Document Zbl 1062.65073

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E. M.</a>; <a href="/authors/el-sayed.ahmed-mohamed-ahmed" title="Author Profile">El-Sayed, A. M. A.</a>; <a href="/authors/el-saka.hala-a-a" title="Author Profile">El-Saka, H. A. A.</a></div> <h2 class="title"> <strong>Numerical methods for multi-term fractional (arbitrary) orders differential equations.</strong> <i>(English)</i> <a class="label nowrap" href="/1062.65073">Zbl 1062.65073</a> </h2> <div class="source"> <a href="/serials/331" title="Journal Profile">Appl. Math. Comput.</a> <a href="/?q=in%3A113015" title="Articles in this Issue">160, No. 3, 683-699 (2005)</a>. </div> <div class="abstract">Summary: Our main concern here is to give a numerical scheme to solve a nonlinear multi-term fractional (arbitrary) orders differential equation.</div> <div class="clear"></div> <br> <div class="citations"><div class="clear"><a href="/?q=rf%3A2136859">Cited in <strong>65</strong> Documents</a></div></div> <div class="classification"> <h3>MSC:</h3> <table><tr> <td> <a class="mono" href="/classification/?q=cc%3A65L05" title="MSC2020">65L05</a> </td> <td class="space"> Numerical methods for initial value problems involving ordinary differential equations </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A65L06" title="MSC2020">65L06</a> </td> <td class="space"> Multistep, Runge-Kutta and extrapolation methods for ordinary 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%3A34A34" title="MSC2020">34A34</a> </td> <td class="space"> Nonlinear ordinary differential equations and systems </td> </tr></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3AMulti-term+fractional+differential+equation">Multi-term fractional differential equation</a>; <a href="/?q=ut%3ACaputo+derivative">Caputo derivative</a>; <a href="/?q=ut%3AEuler%27s+method">Euler’s method</a>; <a href="/?q=ut%3AProduct+trapezoidal+quadrature+formula">Product trapezoidal quadrature formula</a>; <a href="/?q=ut%3AProduct+rectangle+rule">Product rectangle rule</a>; <a href="/?q=ut%3Anumerical+example">numerical example</a></div> <div class="software"> <h3>Software:</h3><a href="/software/24209">FracPECE</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 1062.65073" data-ciurl="/ci/02136859" data-biburl="/bibtex/02136859.bib" data-amsurl="/amsrefs/02136859.bib" data-xmlurl="/xml/02136859.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/02136859.pdf" title="Zbl 1062.65073 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.amc.2003.11.026" aria-label="DOI for “Numerical methods for multi-term fractional (arbitrary) orders differential equations”" title="10.1016/j.amc.2003.11.026">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">Burden, R. L.; Faires, J. D., Numerical Analysis (1993), PWS Publishing Company · <a href="/0788.65001" class="nowrap">Zbl 0788.65001</a></td> </tr><tr> <td>[2]</td> <td class="space">L. Blank, Numerical treatment of differential equations of fractional order, Numerical Analysis Report 287, Manchester Center for Numerical Computational Mathematics, 1996; L. Blank, Numerical treatment of differential equations of fractional order, Numerical Analysis Report 287, Manchester Center for Numerical Computational Mathematics, 1996</td> </tr><tr> <td>[3]</td> <td class="space">Caputo, M., Linear model of dissipation whose $Q$ is almost frequency independent II, Geophys. J. Roy. Astr. Soc, 13, 529-539 (1967)</td> </tr><tr> <td>[4]</td> <td class="space">Diethelm, K., An algorithm for the numerical solution of differential equations of fractional order, Electron. Trans. Numer. Anal., Kent State University, 5, 1-6 (1997) · <a href="/0890.65071" class="nowrap">Zbl 0890.65071</a></td> </tr><tr> <td>[5]</td> <td class="space">Diethelm, K.; Walz, G., Numerical solution of fractional order differential equations by extrapolation, Numer. Alg, 16, 231-253 (1997) · <a href="/0926.65070" class="nowrap">Zbl 0926.65070</a></td> </tr><tr> <td>[6]</td> <td class="space">Diethelm, K.; Freed, A., On the solution of nonlinear fractional order differential equations used in the modelling of viscoplasticity, Scientific Computing in Chemical Engineering II-Computational fluid dynamics, (Keil, F.; Mackens, W.; Voß, H.; Werther, J., Reaction Engineering, and Molecular properties (1999), Springer: Springer Heidelberg), 217-224</td> </tr><tr> <td>[7]</td> <td class="space">Diethelm, K.; Freed, A., The FracPECE subroutine for the numerical solution of differential equations of fractional order, (Heinzel, S.; Plesser, T., Forschung und wissenschaftliches Rechnen 1998 (1999), Gesellschaft für Wisseschaftliche Datenverarbeitung: Gesellschaft für Wisseschaftliche Datenverarbeitung Göttingen), 