Document Zbl 1545.65491

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(2024)</a>. </div> <div class="abstract">Summary: This paper deals with the numerical solution of <span class="zbmathjax-textit">Fredholm</span> integral equations of the second kind with endpoint algebraic singularities and with a kernel of <span class="zbmathjax-textit">Green</span>’s function type. Novel <span class="zbmathjax-textit">Nyström</span> schemes employing the <span class="zbmathjax-textit">Gauss</span> quadrature rule are presented. These methods take into account the lack of smoothness along the diagonal of the kernel and may recover the full convergence rate of smooth kernels. A complete analysis of the stability and convergence is provided, and several numerical tests that illustrate the efficiency and accuracy of various approaches are considered.</div> <div class="clear"></div> <div class="classification"> <h3>MSC:</h3> <table><tr> <td> <a class="mono" href="/classification/?q=cc%3A65R20" title="MSC2020">65R20</a> </td> <td class="space"> Numerical methods for integral equations </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A45B05" title="MSC2020">45B05</a> </td> <td class="space"> Fredholm integral equations </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A42C05" title="MSC2020">42C05</a> </td> <td class="space"> Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A65D30" title="MSC2020">65D30</a> </td> <td class="space"> Numerical integration </td> </tr></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3AFredholm+integral+equation">Fredholm integral equation</a>; <a href="/?q=ut%3AGauss+quadrature+rule">Gauss quadrature rule</a>; <a href="/?q=ut%3AJacobi+polynomial">Jacobi polynomial</a>; <a href="/?q=ut%3Ainterpolatory+projection">interpolatory projection</a>; <a href="/?q=ut%3ANystr%C3%B6m+method">Nyström method</a>; <a href="/?q=ut%3AGreen%27s+kernel">Green’s kernel</a></div>  <div class="modal fade" id="metadataModal" tabindex="-1" role="dialog" aria-labelledby="myModalLabel"> <div class="modal-dialog" role="document"> <div class="modal-content"> <div class="modal-header"> <button type="button" class="close" data-dismiss="modal" aria-label="Close"><span aria-hidden="true">×</span></button> <h4 class="modal-title" id="myModalLabel">Cite</h4> </div> <div class="modal-body"> <div class="form-group"> <label for="select-output" class="control-label">Format</label> <select id="select-output" class="form-control" aria-label="Select Metadata format"></select> </div> <div class="form-group"> <label for="metadataText" class="control-label">Result</label> <textarea class="form-control" id="metadataText" rows="10" style="min-width: 100%;max-width: 100%"></textarea> </div> <div id="metadata-alert" class="alert alert-danger" role="alert" style="display: none;">  </div> </div> <div class="modal-footer"> <button type="button" class="btn btn-primary" onclick="copyMetadata()">Copy to clipboard</button> <button type="button" class="btn btn-default" data-dismiss="modal">Close</button> </div> </div> </div> </div> <div class="functions clearfix"> <div class="function">  <a type="button" class="btn btn-default btn-xs pdf" data-toggle="modal" data-target="#metadataModal" data-itemtype="Zbl" data-itemname="Zbl 1545.65491" data-ciurl="/ci/07901431" data-biburl="/bibtex/07901431.bib" data-amsurl="/amsrefs/07901431.bib" data-xmlurl="/xml/07901431.