Document Zbl 1393.35280

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Model. Nat. Phenom.</a> <a href="/?q=in%3A374750" title="Articles in this Issue">11, No. 3, 1-17 (2016)</a>. </div> <div class="abstract">Summary: In this paper, a special model for the two-dimensional anomalous diffusion is first deduced from the basic continuous time random walk equations in terms of a time- and space-fractional partial differential equation with the Caputo time-fractional derivative of order $\alpha/2$ and the Riesz space-fractional derivative of order $\alpha$. For $\alpha<2$, this $\alpha$-fractional diffusion equation describes the so called Lévy flights that correspond to the continuous time random walk model, where both the mean waiting time and the jump length variance of the diffusing particles are divergent. The fundamental solution to the $\alpha$-fractional diffusion equation is shown to be a two-dimensional probability density function that can be expressed in explicit form in terms of the Mittag-Leffler function depending on the auxiliary variable $| x| /(2\sqrt{t})$ as in the case of the fundamental solution to the classical isotropic diffusion equation. Moreover, we show that the entropy production rate associated with the anomalous diffusion process described by the $\alpha$-fractional diffusion equation is exactly the same as in the case of the classical isotropic diffusion equation. Thus the $\alpha$-fractional diffusion equation can be considered to be a natural generalization of the classical isotropic diffusion equation that exhibits some characteristics of both anomalous and classical diffusion.</div> <div class="clear"></div> <br> <div class="citations"><div class="clear"><a href="/?q=rf%3A6849678">Cited in <strong>18</strong> Documents</a></div></div> <div class="classification"> <h3>MSC:</h3> <table><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%3A35C05" title="MSC2020">35C05</a> </td> <td class="space"> Solutions to PDEs in closed form </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A35E05" title="MSC2020">35E05</a> </td> <td class="space"> Fundamental solutions to PDEs and systems of PDEs with constant coefficients </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A35L05" title="MSC2020">35L05</a> </td> <td class="space"> Wave equation </td> </tr></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3Aanomalous+diffusion">anomalous diffusion</a>; <a href="/?q=ut%3Acontinuous+time+random+walks">continuous time random walks</a>; <a href="/?q=ut%3Afractional+diffusion+equation">fractional diffusion equation</a>; <a href="/?q=ut%3AMellin-Barnes+integral">Mellin-Barnes integral</a>; <a href="/?q=ut%3AMittag-Leffler+function">Mittag-Leffler function</a>; <a href="/?q=ut%3Atwo-dimensional+probability+density+function">two-dimensional probability density function</a>; <a href="/?q=ut%3Aentropy">entropy</a>; <a href="/?q=ut%3Aentropy+production+rate">entropy production rate</a></div> <div class="software"> <h3>Software:</h3><a href="/software/558">Matlab</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 1393.35280" data-ciurl="/ci/06849678" data-biburl="/bibtex/06849678.bib" data-amsurl="/amsrefs/06849678.bib" data-xmlurl="/xml/06849678.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/06849678.pdf" title="Zbl 1393.35280 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/10.1051/mmnp/201611301" aria-label="DOI for “A new fractional calculus model for the two-dimensional anomalous diffusion and its analysis”" title="10.1051/mmnp/201611301">DOI</a> <a class="btn btn-default btn-xs" type="button" href="https://www.mmnp-journal.org/10.1051/mmnp/201611301/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">G. 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