Document Zbl 1023.65134

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EUR 120.56/net; sFr. 190.00 (2002). </div> <div class="abstract">As stated in its preface, this book “is designed as a new source of classical as well as modern topics on the subject of the numerical computation of linear integral equations…[discussing…] the underlying theory of integral equations and numerical analysis of numerical integration and convergence…”. It is primarily written for graduate students.<br class="zbmathjax-paragraph">Its contents is: Chapter 1: Introduction (43 pages): Classification, function spaces, Neumann series and resolvents, the Fredholm alternative. Chapter 2: Eigenvalue problems (45 pages): Linear symmetric equations, residual methods, degenerate kernel methods. Chapter 3: Equations of the second kind (15 pages): Taylor, quadrature and block methods.<br class="zbmathjax-paragraph">Chapter 4: Classical methods for second-kind Fredholm equations (20 pages). Chapter 5: Variational methods (15 pages). Chapter 6: Iteration methods (30 pages). Chapter 7: Singular equations (40 pages): Fredholm and Volterra equations with (weakly) singular kernels, equations with Hilbert kernels, finite part singular equations. Chapter 8: Weakly singular equations (40 pages): Taylor series methods (!), product integration and splines methods.<br class="zbmathjax-paragraph">Chapter 9: Cauchy singular equations (35 pages). Chapter 10: Sinc-Galerkin methods (35 pages): Sinc-Galerkin and Sinc collocation, application to single-layer and double-layer problems. Chapter 11: Equations of the first kind 50 pages). Chapter 12: Inversion of Laplace transforms (40 pages).<br class="zbmathjax-paragraph">This is followed by an appendix of four sections describing quadrature rules, orthogonal polynomials, Whittaker’s cardinal function, and singular integrals, and by a extensive bibliography.<br class="zbmathjax-paragraph">The book consists essentially of a listing of numerical methods for solving various kinds of integral equations. In spite of the mathematical machinery (functional analysis) introduced in Sections 1.3-1.5, the authors do not discuss or analyze the convergence properties of the methods. But more serious is the fact that, with very few exceptions, the presentation in this book reflects the state of the art in the numerical treatment of integral equations of about 1975. The results contained in many of the fundamental books and papers of the last twenty years appear to have had no impact and are not mentioned in the bibliography.<br class="zbmathjax-paragraph">Typical examples are the monographs by <span class="zbmathjax-textit">F. Chatelin</span> [Spectral approximation of linear operator (1983; <a href="/0517.65036">Zbl 0517.65036</a>)], by <span class="zbmathjax-textit">S. Pr枚ssdorf</span> and <span class="zbmathjax-textit">B. Silbermann</span> [Numerical analysis for integral and related operator equations (1991; <a href="/0763.65102">Zbl 0763.65102</a>)], and by <span class="zbmathjax-textit">G. Vainikko</span> [Multidimensional weakly singular integral equations (1993; <a href="/0789.65097">Zbl 0789.65097</a>)], on projection methods for integral equations and on the numerical analysis of (singular) integral equations; the papers by R.S. Anderssen, F. Natterer, H. W. Engl, and many others, and the book by <span class="zbmathjax-textit">A. Kirsch</span> [An introduction to the mathematical theory of inverse problems (1996; <a href="/0865.35004">Zbl 0865.35004</a>)], on Fredholm integral equations of the first kind; and the work by, e.g., W. Wendland, J. Saranen, I. Sloan, J. Elschner and I. Graham on numerical methods for boundary integral equations. This is complemented by many mathematical inaccuracies (e.g. in (1.2.10) where the kernel $k(x,s)$ is complex; in Section 4.6, in the description, and comparison, of Galerkin and collocation methods; the important notion of a compact (integral) operator is not introduced – although the book is intended for graduate students; etc.). Although Galerkin and collocation methods are described in a number of places, the reader is told neither about the general framework nor about the superconvergence properties of these important (projection) methods.<div class="reviewer"> Reviewer: <a href="/authors/?q=rv%3A6225">Hermann Brunner (St.John’s)</a></div> <div class="clearfix"></div></div> <div class="clear"></div> <br> <div class="citations"><div class="clear"><a href="/?q=ci%3A1862452">Cited in <strong>1</strong> Review</a></div><div class="clear"><a href="/?q=rf%3A1862452">Cited in <strong>108</strong> Documents</a></div></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%3A65-02" title="MSC2020">65-02</a> </td> <td class="space"> Research exposition (monographs, survey articles) pertaining to numerical analysis </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A45D05" title="MSC2020">45D05</a> </td> <td class="space"> Volterra integral equations </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A45Exx" title="MSC2020">45Exx</a> </td> <td class="space"> Singular integral equations </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A44A10" title="MSC2020">44A10</a> </td> <td class="space"> Laplace transform </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A65R10" title="MSC2020">65R10</a> </td> <td class="space"> Numerical methods for integral transforms </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A45C05" title="MSC2020">45C05</a> </td> <td class="space"> Eigenvalue problems 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></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3Atextbook">textbook</a>; <a href="/?q=ut%3Aeigenvalue+problems">eigenvalue problems</a>; <a href="/?q=ut%3Avariational+methods">variational methods</a>; <a href="/?q=ut%3Aiteration+methods">iteration methods</a>; <a href="/?q=ut%3Asingular+equations">singular equations</a>; <a href="/?q=ut%3Aweakly+singular+equations">weakly singular equations</a>; <a href="/?q=ut%3Aresidual+methods">residual methods</a>; <a href="/?q=ut%3Adegenerate+kernel+methods">degenerate kernel methods</a>; <a href="/?q=ut%3AFredholm+equations">Fredholm equations</a>; <a href="/?q=ut%3AVolterra+equations">Volterra equations</a>; <a href="/?q=ut%3ATaylor+series+methods">Taylor series methods</a>; <a href="/?q=ut%3Aproduct+integration">product integration</a>; <a href="/?q=ut%3Asplines+methods">splines methods</a>; <a href="/?q=ut%3Asinc-Galerkin+methods">sinc-Galerkin methods</a>; <a href="/?q=ut%3Asinc+collocation">sinc collocation</a>; <a href="/?q=ut%3ALaplace+transforms">Laplace transforms</a>; <a href="/?q=ut%3Abibliography">bibliography</a>; <a href="/?q=ut%3Aintegral+equations">integral equations</a></div><div class="keywords"> <h3>Citations:</h3><a href="/0517.65036">Zbl 0517.65036</a>; <a href="/0763.65102">Zbl 0763.65102</a>; <a href="/0789.65097">Zbl 0789.65097</a>; <a href="/0865.35004">Zbl 0865.35004</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 1023.65134" data-ciurl="/ci/01862452" data-biburl="/bibtex/01862452.bib" data-amsurl="/amsrefs/01862452.bib" data-xmlurl="/xml/01862452.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/01862452.pdf" title="Zbl 1023.65134 as PDF">Review PDF</a> </div> <div class="fulltexts"> </div> <div class="sfx" style="float: right;"> </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 = ""; 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