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/07901431.pdf" title="Zbl 1545.65491 as PDF">Review PDF</a> </div> <div class="fulltexts"> <span class="fulltext">Full Text:</span> <a class="btn btn-default btn-xs" type="button" href="https://doi.org/10.1007/s10092-024-00593-1" aria-label="DOI for “Fredholm integral equations with non-smooth kernels in weighted spaces: Nyström approximations, stability and convergence”" title="10.1007/s10092-024-00593-1">DOI</a> </div> <div class="sfx" style="float: right;"> </div> </div> <div class="references"> <h3>References:</h3> <table><tr> <td>[1]</td> <td class="space">Alba, PD; Fermo, L.; Rodriguez, G., Solution of second kind Fredholm integral equations by means of Gauss and anti-Gauss quadrature rules, Numer. Math., 146, 699-728, 2020 · <a href="/1458.65158" class="nowrap">Zbl 1458.65158</a> · <a href="https://doi.org/10.1007/s00211-020-01163-7" class="nowrap">doi:10.1007/s00211-020-01163-7</a></td> </tr><tr> <td>[2]</td> <td class="space">Allouch, C., Collocation and modified collocation methods for solving second kind Fredholm integral equations in weighted spaces, Appl. Numer. Math., 198, 202-216, 2024 · <a href="/07856302" class="nowrap">Zbl 07856302</a> · <a href="https://doi.org/10.1016/j.apnum.2024.01.011" class="nowrap">doi:10.1016/j.apnum.2024.01.011</a></td> </tr><tr> <td>[3]</td> <td class="space">Allouch, C., Fast and accurate solvers for weakly singular Volterra integral equations in weighted spaces, J. Comp. Appl. Math., 438, 2024 · <a href="/1525.65133" class="nowrap">Zbl 1525.65133</a> · <a href="https://doi.org/10.1016/j.cam.2023.115535" class="nowrap">doi:10.1016/j.cam.2023.115535</a></td> </tr><tr> <td>[4]</td> <td class="space">Allouch, C.; Sbibih, D.; Tahrichi, M., Spectral approximation methods for Fredholm integral equations with non-smooth kernels, Math. Model. Anal., 27, 4, 652-667, 2022 · <a href="/1505.65313" class="nowrap">Zbl 1505.65313</a> · <a href="https://doi.org/10.3846/mma.2022.16029" class="nowrap">doi:10.3846/mma.2022.16029</a></td> </tr><tr> <td>[5]</td> <td class="space">Allouch, C.; Sbibih, D.; Tahrichi, M., Spectral approximation methods for nonlinear integral equations with non-smooth kernels, Calcolo, 60, 23, 2023 · <a href="/1514.45008" class="nowrap">Zbl 1514.45008</a> · <a href="https://doi.org/10.1007/s10092-023-00519-3" class="nowrap">doi:10.1007/s10092-023-00519-3</a></td> </tr><tr> <td>[6]</td> <td class="space">Allouch, C.; Sbibih, D.; Tahrichi, M., Superconvergent Nyström and degenerate kernel methods for integral equations of the second kind, J. int. eqns. appl., 57, 3-20, 2012 · <a href="/1453.65444" class="nowrap">Zbl 1453.65444</a></td> </tr><tr> <td>[7]</td> <td class="space">Allouch, C.; Tahrichi, M., Discrete superconvergent Nyström method for integral equations and eigenvalue problems, Math. Comput. Simul., 118, 17-29, 2015 · <a href="/1540.65541" class="nowrap">Zbl 1540.65541</a> · <a href="https://doi.org/10.1016/j.matcom.2014.11.010" class="nowrap">doi:10.1016/j.matcom.2014.11.010</a></td> </tr><tr> <td>[8]</td> <td class="space">Atkinson, KE, The Numerical Solution of Integral Equations of the Second Kind, 1997, Cambridge: Cambridge University Press, Cambridge · <a href="/0899.65077" class="nowrap">Zbl 0899.65077</a> · <a href="https://doi.org/10.1017/CBO9780511626340" class="nowrap">doi:10.1017/CBO9780511626340</a></td> </tr><tr> <td>[9]</td> <td class="space">Atkinson, KE; Potra, FA, On the Discrete Galerkin Method for Fredholm Integral Equations of the Second Kind, IMA J. Numer. Anal., 9, 385-403, 1989 · <a href="/0678.65093" class="nowrap">Zbl 0678.65093</a> · <a href="https://doi.org/10.1093/imanum/9.3.385" class="nowrap">doi:10.1093/imanum/9.3.385</a></td> </tr><tr> <td>[10]</td> <td class="space">Atkinson, KE; Potra, FA, The Discrete Galerkin Method for Nonlinear Integral Equations, IMA J. Numer. Anal., 1, 1, 385-403, 1988 · <a href="/0678.65093" class="nowrap">Zbl 0678.65093</a></td> </tr><tr> <td>[11]</td> <td class="space">Bouda, H.; Allouch, C.; Tahrichi, M., Legendre Superconvergent Degenerate Kernel and Nyström Methods for Fredholm Integral