Document Zbl 1388.35095

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Comput. Phys.</a> <a href="/?q=in%3A214341" title="Articles in this Issue">227, No. 2, 1510-1522 (2007)</a>. </div> <div class="abstract">Summary: Anomalous diffusion is one of the most ubiquitous phenomena in nature, and it is present in a wide variety of physical situations, for instance, transport of fluid in porous media, diffusion of plasma, diffusion at liquid surfaces, etc. The fractional approach proved to be highly effective in a rich variety of scenarios such as continuous time random walk models, generalized Langevin equations, or the generalized master equation. To investigate the subdiffusion of anomalous diffusion, it would be useful to study a time fractional Fokker-Planck equation. In this paper, firstly the time fractional, the sense of Riemann-Liouville derivative, Fokker-Planck equation is transformed into a time fractional ordinary differential equation (FODE) in the sense of Caputo derivative by discretizing the spatial derivatives and using the properties of Riemann-Liouville derivative and Caputo derivative. Then combining the predictor-corrector approach with the method of lines, the algorithm is designed for numerically solving FODE with the numerical error $O(k^{\min\{1+2\alpha ,2\}})+O(h^{2})$, and the corresponding stability condition is got. The effectiveness of this numerical algorithm is evaluated by comparing its numerical results for $\alpha =1.0$ with the ones of directly discretizing classical Fokker-Planck equation, some numerical results for time fractional Fokker-Planck equation with several different fractional orders are demonstrated and compared with each other, moreover for $\alpha =0.8$ the convergent order in space is confirmed and the numerical results with different time step sizes are shown.</div> <div class="clear"></div> <br> <div class="citations"><div class="clear"><a href="/?q=rf%3A5221351">Cited in <strong>141</strong> Documents</a></div></div> <div class="classification"> <h3>MSC:</h3> <table><tr> <td> <a class="mono" href="/classification/?q=cc%3A35K57" title="MSC2020">35K57</a> </td> <td class="space"> Reaction-diffusion 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%3A35Q84" title="MSC2020">35Q84</a> </td> <td class="space"> Fokker-Planck equations </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A65M20" title="MSC2020">65M20</a> </td> <td class="space"> Method of lines for initial value and initial-boundary value problems involving PDEs </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A82B35" title="MSC2020">82B35</a> </td> <td class="space"> Irreversible thermodynamics, including Onsager-Machlup theory </td> </tr><tr> <td class="mono"> 82B80 </td> <td class="space"> Numerical methods in equilibrium statistical mechanics (MSC2010) </td> </tr></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3ARiemann-Liouville+derivative">Riemann-Liouville derivative</a>; <a href="/?q=ut%3ACaputo+derivative">Caputo derivative</a>; <a href="/?q=ut%3AFokker-Planck+equation">Fokker-Planck equation</a>; <a href="/?q=ut%3Asubdiffusion">subdiffusion</a>; <a href="/?q=ut%3Apredictor-corrector+approach">predictor-corrector approach</a>; <a href="/?q=ut%3Amethod+of+lines">method of lines</a>; <a href="/?q=ut%3Aconvergent+order">convergent order</a>; <a href="/?q=ut%3Anumerical+stability">numerical stability</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 1388.35095" data-ciurl="/ci/05221351" data-biburl="/bibtex/05221351.bib" data-amsurl="/amsrefs/05221351.bib" data-xmlurl="/xml/05221351.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/05221351.pdf" title="Zbl 1388.35095 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.jcp.2007.09.015" aria-label="DOI for “Numerical algorithm for the time fractional Fokker-Planck equation”" title="10.1016/j.jcp.2007.09.015">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">Agrawal, Om P.; Tenreiro Machado, J. 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