The Convergence of Collocation Solutions in Continuous Piecewise Polynomial Spaces for Weakly Singular Volterra Integral Equations

Hui Liang; Hermann Brunner

doi:10.1137/19M1245062

The Convergence of Collocation Solutions in Continuous Piecewise Polynomial Spaces for Weakly Singular Volterra Integral Equations

Hui Liang
, Hermann Brunner^*

^*Corresponding author for this work

Department of Mathematics

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

37 Citations (Scopus)

92 Downloads (Pure)

Abstract

Collocation solutions by globally continuous piecewise polynomials to second-kind Volterra integral equations (VIEs) with smooth kernels are uniformly convergent only for certain sets of collocation points. In this paper we establish the analogous convergence theory for VIEs with weakly singular kernels, for both uniform and graded meshes.

Original language	English
Pages (from-to)	1875-1896
Number of pages	22
Journal	SIAM Journal on Numerical Analysis
Volume	57
Issue number	4
DOIs	https://doi.org/10.1137/19M1245062
Publication status	Published - 30 Jul 2019

User-Defined Keywords

Collocation solutions
Globally continuous piecewise polynomials
Uniform convergence
Volterra integral equations
Weakly singular kernels

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10.1137/19M1245062

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title = "The Convergence of Collocation Solutions in Continuous Piecewise Polynomial Spaces for Weakly Singular Volterra Integral Equations",

abstract = "Collocation solutions by globally continuous piecewise polynomials to second-kind Volterra integral equations (VIEs) with smooth kernels are uniformly convergent only for certain sets of collocation points. In this paper we establish the analogous convergence theory for VIEs with weakly singular kernels, for both uniform and graded meshes.",

keywords = "Collocation solutions, Globally continuous piecewise polynomials, Uniform convergence, Volterra integral equations, Weakly singular kernels",

author = "Hui Liang and Hermann Brunner",

note = "Funding Information: The work of the first author was supported by the National Natural Science Foundation of China grants 11771128 and 11101130, by the China Scholarship Council grant 201708230219, and by the University Nursing Program for Young Scholars with Creative Talents in Heilongjiang Province grant UNPYSCT-2016020. The work of the second author was supported by the Hong Kong Research Grants Council GRF grants HKBU 200113 and 12300014. Publisher copyright: {\textcopyright} 2019, Society for Industrial and Applied Mathematics",

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AU - Liang, Hui

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N1 - Funding Information: The work of the first author was supported by the National Natural Science Foundation of China grants 11771128 and 11101130, by the China Scholarship Council grant 201708230219, and by the University Nursing Program for Young Scholars with Creative Talents in Heilongjiang Province grant UNPYSCT-2016020. The work of the second author was supported by the Hong Kong Research Grants Council GRF grants HKBU 200113 and 12300014. Publisher copyright: © 2019, Society for Industrial and Applied Mathematics

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N2 - Collocation solutions by globally continuous piecewise polynomials to second-kind Volterra integral equations (VIEs) with smooth kernels are uniformly convergent only for certain sets of collocation points. In this paper we establish the analogous convergence theory for VIEs with weakly singular kernels, for both uniform and graded meshes.

AB - Collocation solutions by globally continuous piecewise polynomials to second-kind Volterra integral equations (VIEs) with smooth kernels are uniformly convergent only for certain sets of collocation points. In this paper we establish the analogous convergence theory for VIEs with weakly singular kernels, for both uniform and graded meshes.

KW - Collocation solutions

KW - Globally continuous piecewise polynomials

KW - Uniform convergence

KW - Volterra integral equations

KW - Weakly singular kernels

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DO - 10.1137/19M1245062

M3 - Journal article

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JO - SIAM Journal on Numerical Analysis

JF - SIAM Journal on Numerical Analysis

IS - 4

ER -

The Convergence of Collocation Solutions in Continuous Piecewise Polynomial Spaces for Weakly Singular Volterra Integral Equations

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