Blow-Up Behavior of Collocation Solutions to Hammerstein-Type Volterra Integral Equations

Z. W. Yang; H. Brunner

doi:10.1137/12088238X

Blow-Up Behavior of Collocation Solutions to Hammerstein-Type Volterra Integral Equations

Z. W. Yang, H. Brunner

Department of Mathematics

Research output: Contribution to journal › Journal article › peer-review

11 Citations (Scopus)

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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.

Original language	English
Pages (from-to)	2260-2282
Number of pages	23
Journal	SIAM Journal on Numerical Analysis
Volume	51
Issue number	4
DOIs	https://doi.org/10.1137/12088238X
Publication status	Published - 1 Aug 2013

Scopus Subject Areas

Numerical Analysis
Computational Mathematics
Applied Mathematics

User-Defined Keywords

Adaptive stepsize
Collocation methods
Convergence of numerical blow-up time
Finite-time blow-up
Nonlinear Volterra integral equations

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10.1137/12088238X

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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.",

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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.

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Blow-Up Behavior of Collocation Solutions to Hammerstein-Type Volterra Integral Equations

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