Document Zbl 1480.65253

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Math. Modelling</a> <a href="/?q=in%3A446504" title="Articles in this Issue">59, 441-463 (2018)</a>. </div> <div class="abstract">Summary: Fractional differential equations are powerful tools to model the non-locality and spatial heterogeneity evident in many real-world problems. Although numerous numerical methods have been proposed, most of them are limited to regular domains and uniform meshes. For irregular convex domains, the treatment of the space fractional derivative becomes more challenging and the general methods are no longer feasible. In this work, we propose a novel numerical technique based on the Galerkin finite element method (FEM) with an unstructured mesh to deal with the space fractional derivative on arbitrarily shaped convex and non-convex domains, which is the most original and significant contribution of this paper. Moreover, we present a second order finite difference scheme for the temporal fractional derivative. In addition, the stability and convergence of the method are discussed and numerical examples on different irregular convex domains and non-convex domains illustrate the reliability of the method. We also extend the theory and develop a computational model for the case of a multiply-connected domain. Finally, to demonstrate the versatility and applicability of our method, we solve the coupled two-dimensional fractional Bloch-Torrey equation on a human brain-like domain and exhibit the effects of the time and space fractional indices on the behaviour of the transverse magnetization.</div> <div class="clear"></div> <br> <div class="citations"><div class="clear"><a href="/?q=rf%3A7167919">Cited in <strong>41</strong> Documents</a></div></div> <div class="classification"> <h3>MSC:</h3> <table><tr> <td> <a class="mono" href="/classification/?q=cc%3A65M60" title="MSC2020">65M60</a> </td> <td class="space"> Finite element, Rayleigh-Ritz and Galerkin 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%3A65M06" title="MSC2020">65M06</a> </td> <td class="space"> Finite difference methods for initial value and initial-boundary value problems involving PDEs </td> </tr></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3Afinite+element+method">finite element method</a>; <a href="/?q=ut%3Aunstructured+mesh">unstructured mesh</a>; <a href="/?q=ut%3ARiesz+fractional+derivative">Riesz fractional derivative</a>; <a href="/?q=ut%3Airregular+domains">irregular domains</a>; <a href="/?q=ut%3Atwo-dimensional">two-dimensional</a>; <a href="/?q=ut%3Atime-space+fractional+diffusion+equation">time-space fractional diffusion equation</a></div> <div class="software"> <h3>Software:</h3><a href="/software/366">Gmsh</a>; <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 1480.65253" data-ciurl="/ci/07167919" data-biburl="/bibtex/07167919.bib" data-amsurl="/amsrefs/07167919.bib" data-xmlurl="/xml/07167919.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/07167919.pdf" title="Zbl 1480.65253 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/10.1016/j.apm.2018.01.044" aria-label="DOI for “Unstructured mesh finite difference/finite element method for the 2D time-space Riesz fractional diffusion equation on irregular convex domains”" title="10.1016/j.apm.2018.01.044">DOI</a> <a class="btn btn-default btn-xs" type="button" href="https://eprints.qut.edu.au/116276/1/1-s2.0-S0307904X18300568-main.pdf" title="Full Text Link">Link</a> </div> <div class="sfx" style="float: right;"> </div> </div> <div class="references"> <h3>References:</h3> <table><tr> <td>[1]</td> <td class="space">Sokolov, I. 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