Xu, Xiaoli; Xiao, Yu; Song, Hui Ming Smoothing transformation and collocation methods for third-kind linear Volterra integral equations. (English) Zbl 1472.65168 J. Integral Equations Appl. 32, No. 3, 361-375 (2020). 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. Cited in 1 Document MSC: 65R20 Numerical methods for integral equations 45D05 Volterra integral equations Keywords:Volterra integral equations; smoothing transformation; piecewise polynomial collocation methods; solvability; convergence order PDFBibTeX XMLCite \textit{X. Xu} et al., J. Integral Equations Appl. 32, No. 3, 361--375 (2020; Zbl 1472.65168) Full Text: DOI Euclid References: [1] S. S. Allaei, Z.-W. Yang, and H. Brunner, “Collocation methods for third-kind VIEs”, IMA J. Numer. Anal. 37:3 (2017), 1104-1124. Mathematical Reviews (MathSciNet): MR3671489 Zentralblatt MATH: 1433.65345 · Zbl 1433.65345 [2] H. Beyrami, T. Lotfi, and K. Mahdiani, “Stability and error analysis of the reproducing kernel Hilbert space method for the solution of weakly singular Volterra integral equation on graded mesh”, Appl. Numer. Math. 120 (2017), 197-214. 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