Document Zbl 1457.65086

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Appl. Math.</a> <a href="/?q=in%3A371302" title="Articles in this Issue">5, No. 3, 273-300 (2015)</a>. </div> <div class="abstract">Summary: The inverse problem of identifying the time-independent source term and initial value simultaneously for a time-fractional diffusion equation is investigated. This inverse problem is reformulated into an operator equation based on the Fourier method. Under a certain smoothness assumption, conditional stability is established. A standard Tikhonov regularisation method is proposed to solve the inverse problem. Furthermore, the convergence rate is given for an <span class="zbmathjax-textit">a priori</span> and <span class="zbmathjax-textit">a posteriori</span> regularisation parameter choice rule, respectively. Several numerical examples, including one-dimensional and two-dimensional cases, show the efficiency of our proposed method.</div> <div class="clear"></div> <br> <div class="citations"><div class="clear"><a href="/?q=rf%3A6797001">Cited in <strong>21</strong> Documents</a></div></div> <div class="classification"> <h3>MSC:</h3> <table><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%3A65M12" title="MSC2020">65M12</a> </td> <td class="space"> Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A65J20" title="MSC2020">65J20</a> </td> <td class="space"> Numerical solutions of ill-posed problems in abstract spaces; regularization </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A49M15" title="MSC2020">49M15</a> </td> <td class="space"> Newton-type methods </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A49N45" title="MSC2020">49N45</a> </td> <td class="space"> Inverse problems in optimal control </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A35B65" title="MSC2020">35B65</a> </td> <td class="space"> Smoothness and regularity of solutions to PDEs </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A35R11" title="MSC2020">35R11</a> </td> <td class="space"> Fractional partial differential equations </td> </tr></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3Atime-fractional+diffusion+equation">time-fractional diffusion equation</a>; <a href="/?q=ut%3Aconditional+stability">conditional stability</a>; <a href="/?q=ut%3ATikhonov+regularisation">Tikhonov regularisation</a>; <a href="/?q=ut%3AMorozov+discrepancy+principle">Morozov discrepancy principle</a>; <a href="/?q=ut%3Aconvergence+rate">convergence rate</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 1457.65086" data-ciurl="/ci/06797001" data-biburl="/bibtex/06797001.bib" data-amsurl="/amsrefs/06797001.bib" data-xmlurl="/xml/06797001.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/06797001.pdf" title="Zbl 1457.65086 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/10.4208/eajam.310315.030715a" aria-label="DOI for “Tikhonov regularisation method for simultaneous inversion of the source term and initial data in a time-fractional diffusion equation”" title="10.4208/eajam.310315.030715a">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">[1]KenichiS. and MasahiroY., Initial value/boundary value problems for fractional diffusion-wave equations and applications to some inverse problems, J. 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