Blow-up behavior of collocation solutions to Hammerstein-type volterra integral equations

Z.W. Yang, Hermann Brunner

Research output: Contribution to journalArticle

6 Citations (Scopus)

Abstract

We analyze the blow-up behavior of one-parameter collocation solutions for Hammerstein-type Volterra integral equations (VIEs) whose solutions may blow up in finite time. To approximate such solutions (and the corresponding blow-up time), we will introduce an adaptive stepsize strategy that guarantees the existence of collocation solutions whose blow-up behavior is the same as the one for the exact solution. Based on the local convergence of the collocation methods for VIEs, we present the convergence analysis for the numerical blow-up time. Numerical experiments illustrate the analysis.



LanguageEnglish
Pages2260-2282
Number of pages23
JournalSIAM Journal on Numerical Analysis
Volume51
Issue number4
DOIs
Publication statusPublished - 1 Aug 2013

Fingerprint

Blow-up Time
Volterra Integral Equations
Collocation
Blow-up
Integral equations
Blow-up Solution
Local Convergence
Collocation Method
Convergence Analysis
Approximate Solution
Exact Solution
Numerical Experiment
Experiments
Strategy

Keywords

  • nonlinear Volterra integral equations
  • finite-time blow-up
  • collocation methods
  • adaptive stepsize
  • convergence of numerical blow-up time

Cite this

@article{1efcb9236ed54f04861e98ff3ad8d677,
title = "Blow-up behavior of collocation solutions to Hammerstein-type volterra integral equations",
abstract = "We analyze the blow-up behavior of one-parameter collocation solutions for Hammerstein-type Volterra integral equations (VIEs) whose solutions may blow up in finite time. To approximate such solutions (and the corresponding blow-up time), we will introduce an adaptive stepsize strategy that guarantees the existence of collocation solutions whose blow-up behavior is the same as the one for the exact solution. Based on the local convergence of the collocation methods for VIEs, we present the convergence analysis for the numerical blow-up time. Numerical experiments illustrate the analysis.",
keywords = "nonlinear Volterra integral equations, finite-time blow-up, collocation methods, adaptive stepsize, convergence of numerical blow-up time",
author = "Z.W. Yang and Hermann Brunner",
year = "2013",
month = "8",
day = "1",
doi = "10.1137/12088238X",
language = "English",
volume = "51",
pages = "2260--2282",
journal = "SIAM Journal on Numerical Analysis",
issn = "0036-1429",
number = "4",

}

Blow-up behavior of collocation solutions to Hammerstein-type volterra integral equations. / Yang, Z.W.; Brunner, Hermann.

In: SIAM Journal on Numerical Analysis, Vol. 51, No. 4, 01.08.2013, p. 2260-2282.

Research output: Contribution to journalArticle

TY - JOUR

T1 - Blow-up behavior of collocation solutions to Hammerstein-type volterra integral equations

AU - Yang, Z.W.

AU - Brunner, Hermann

PY - 2013/8/1

Y1 - 2013/8/1

N2 - We analyze the blow-up behavior of one-parameter collocation solutions for Hammerstein-type Volterra integral equations (VIEs) whose solutions may blow up in finite time. To approximate such solutions (and the corresponding blow-up time), we will introduce an adaptive stepsize strategy that guarantees the existence of collocation solutions whose blow-up behavior is the same as the one for the exact solution. Based on the local convergence of the collocation methods for VIEs, we present the convergence analysis for the numerical blow-up time. Numerical experiments illustrate the analysis.

AB - We analyze the blow-up behavior of one-parameter collocation solutions for Hammerstein-type Volterra integral equations (VIEs) whose solutions may blow up in finite time. To approximate such solutions (and the corresponding blow-up time), we will introduce an adaptive stepsize strategy that guarantees the existence of collocation solutions whose blow-up behavior is the same as the one for the exact solution. Based on the local convergence of the collocation methods for VIEs, we present the convergence analysis for the numerical blow-up time. Numerical experiments illustrate the analysis.

KW - nonlinear Volterra integral equations

KW - finite-time blow-up

KW - collocation methods

KW - adaptive stepsize

KW - convergence of numerical blow-up time

U2 - 10.1137/12088238X

DO - 10.1137/12088238X

M3 - Article

VL - 51

SP - 2260

EP - 2282

JO - SIAM Journal on Numerical Analysis

T2 - SIAM Journal on Numerical Analysis

JF - SIAM Journal on Numerical Analysis

SN - 0036-1429

IS - 4

ER -