Document Zbl 1420.65087

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Algorithms</a> <a href="/?q=in%3A404637" title="Articles in this Issue">78, No. 2, 485-511 (2018)</a>. </div> <div class="abstract">The article presents numerical schemes for time-fractional nonlinear Klein-Gordon equations. The second-order approximation of the fractional time derivative of order 2 is based on formulas derived earlier in [<span class="zbmathjax-textit">H.-l. Liao</span> et al., J. Sci. Comput. 69, No. 3, 1144–1164 (2016; <a href="/1371.65082">Zbl 1371.65082</a>)]. The first scheme is a linearized scheme, where the nonlinear term is treated explicitly. Nonetheless, the stability and convergence seems to be unconditional, as long as the nonlinearity is Lipschitz continuous. The second scheme is a compact scheme of order 4 in space. Numerical tests confirm the expected theoretical results.<div class="reviewer"> Reviewer: <a href="/authors/?q=rv%3A16743">Aziz Takhirov (Edmonton)</a></div> <div class="clearfix"></div></div> <div class="clear"></div> <br> <div class="citations"><div class="clear"><a href="/?q=ci%3A6894097">Cited in <strong>1</strong> Review</a></div><div class="clear"><a href="/?q=rf%3A6894097">Cited in <strong>25</strong> Documents</a></div></div> <div class="classification"> <h3>MSC:</h3> <table><tr> <td> <a class="mono" href="/classification/?q=cc%3A65M06" title="MSC2020">65M06</a> </td> <td class="space"> Finite difference methods for initial value and initial-boundary value problems involving 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><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%3A35Q53" title="MSC2020">35Q53</a> </td> <td class="space"> KdV equations (Korteweg-de Vries equations) </td> </tr></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3Anonlinear+Klein-Gordon+equations">nonlinear Klein-Gordon equations</a>; <a href="/?q=ut%3Atime-fractional+differential+equations">time-fractional differential equations</a>; <a href="/?q=ut%3Afinite+differences">finite differences</a></div><div class="keywords"> <h3>Citations:</h3><a href="/1371.65082">Zbl 1371.65082</a></div> <div class="software"> <h3>Software:</h3><a href="/software/8377">FODE</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 1420.65087" data-ciurl="/ci/06894097" data-biburl="/bibtex/06894097.bib" data-amsurl="/amsrefs/06894097.bib" data-xmlurl="/xml/06894097.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/06894097.pdf" title="Zbl 1420.65087 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/s11075-017-0385-y" aria-label="DOI for “A linearized second-order scheme for nonlinear time fractional Klein-Gordon type equations”" title="10.1007/s11075-017-0385-y">DOI</a> <a class="btn btn-default btn-xs" type="button" href="https://arxiv.org/abs/1705.08057"title="Note: arXiv document may differ from published version">arXiv</a> </div> <div class="sfx" style="float: right;"> </div> </div> <div class="references"> <h3>References:</h3> <table><tr> <td>[1]</td> <td class="space">Barkai, E; Metzler, R; Klafter, J, From continuous time random walks to the fractional Fokker-Planck equation, Phys. 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