Fully discretized collocation methods for nonlinear singular Volterra integral equations
type: Item
instance of: scholarly article
title: Fully discretized collocation methods for nonlinear singular Volterra integral equations
zbMATH Open document ID: 1269.65136
author: M. Teresa Diogo
author: Jingtang Ma
author: Magda Rebelo
published in: Journal of Computational and Applied Mathematics
publication date: 2013-04-18T00:00:00Z
review text: The authors consider a nonlinear weakly singular Volterra integral equation as \[ F^4(t) = -\frac{1}{2\sqrt{t}}\int\limits^t_0 \frac{F'(s)}{(t^{3/2}-s^{3/2})^{1/3}}ds, ~\, t > 0. \tag{1} \] A series expansion for the solution is obtained and shown to be convergent in a neighbourhood of the origin. Owing to the singularity of the solution at the origin, the global convergence order of product integration and collocation methods is not optimal. However, the optimal orders can be recovered if fully discretized collocation methods based on graded meshes are used. A theoretical proof is given and some numerical results are presented which illustrate the performance of the methods.
Mathematics Subject Classification ID: 65R20
Mathematics Subject Classification ID: 45D05
Mathematics Subject Classification ID: 45G05
zbMATH DE Number: 6155443
zbMATH Keywords: cordial equation
zbMATH Keywords: fully discretized collocation methods
zbMATH Keywords: graded meshes
zbMATH Keywords: nonlinear weakly singular Volterra integral equation
zbMATH Keywords: series expansion
zbMATH Keywords: global convergence
zbMATH Keywords: product integration
zbMATH Keywords: numerical results
reviewed by: Ilia V. Boikov
MaRDI profile type: Publication
full work available at URL: j.cam.2013.01.002
OpenAlex ID: W2035685331
DOI: 10.1016/J.CAM.2013.01.002
description: scientific article; zbMATH DE number 6155443