Document Zbl 07987117

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Appl. Math.</a> <a href="/?q=in%3A520457" title="Articles in this Issue">44, No. 1, Paper No. 115, 31 p. (2025)</a>. </div> <div class="abstract">Summary: This paper presents two high-order compact difference schemes to discuss the numerical solution of the one-dimensional and two-dimensional multi-term time-fractional convection-diffusion equation of the Sobolev type based on the Caputo fractional derivative. For this purpose, we employ the L2 formula for the temporal discretization of the Caputo fractional derivatives and introduce a new compact difference operator for the space discretization. The proposed schemes transform the original problem into a system of algebraic equation. We present a novel analysis of the convergence and theoretical stability of both methods. The difference schemes have fourth-order and $3 - \max\{\alpha_r, \beta_r\}$ order of accuracy in space and time respectively, where $\alpha_r$ and $\beta_r$ are the orders of the fractional derivatives in the multi-term convection-diffusion model. Additionally, we construct an L2-type numerical scheme on non-uniform graded meshes to address problems where the solution exhibits weak initial regularity. Some test functions are provided to demonstrate the accuracy and efficiency of the proposed schemes. The numerical results of inclusive examples confirm the proposed schemes’ theoretical results and illustrate their applicability and efficiency.</div> <div class="clear"></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%3A26A33" title="MSC2020">26A33</a> </td> <td class="space"> Fractional derivatives and integrals </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A34A12" title="MSC2020">34A12</a> </td> <td class="space"> Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions to ordinary differential equations </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%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%3Acompact+difference+scheme">compact difference scheme</a>; <a href="/?q=ut%3Amulti-term+Sobolev+type+equation">multi-term Sobolev type equation</a>; <a href="/?q=ut%3ACaputo+time-fractional+derivative">Caputo time-fractional derivative</a>; <a href="/?q=ut%3AL2+formula">L2 formula</a>; <a href="/?q=ut%3Astability+and+convergence">stability and 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 07987117" data-ciurl="/ci/07987117" data-biburl="/bibtex/07987117.bib" data-amsurl="/amsrefs/07987117.bib" data-xmlurl="/xml/07987117.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/07987117.pdf" title="Zbl 07987117 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/10.1007/s40314-024-03077-8" aria-label="DOI for “A high-order compact difference scheme for the multi-term time-fractional Sobolev-type convection-diffusion equation”" title="10.1007/s40314-024-03077-8">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">Abro, KA; 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