Equations, Sahand Commun. Math. Anal., 20, 1, 45-60, 2023 · <a href="/1538.65613" class="nowrap">Zbl 1538.65613</a></td> </tr><tr> <td>[12]</td> <td class="space">Chatelin, F.; Lebbar, R., Superconvergence results for the iterated projection method applied to a Fredholm integral equation of the second kind and the corresponding eigenvalue problem, J. Int. Eqns. Appl., 6, 71-91, 1984 · <a href="/0528.65074" class="nowrap">Zbl 0528.65074</a></td> </tr><tr> <td>[13]</td> <td class="space">Cubillos, O., Integral operators with Green’s function type kernel, J. Appl. Math. Comput., 10, 25-31, 1984 · <a href="/0539.65094" class="nowrap">Zbl 0539.65094</a> · <a href="https://doi.org/10.1016/0377-0427(84)90065-7" class="nowrap">doi:10.1016/0377-0427(84)90065-7</a></td> </tr><tr> <td>[14]</td> <td class="space">Ditzian, Z.; Totik, W., Moduli of smoothness, 1987, New York Berlin Heidelberg London Paris Tokyo: SCMG Springer-Verlag, New York Berlin Heidelberg London Paris Tokyo · <a href="/0666.41001" class="nowrap">Zbl 0666.41001</a> · <a href="https://doi.org/10.1007/978-1-4612-4778-4" class="nowrap">doi:10.1007/978-1-4612-4778-4</a></td> </tr><tr> <td>[15]</td> <td class="space">Ditzian, Z.; Totik, W., Remarks on Besov spaces and best polynomial approximation, Proc. Am. Math. Soc., 104, 4, 1059-1066, 1988 · <a href="/0692.41008" class="nowrap">Zbl 0692.41008</a> · <a href="https://doi.org/10.1090/S0002-9939-1988-0969047-5" class="nowrap">doi:10.1090/S0002-9939-1988-0969047-5</a></td> </tr><tr> <td>[16]</td> <td class="space">Fermo, L.; Russo, MG, Numerical methods for Fredholm integral equations with singular right-hand sides, Adv. Comput. Math., 33, 305-330, 2010 · <a href="/1204.65153" class="nowrap">Zbl 1204.65153</a> · <a href="https://doi.org/10.1007/s10444-009-9137-4" class="nowrap">doi:10.1007/s10444-009-9137-4</a></td> </tr><tr> <td>[17]</td> <td class="space">Kulkarni, RP, On improvement of the iterated Galerkin solution of the second kind integral equations, J. Numer. Math., 13, 205-218, 2005 · <a href="/1088.65115" class="nowrap">Zbl 1088.65115</a> · <a href="https://doi.org/10.1515/156939505774286139" class="nowrap">doi:10.1515/156939505774286139</a></td> </tr><tr> <td>[18]</td> <td class="space">Mastroianni, G.; Milovanovic, GV, Interpolation Processes: Basic Theory and Applications, 2008, Berlin: Springer Monographs in Mathematics. Springer, Berlin · <a href="/1154.41001" class="nowrap">Zbl 1154.41001</a> · <a href="https://doi.org/10.1007/978-3-540-68349-0" class="nowrap">doi:10.1007/978-3-540-68349-0</a></td> </tr><tr> <td>[19]</td> <td class="space">Mastroianni, G.; Russo, MG; Themistoclakis, W., Numerical Methods for Cauchy Singular Integral Equations in Spaces of Weighted Continuous Functions, Oper. Theory Adv. Appl., 160, 311-336, 2005 · <a href="/1084.65131" class="nowrap">Zbl 1084.65131</a></td> </tr><tr> <td>[20]</td> <td class="space">Mezzanotte, D.; Occorsio, D.; Russo, MG, Combining Nyström Methods for a Fast Solution of Fredholm Integral Equations of the Second Kind, Mathematics, 9, 21, 26-52, 2021 · <a href="https://doi.org/10.3390/math9212652" class="nowrap">doi:10.3390/math9212652</a></td> </tr><tr> <td>[21]</td> <td class="space">Nevai, P., Mean convergence of Lagrange interpolation, III, Trans. Amer. Math. Soc., 282, 2, 669-698, 1984 · <a href="/0577.41001" class="nowrap">Zbl 0577.41001</a> · <a href="https://doi.org/10.1090/S0002-9947-1984-0732113-4" class="nowrap">doi:10.1090/S0002-9947-1984-0732113-4</a></td> </tr><tr> <td>[22]</td> <td class="space">Occorsio, D.; Russo, MG, A mixed collocation scheme for solving second kind Fredholm integral equations in $[-1,1]$, Electron. Trans. Numer. (ETNA), 54, 443-459, 2021 · <a href="/1470.65216" class="nowrap">Zbl 1470.65216</a> · <a