Document Zbl 1311.65114

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Math. Comput.</a> <a href="/?q=in%3A312337" title="Articles in this Issue">219, No. 8, 4082-4095 (2012)</a>. </div> <div class="abstract">Summary: The space and time fractional Bloch-Torrey equation (ST-FBTE) has been used to study anomalous diffusion in the human brain. Numerical methods for solving ST-FBTE in three-dimensions are computationally demanding. In this paper, we propose a computationally effective fractional alternating direction method (FADM) to overcome this problem. We consider ST-FBTE on a finite domain where the time and space derivatives are replaced by the Caputo-Djrbashian and the sequential Riesz fractional derivatives, respectively. The stability and convergence properties of the FADM are discussed. Finally, some numerical results for ST-FBTE are given to confirm our theoretical findings.</div> <div class="clear"></div> <br> <div class="citations"><div class="clear"><a href="/?q=rf%3A6436090">Cited in <strong>39</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%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%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%3Afractional+Bloch-Torrey+equation">fractional Bloch-Torrey equation</a>; <a href="/?q=ut%3Afractional+calculus">fractional calculus</a>; <a href="/?q=ut%3Aimplicit+numerical+method">implicit numerical method</a>; <a href="/?q=ut%3Aalternating+direction+method">alternating direction method</a>; <a href="/?q=ut%3Astability">stability</a>; <a href="/?q=ut%3Aconvergence">convergence</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 1311.65114" data-ciurl="/ci/06436090" data-biburl="/bibtex/06436090.bib" data-amsurl="/amsrefs/06436090.bib" data-xmlurl="/xml/06436090.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/06436090.pdf" title="Zbl 1311.65114 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.amc.2012.10.056" aria-label="DOI for “A computationally effective alternating direction method for the space and time fractional Bloch-Torrey equation in 3-D”" title="10.1016/j.amc.2012.10.056">DOI</a> <a class="btn btn-default btn-xs" type="button" href="https://eprints.qut.edu.au/60023/1/Liu30_revised_AMC_FDA12_y12m10d15.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">Magin, R.; Abdullah, O.; Baleanu, D.; Zhou, X., Anomalous diffusion expressed through fractional order differential operators in the Bloch-Torrey equation, Journal of Magnetic Resonance, 190, 2, 255-270 (2008)</td> </tr><tr> <td>[2]</td> <td class="space">Q. 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