMATH Open document ID: 0883.45003
DOI: 10.1016/S0168-9274(97)00014-7
published in: Applied Numerical Mathematics
publication date: 1997-07-17T00:00:00Z
review text: The paper contains a survey of results concerning the second kind Volterra integral equations of the form \[ \varepsilon y(t)=g(t)+ \int_0^t k(t,s,y(s))ds, \qquad 0\leq t\leq T,\tag{1} \] and the Volterra integro-differential equation of the form \[ \varepsilon y'(t)= f\Biggl( t,y(t), \int_0^t k(t,s,y(s))ds \Biggr), \quad 0<t\leq T, \qquad y(0)=y_0.\tag{2} \] Both in (1) and (2) \(\varepsilon\) is a parameter from the interval \((0,1]\). Several results concerning the existence, uniqueness, qualitative properties and asymptotic expansion of solutions of the problems (1) and (2) are presented and described. A review of numerical methods for singularly perturbed Volterra equations is also added. Moreover, the paper contains two original results associated with asymptotic expansions of solutions of linear convolution equations and with a numerical method for the solution of weakly singular Volterra integral equations.
Mathematics Subject Classification ID: 45G10
Mathematics Subject Classification ID: 45G05
Mathematics Subject Classification ID: 45J05
Mathematics Subject Classification ID: 65R20
Mathematics Subject Classification ID: 45M05
zbMATH DE Number: 1036647
zbMATH Keywords: nonlinear Volterra equations
zbMATH Keywords: singular perturbation
zbMATH Keywords: integro-differential equation
zbMATH Keywords: asymptotic expansion
zbMATH Keywords: linear convolution equations
zbMATH Keywords: weakly singular
author: J.-P. Kauthen
reviewed by: Józef Banaś
describes a project that uses: RODAS
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1036647