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Smoothing transformation and collocation methods for third-kind linear Volterra integral equations. (English) Zbl 1472.65168

Summary: In 2016, Sonia et al. first considered the convergence order for the third-kind linear Volterra integral equations (VIEs) based on the assumption that solutions are smooth. For the third-kind linear VIEs with nonsmooth solutions, we construct high-order numerical algorithms and discuss the convergence order. By introducing a new suitable independent variable, we obtain a transformed equation with a smooth exact solution. Then the solvability of the transformed equation is investigated on the basis of piecewise polynomial collocation methods. Meanwhile, the convergence order of the collocation solution is given. Furthermore, based on the inverse transformation, we get the convergence order of the original equation. Numerical simulations are finally presented to demonstrate the effectiveness of the theoretical results.

MSC:

65R20 Numerical methods for integral equations
45D05 Volterra integral equations
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Full Text: DOI Euclid

References:

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Zentralblatt MATH: 1433.65345
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Zentralblatt MATH: 1370.65075
Digital Object Identifier: doi:10.1016/j.apnum.2017.05.010
· Zbl 1370.65075 · doi:10.1016/j.apnum.2017.05.010
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Digital Object Identifier: doi:10.1017/S0308210500028432
· Zbl 0807.65141 · doi:10.1017/S0308210500028432
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Zentralblatt MATH: 1376.65158
Digital Object Identifier: doi:10.1216/JIE-2017-29-3-401
Project Euclid: euclid.jiea/1502676096
· Zbl 1376.65158 · doi:10.1216/JIE-2017-29-3-401
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· Zbl 1357.65105 · doi:10.1016/j.cam.2016.11.022
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Zentralblatt MATH: 1329.45001
Digital Object Identifier: doi:10.1216/JIE-2015-27-3-325
Project Euclid: euclid.jiea/1450388938
· Zbl 1329.45001 · doi:10.1216/JIE-2015-27-3-325
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Zentralblatt MATH: 1434.65320
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· Zbl 1434.65320 · doi:10.1007/s10092-019-0304-9
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