Fast numerical solution of weakly singular Volterra integral equations
type: Item
instance of: scholarly article
title: Fast numerical solution of weakly singular Volterra integral equations
zbMATH Open document ID: 0654.65091
DOI: 10.1016/0377-0427(88)90332-9
author: M. Schlichte
author: Ernst Hairer
author: Christian Lubich
published in: Journal of Computational and Applied Mathematics
publication date: 1988-01-01T00:00:00Z
review text: The authors present an algorithm for the solution of nonlinear, weakly singular Volterra integral equations of the form \[ y(t)=f(t)+1/\sqrt{\pi}\int^{t}_{0}(t-s)^{-1/2}k(t-s)g(s,y(s))ds, \] whose solution is assumed to be a smooth function of \(\sqrt{t}\). This algorithm is based on the fractional backward differentiation formula of order four introduced by the second author [Math. Comput. 45, 463-469 (1985; Zbl 0584.65090)]. The resulting discrete equations are solved by Newton's method; FFT techniques are employed in the generation of the quadrature weights and the computation of the convolution terms. Two numerical illustrations and the listing of the FORTRAN subroutine ABELFT conclude the paper.
Mathematics Subject Classification ID: 65R20
Mathematics Subject Classification ID: 45G05
zbMATH DE Number: 4068227
zbMATH Keywords: Abel-Volterra convolution
zbMATH Keywords: integral equation
zbMATH Keywords: fractional multistep method
zbMATH Keywords: numerical examples
zbMATH Keywords: fast Fourier transform techniques
zbMATH Keywords: nonlinear, weakly singular Volterra integral equations
zbMATH Keywords: fractional backward differentiation formula
zbMATH Keywords: Newton's method
zbMATH Keywords: FORTRAN subroutine
reviewed by: Götz Brunner
MaRDI profile type: Publication
full work available at URL: 0377-0427(88)90332-9
OpenAlex ID: W2021616075
cites work: Fast Numerical Solution of Nonlinear Volterra Convolution Equations
cites work: Q3699980
cites work: A nonlinear Volterra equation arising in the theory of superfluidity
cites work: Discretized Fractional Calculus
cites work: Fractional Linear Multistep Methods for Abel-Volterra Integral Equations of the Second Kind
cites work: Q3732871
cites work: Fractional Linear Multistep Methods for Abel-Volterra Integral Equations of the First Kind
description: scientific article; zbMATH DE number 4068227