href="https://doi.org/10.1553/etna_vol54s443" class="nowrap">doi:10.1553/etna_vol54s443</a></td> </tr><tr> <td>[23]</td> <td class="space">Sloan, IH, A quadrature-based approach to improving the collocation method, Numer. Math., 54, 41-56, 1988 · <a href="/0668.65111" class="nowrap">Zbl 0668.65111</a> · <a href="https://doi.org/10.1007/BF01403890" class="nowrap">doi:10.1007/BF01403890</a></td> </tr><tr> <td>[24]</td> <td class="space">Szegő, G.: Orthogonal Polynomials, Amer. Math. Soc. Colloq. Publ. 23, 4th ed., Providence, RI, (1975) · <a href="/0305.42011" class="nowrap">Zbl 0305.42011</a></td> </tr><tr> <td>[25]</td> <td class="space">Sloan, I., Improvement by iteration for compact operator equations, Math. Comput., 30, 758-764, 1976 · <a href="/0343.45010" class="nowrap">Zbl 0343.45010</a> · <a href="https://doi.org/10.1090/S0025-5718-1976-0474802-4" class="nowrap">doi:10.1090/S0025-5718-1976-0474802-4</a></td> </tr></table> <div class="reference_disclaimer"> This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching. </div> </div></article> </div></div> </div> </div> <div class="clearfix"></div> </div> </div> <div id="foot"><div class="copyright"> © 2025 <a target="fiz" href="https://www.fiz-karlsruhe.de/en">FIZ Karlsruhe GmbH</a> <a href="/privacy-policy/">Privacy Policy</a> <a href="/legal-notices/">Legal Notices</a> <a href="/terms-conditions/">Terms & Conditions</a> <div class="info"> <ul class="nav"> <li class="mastodon"> <a href="https://mathstodon.xyz/@zbMATH" target="_blank" class="no-new-tab-icon"> <img src="/static/mastodon.png" title="zbMATH at Mathstodon (opens in new tab)" alt="Mastodon logo"> </a> </li> </ul> </div> </div> <div class="clearfix" style="height: 0px;"></div> </div> </div> <script src="https://static.zbmath.org/contrib/jquery/1.9.1/jquery.min.js"></script> <script src="https://static.zbmath.org/contrib/jquery-caret/1.5.2/jquery.caret.min.js"></script> <script src="/static/js/jquery-ui-1.10.1.custom.min.js"></script> <script src="https://static.zbmath.org/contrib/bootstrap/v3.3.7zb1/js/bootstrap.min.js"></script> <script src="https://static.zbmath.org/contrib/bootstrap-lightbox/v0.7.0/bootstrap-lightbox.min.js"></script> <script src="https://static.zbmath.org/contrib/retina/unknown/retina.js"></script> <script src="https://static.zbmath.org/contrib/bootstrap-select/v1.13.14/js/bootstrap-select.min.js"></script> <script> var SCRIPT_ROOT = ""; </script> <script src="/static/scripts.js?v=20240926"> </script> <script src="https://static.zbmath.org/contrib/mathjax/2.7.1/MathJax.js?config=TeX-AMS-MML_HTMLorMML"></script> <script type="text/x-mathjax-config"> MathJax.Hub.Config({ "HTML-CSS": { preferredFont: "TeX", availableFonts: [ "STIX", "TeX" ], linebreaks: { automatic: true }, EqnChunk: (MathJax.Hub.Browser.isMobile ? 10 : 50) }, tex2jax: { processEscapes: true, ignoreClass: "tex2jax_ignore|dno" }, TeX: { Macros: { Aut: "\\operatorname{Aut}", Hom: "\\operatorname{Hom}" }, noUndefined: { attributes: { mathcolor: "#039", //"red", mathbackground: "white", //"#FFEEEE", mathsize: "90%" } } }, messageStyle: "none" }); </script> <script type="text/javascript"> $(document).ready(function() { $("#MathInput").stop(true, true).keyup(function() { $.ajax({ url: "/mwsq/", type: "POST", data: { query : $("#MathInput").val() }, dataType: "text" }) .done(function(xml) { $("#MathPreview").html(xml); $(window).resize(); }); }); var press = jQuery.Event("keyup"); press.ctrlKey = false; press.which = 40; $("#MathInput").trigger(press); }); </script> <div id="new_tab_icon" style="display: none"> <span class="glyphicon glyphicon-new-window" aria-hidden="true"></span><span class="sr-only">(opens in new tab)</span></div> </body> </html>

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