Document Zbl 1428.34001

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Calc. Appl. Anal.</a> <a href="/?q=in%3A439540" title="Articles in this Issue">22, No. 1, 27-59 (2019)</a>. </div> <div class="abstract">Summary: Variable-order (VO) fractional differential equations (FDEs) with a time $(t)$, space $(x)$ or other variables dependent order have been successfully applied to investigate time and/or space dependent dynamics. This study aims to provide a survey of the recent relevant literature and findings in primary definitions, models, numerical methods and their applications. This review first offers an overview over the existing definitions proposed from different physical and application backgrounds, and then reviews several widely used numerical schemes in simulation. Moreover, as a powerful mathematical tool, the VO-FDE models have been remarkably acknowledged as an alternative and precise approach in effectively describing real-world phenomena. Hereby, we also make a brief summary on different physical models and typical applications. This review is expected to help the readers for the selection of appropriate definition, model and numerical method to solve specific physical and engineering problems.</div> <div class="clear"></div> <br> <div class="citations"><div class="clear"><a href="/?q=rf%3A7115416">Cited in <strong>145</strong> Documents</a></div></div> <div class="classification"> <h3>MSC:</h3> <table><tr> <td> <a class="mono" href="/classification/?q=cc%3A34-02" title="MSC2020">34-02</a> </td> <td class="space"> Research exposition (monographs, survey articles) pertaining to ordinary differential equations </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A26A33" title="MSC2020">26A33</a> </td> <td class="space"> Fractional derivatives and integrals </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A34A08" title="MSC2020">34A08</a> </td> <td class="space"> Fractional ordinary differential equations </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A34A45" title="MSC2020">34A45</a> </td> <td class="space"> Theoretical approximation of solutions to ordinary differential equations </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%3A65-02" title="MSC2020">65-02</a> </td> <td class="space"> Research exposition (monographs, survey articles) pertaining to numerical analysis </td> </tr></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3Avariable-order">variable-order</a>; <a href="/?q=ut%3Afractional+calculus">fractional calculus</a>; <a href="/?q=ut%3Afractional+differential+equations">fractional differential equations</a>; <a href="/?q=ut%3Anumerical+methods">numerical methods</a>; <a href="/?q=ut%3Aapplications">applications</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 1428.34001" data-ciurl="/ci/07115416" data-biburl="/bibtex/07115416.bib" data-amsurl="/amsrefs/07115416.bib" data-xmlurl="/xml/07115416.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/07115416.pdf" title="Zbl 1428.34001 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/10.1515/fca-2019-0003" aria-label="DOI for “A review on variable-order fractional differential equations: mathematical foundations, physical models, numerical methods and applications”" title="10.1515/fca-2019-0003">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">R. 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