Document Zbl 1133.35436

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Tomas</a>; <a href="/authors/lesnic.daniel" title="Author Profile">Lesnic, Daniel</a></div> <h2 class="title"> <strong>A procedure for determining a spacewise dependent heat source and the initial temperature.</strong> <i>(English)</i> <a class="label nowrap" href="/1133.35436">Zbl 1133.35436</a> </h2> <div class="source"> <a href="/serials/26" title="Journal Profile">Appl. Anal.</a> <a href="/?q=in%3A218615" title="Articles in this Issue">87, No. 3, 265-276 (2008)</a>. </div> <div class="abstract">Summary: The inverse problem of determining a spacewise dependent heat source, together with the initial temperature for the parabolic heat equation, using the usual conditions of the direct problem and information from two supplementary temperature measurements at different instants of time is studied. These spacewise dependent temperature measurements ensure that this inverse problem has a unique solution, despite the solution being unstable, hence the problem is ill-posed. We propose an iterative algorithm for the stable reconstruction of both the initial data and the source based on a sequence of well-posed direct problems for the parabolic heat equation, which are solved at each iteration step using the boundary element method. The instability is overcome by stopping the iterations at the first iteration for which the discrepancy principle is satisfied. Numerical results are presented for a typical benchmark test example, which has the input measured data perturbed by increasing amounts of random noise. The numerical results show that the proposed procedure gives accurate numerical approximations in relatively few iterations.</div> <div class="clear"></div> <br> <div class="citations"><div class="clear"><a href="/?q=rf%3A5270187">Cited in <strong>61</strong> Documents</a></div></div> <div class="classification"> <h3>MSC:</h3> <table><tr> <td> <a class="mono" href="/classification/?q=cc%3A35R25" title="MSC2020">35R25</a> </td> <td class="space"> Ill-posed problems for PDEs </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A65M32" title="MSC2020">65M32</a> </td> <td class="space"> Numerical methods for inverse problems for initial value and initial-boundary value problems involving PDEs </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A35K05" title="MSC2020">35K05</a> </td> <td class="space"> Heat equation </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A47A52" title="MSC2020">47A52</a> </td> <td class="space"> Linear operators and ill-posed problems, regularization </td> </tr></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3Aboundary+element+method">boundary element method</a>; <a href="/?q=ut%3Adiscrepancy+principle">discrepancy principle</a>; <a href="/?q=ut%3Aheat+source">heat source</a>; <a href="/?q=ut%3Ainitial+condition">initial condition</a>; <a href="/?q=ut%3Ainverse+problem">inverse problem</a>; <a href="/?q=ut%3Aiterative+regularization">iterative regularization</a>; <a href="/?q=ut%3Aparabolic+heat+equation">parabolic heat equation</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 1133.35436" data-ciurl="/ci/05270187" data-biburl="/bibtex/05270187.bib" data-amsurl="/amsrefs/05270187.bib" data-xmlurl="/xml/05270187.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/05270187.pdf" title="Zbl 1133.35436 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/10.1080/00036810701858193" aria-label="DOI for “A procedure for determining a spacewise dependent heat source and the initial temperature”" title="10.1080/00036810701858193">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">Cannon JR, Numerical Solutions of Nonlinear Differential Equations pp 21– (1966)</td> </tr><tr> <td>[2]</td> <td class="space">DOI: 10.1016/0022-247X(79)90238-5 · <a href="/0425.35061" class="nowrap">Zbl 0425.35061</a> · <a href="https://doi.org/10.1016/0022-247X(79)90238-5" class="nowrap">doi:10.1016/0022-247X(79)90238-5</a></td> </tr><tr> <td>[3]</td> <td class="space">Lattès R, The Method of Quasi-Reversibility, Applications to Partial Differential Equations (1969) · <a href="/1220.65002" class="nowrap">Zbl 1220.65002</a></td> </tr><tr> <td>[4]</td> <td class="space">DOI: 10.1016/S0017-9310(00)00235-0 · <a href="/0979.80008" class="nowrap">Zbl 0979.80008</a> · <a href="https://doi.org/10.1016/S0017-9310(00)00235-0" class="nowrap">doi:10.1016/S0017-9310(00)00235-0</a></td> </tr><tr> <td>[5]</td> <td class="space">DOI: 10.1137/0705024 · <a href="/0176.15403" class="nowrap">Zbl 0176.15403</a> · <a href="https://doi.org/10.1137/0705024" class="nowrap">doi:10.1137/0705024</a></td> </tr><tr> <td>[6]</td> <td class="space">Prilepko AI, Diff. 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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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