Recent zbMATH articles in MSC 45Ehttps://www.zbmath.org/atom/cc/45E2021-04-16T16:22:00+00:00WerkzeugOn the approximate solution of a class of nonlinear multidimensional weakly singular integral equations.https://www.zbmath.org/1456.651812021-04-16T16:22:00+00:00"Agachev, Yu. R."https://www.zbmath.org/authors/?q=ai:agachev.yurii-romanovich"Gubaidullina, R. K."https://www.zbmath.org/authors/?q=ai:gubaidullina.r-kSummary: In this work, for a class of nonlinear weakly singular integral equations defined in the space of quadratically summable functions in a circle, the rationale for the general projection method is given. Using the obtained general results, the convergence of the well-known Galerkin method is proved.The fine error estimation of collocation methods on uniform meshes for weakly singular Volterra integral equations.https://www.zbmath.org/1456.651842021-04-16T16:22:00+00:00"Liang, Hui"https://www.zbmath.org/authors/?q=ai:liang.hui"Brunner, Hermann"https://www.zbmath.org/authors/?q=ai:brunner.hermannSummary: It is well known that for the second-kind Volterra integral equations (VIEs) with weakly singular kernel, if we use piecewise polynomial collocation methods of degree \(m\) to solve it numerically, due to the weak singularity of the solution at the initial time \(t=0\), only \(1-\alpha\) global convergence order can be obtained on uniform meshes, comparing with \(m\) global convergence order for VIEs with smooth kernel. However, in this paper, we will see that at mesh points, the convergence order can be improved, and it is better and better as \(n\) increasing. In particular, 1 order can be recovered for \(m=1\) at the endpoint. Some superconvergence results are obtained for iterated collocation methods, and a representative numerical example is presented to illustrate the obtained theoretical results.