Document Zbl 1417.65181

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Methods Partial Differ. Equations</a> <a href="/?q=in%3A413102" title="Articles in this Issue">34, No. 5, 1781-1798 (2018)</a>. </div> <div class="abstract">Summary: The main aim of this research article is to propose and analyze a Legendre wavelet collocation method (LWCM) for the nonlinear weakly singular partial integro-differential equation (SPIDE) arising from viscoelasticity subject to the given initial and boundary conditions. This problem can be found in the mathematical modeling of physical phenomena involving viscoelastic forces. Operational matrix of integration of Legendre wavelets along with collocation method are utilized to reduce the original SPIDE into the nonlinear system of algebraic equations. Some numerical results are presented to simplify applications of operational matrix formulation and reduce the computational cost. Convergence analysis, numerical stability and rate of convergence (C-order) of the proposed method are also investigated by considering a test function. Numerical results confirm the predicted convergence rates and also exhibit optimal accuracy in the $L^2$ and $L^{\infty}$ norms. Finally, we compare the proposed LWCM with well-known Crank-Nicolson and Crandall’s methods (for instance, see Table 4).</div> <div class="clear"></div> <br> <div class="citations"><div class="clear"><a href="/?q=rf%3A7023982">Cited in <strong>12</strong> Documents</a></div></div> <div class="classification"> <h3>MSC:</h3> <table><tr> <td> <a class="mono" href="/classification/?q=cc%3A65M70" title="MSC2020">65M70</a> </td> <td class="space"> Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A65T60" title="MSC2020">65T60</a> </td> <td class="space"> Numerical methods for wavelets </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%3A35R09" title="MSC2020">35R09</a> </td> <td class="space"> Integro-partial differential equations </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A45J05" title="MSC2020">45J05</a> </td> <td class="space"> Integro-ordinary differential equations </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A35Q35" title="MSC2020">35Q35</a> </td> <td class="space"> PDEs in connection with fluid mechanics </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A76A10" title="MSC2020">76A10</a> </td> <td class="space"> Viscoelastic fluids </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><tr> <td> <a class="mono" href="/classification/?q=cc%3A42C10" title="MSC2020">42C10</a> </td> <td class="space"> Fourier series in special orthogonal functions (Legendre polynomials, Walsh functions, etc.) </td> </tr></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3Apartial+integro-differential+equation">partial integro-differential equation</a>; <a href="/?q=ut%3Atwo+dimensional+Legendre+wavelets">two dimensional Legendre wavelets</a>; <a href="/?q=ut%3Aoperational+matrix+of+integration">operational matrix of integration</a>; <a href="/?q=ut%3Acollocation+method">collocation method</a>; <a href="/?q=ut%3Aconvergence+analysis">convergence analysis</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 1417.65181" data-ciurl="/ci/07023982" data-biburl="/bibtex/07023982.bib" data-amsurl="/amsrefs/07023982.bib" data-xmlurl="/xml/07023982.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/07023982.pdf" title="Zbl 1417.65181 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/10.1002/num.22245" aria-label="DOI for “Convergence rate of collocation method based on wavelet for nonlinear weakly singular partial integro-differential equation arising from viscoelasticity”" title="10.1002/num.22245">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">Tohidi, E.; Toutounian, F., Convergence analysis of Bernoulli matrix approach for one‐dimensional matrix hyperbolic equations of first order, Comput. 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