Document Zbl 1481.65149

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Math. Modelling</a> <a href="/?q=in%3A472228" title="Articles in this Issue">89, Part 2, 1721-1746 (2021)</a>. </div> <div class="abstract">Summary: Stable multistep schemes based on Caputo fractional derivative approximation are presented for solving 1D and 2D nonlinear fractional model arising from dielectric media. We approximate Caputo fractional derivatives in time with a multistep scheme of order $\mathcal{O}(\tau^{3 - \alpha})$ & $\mathcal{O}(\tau^{3 - \beta})$, $1 < \beta < \alpha < 2$, spatial Laplacian operator with a central difference scheme, and nonlinear source term $g (\mathscr{B})$ by using Taylor series. The discretization of the problem results in a linear system of equations that is tridiagonal and penta-diagonal for 1D and 2D case, respectively. The unique solvability and unconditional stability are derived for both cases. The convergence of schemes is established with the help of optimal error bounds. Further, we establish that the order of convergence for 1D case is $\mathcal{O}(\tau^{3 - \alpha} + \tau^{3 - \beta} + h^2)$ and for 2D case is $\mathcal{O}(\tau^{3 - \alpha} + \tau^{3 - \beta} + h_x^2 + h_y^2)$. Moreover, the stability of our schemes are verified numerically by adding some linear and nonlinear noisy inputs. Finally, four test functions are investigated to show the effectiveness and stability of our schemes. The method is simple, easy to implement, and yields very accurate results.</div> <div class="clear"></div> <br> <div class="citations"><div class="clear"><a href="/?q=rf%3A7426721">Cited in <strong>8</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%3ACaputo+fractional+derivative">Caputo fractional derivative</a>; <a href="/?q=ut%3Aelectromagnetic+waves+model">electromagnetic waves model</a>; <a href="/?q=ut%3Afinite+difference+method">finite difference method</a>; <a href="/?q=ut%3Aunconditional+stability">unconditional stability</a>; <a href="/?q=ut%3Aconvergence+analysis">convergence analysis</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 1481.65149" data-ciurl="/ci/07426721" data-biburl="/bibtex/07426721.bib" data-amsurl="/amsrefs/07426721.bib" data-xmlurl="/xml/07426721.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/07426721.pdf" title="Zbl 1481.65149 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.2020.08.038" aria-label="DOI for “Stability and convergence of multistep schemes for 1D and 2D fractional model with nonlinear source term”" title="10.1016/j.apm.2020.08.038">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">Torvik, P. 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