{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,1]],"date-time":"2026-03-01T12:18:12Z","timestamp":1772367492281,"version":"3.50.1"},"reference-count":12,"publisher":"Cambridge University Press (CUP)","issue":"2","license":[{"start":{"date-parts":[[2011,11,14]],"date-time":"2011-11-14T00:00:00Z","timestamp":1321228800000},"content-version":"unspecified","delay-in-days":6526,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Proceedings of the Royal Society of Edinburgh: Section A Mathematics"],"published-print":{"date-parts":[[1994]]},"abstract":"<jats:p>In this paper it is shown that the use of uniform meshes leads to optimal convergence rates provided that the analytic solutions of a particular class of Volterra integral equations (VIEs) are smooth. If the exact solutions are not smooth, however, suitable transformations can be made so that the new VIEs possess smooth solutions. Spline collocation methods with uniform meshes applied to these new VIEs are then shown to be able to yield optimal (global) convergence rates. The general theory is applied to a typical case, i.e. the integral kernels consisting of the singular term (<jats:italic>t<\/jats:italic> \u2212 <jats:italic>s<\/jats:italic>) <jats:sup>\u2212\u00bd.<\/jats:sup><\/jats:p>","DOI":"10.1017\/s0308210500028432","type":"journal-article","created":{"date-parts":[[2011,11,14]],"date-time":"2011-11-14T08:10:50Z","timestamp":1321258250000},"page":"199-210","source":"Crossref","is-referenced-by-count":51,"title":["Collocation methods for second-kind Volterra integral equations with weakly singular kernels"],"prefix":"10.1017","volume":"124","author":[{"given":"Teresa","family":"Diogo","sequence":"first","affiliation":[]},{"given":"Sean","family":"McKee","sequence":"additional","affiliation":[]},{"given":"Tao","family":"Tang","sequence":"additional","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2011,11,14]]},"reference":[{"key":"S0308210500028432_ref001","unstructured":"1 Abdalkhani J. . Collocation and Runge\u2013Kutta-type methods for Volterra integral equations with weakly singular kernels (Ph.D. Thesis, Dalhousie University, Halifax, N.S., 1982)."},{"key":"S0308210500028432_ref011","doi-asserted-by":"publisher","DOI":"10.1017\/S0308210500018485"},{"key":"S0308210500028432_ref008","doi-asserted-by":"publisher","DOI":"10.1002\/zamm.19820620902"},{"key":"S0308210500028432_ref004","first-page":"57","article-title":"The approximate solution of Volterra equations with nonsmooth solutions","volume":"27","author":"Brunner","year":"1985","journal-title":"Utilitas Math."},{"key":"S0308210500028432_ref003","doi-asserted-by":"publisher","DOI":"10.1090\/S0025-5718-1985-0804933-3"},{"key":"S0308210500028432_ref009","doi-asserted-by":"publisher","DOI":"10.1137\/0502022"},{"key":"S0308210500028432_ref012","volume-title":"On collocation methods for second-kind Volterra integral equations with weakly singular kernels","author":"Tang","year":"1990"},{"key":"S0308210500028432_ref002","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9939-1969-0234228-3"},{"key":"S0308210500028432_ref010","doi-asserted-by":"publisher","DOI":"10.1017\/S0308210500018473"},{"key":"S0308210500028432_ref007","doi-asserted-by":"publisher","DOI":"10.1002\/zamm.19860661107"},{"key":"S0308210500028432_ref005","volume-title":"The Numerical Solutions of Volterra Equations","author":"Brunner","year":"1986"},{"key":"S0308210500028432_ref006","doi-asserted-by":"publisher","DOI":"10.1016\/0377-0427(84)90027-X"}],"container-title":["Proceedings of the Royal Society of Edinburgh: Section A Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0308210500028432","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,15]],"date-time":"2019-05-15T15:38:38Z","timestamp":1557934718000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0308210500028432\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1994]]},"references-count":12,"journal-issue":{"issue":"2","published-print":{"date-parts":[[1994]]}},"alternative-id":["S0308210500028432"],"URL":"https:\/\/doi.org\/10.1017\/s0308210500028432","relation":{},"ISSN":["0308-2105","1473-7124"],"issn-type":[{"value":"0308-2105","type":"print"},{"value":"1473-7124","type":"electronic"}],"subject":[],"published":{"date-parts":[[1994]]}}}