57-71</td> </tr><tr> <td>[8]</td> <td class="space">K. Diethelm, N.J. Ford, The numerical solution of linear and non-linear fractional differential equations involving fractional derivatives of several orders, Numerical Analysis Report 379, Manchester Centre for Computational Mathematics, 2001; K. Diethelm, N.J. Ford, The numerical solution of linear and non-linear fractional differential equations involving fractional derivatives of several orders, Numerical Analysis Report 379, Manchester Centre for Computational Mathematics, 2001</td> </tr><tr> <td>[9]</td> <td class="space">K. Diethelm, Predictor-corrector strategies for single and multi-term fractional differential equations, Proceedings Hercma 2001; K. Diethelm, Predictor-corrector strategies for single and multi-term fractional differential equations, Proceedings Hercma 2001</td> </tr><tr> <td>[10]</td> <td class="space">K. Diethelm, N.J. Ford, A.D. Freed, Detailed error analysis for a fractional Adams method, Numer. Algorithms, to appear; K. Diethelm, N.J. Ford, A.D. Freed, Detailed error analysis for a fractional Adams method, Numer. Algorithms, to appear · <a href="/1055.65098" class="nowrap">Zbl 1055.65098</a></td> </tr><tr> <td>[11]</td> <td class="space">Diethelm, K.; Ford, N. J.; Freed, A. D., A predictor-corrector approach for the numerical solution of fractional differential equations, Nonlinear Dynamics, 29, 3-22 (2002) · <a href="/1009.65049" class="nowrap">Zbl 1009.65049</a></td> </tr><tr> <td>[12]</td> <td class="space">Diethelm, K.; Ford, N. J., Numerical solution of the Bagley-Torvik equation, BIT, 42, 490-507 (2002) · <a href="/1035.65067" class="nowrap">Zbl 1035.65067</a></td> </tr><tr> <td>[13]</td> <td class="space">K. Diethelm, Y. Luchko, Numerical solution of linear multi-term initial value problems of fractional order, J. Comput. Anal. Appl. (2003), to appear; K. Diethelm, Y. Luchko, Numerical solution of linear multi-term initial value problems of fractional order, J. Comput. Anal. Appl. (2003), to appear · <a href="/1083.65064" class="nowrap">Zbl 1083.65064</a></td> </tr><tr> <td>[14]</td> <td class="space">El-Sayed, A. M.A., Linear differential equations of fractional orders, Appl. Math. and Comput, 55, 1-12 (1993) · <a href="/0772.34013" class="nowrap">Zbl 0772.34013</a></td> </tr><tr> <td>[15]</td> <td class="space">El-Sayed, A. M.A., Nonlinear functional differential equations of arbitrary orders, Nonlinear Anal.: Theory Meth. Applicat, 33, 2, 181-186 (1998) · <a href="/0934.34055" class="nowrap">Zbl 0934.34055</a></td> </tr><tr> <td>[16]</td> <td class="space">J.T. Edwards, N.J. Ford, A.C. Simpson, The numerical solution of linear multi-term fractional differential equations: systems of equations, Manchester Center for Numerical Computational Mathematics, 2002; J.T. Edwards, N.J. Ford, A.C. Simpson, The numerical solution of linear multi-term fractional differential equations: systems of equations, Manchester Center for Numerical Computational Mathematics, 2002 · <a href="/1019.65048" class="nowrap">Zbl 1019.65048</a></td> </tr><tr> <td>[17]</td> <td class="space">N.J. Ford, A.C. Simpson, The approximate solution of fractional differential equations of order greater than 1, Numerical Analysis Report 386, Manchester Center for Numerical Computational Mathematics, 2001; N.J. Ford, A.C. Simpson, The approximate solution of fractional differential equations of order greater than 1, Numerical Analysis Report 386, Manchester Center for Numerical Computational Mathematics, 2001</td> </tr><tr> <td>[18]</td> <td class="space">I.M. Gelfand, G.E. Shilov, Generalized Functions, vol. 1, Moscow, 1958; I.M. Gelfand, G.E. Shilov, Generalized Functions, vol. 1, Moscow, 1958 · <a href="/0091.11103" class="nowrap">Zbl 0091.11103</a></td> </tr><tr> <td>[19]</td> <td class="space">Gorenflo, R.; Mainardi, F., Fractional calculus: integral and differential equations of fractional order, (Carpinteri, A.; Mainardi, F., Fractals and Fractional Calculus in Continuum Mechanics (1997), Springer: Springer Wien), 223-276 · <a href="/1438.26010" class="nowrap">Zbl 1438.26010</a></td> </tr><tr> <td>[20]</td> <td class="space">Lubich, C., Discretized fractional calculus, SIAM J. Math. Anal, 17, 3, 704-719 (1986) · <a href="/0624.65015" class="nowrap">Zbl 0624.65015</a></td> </tr><tr> <td>[21]</td> <td class="space">J. Leszczynski, M. Ciesielski, A numerical method for solution of ordinary differential equations of fractional order, V 1, 26 February 2002; J. Leszczynski, M. Ciesielski, A numerical method for solution of ordinary differential equations of fractional order, V 1, 26 February 2002 · <a href="/1057.65507" class="nowrap">Zbl 1057.65507</a></td> </tr><tr> <td>[22]</td> <td class="space">Miller, K. S.; Ross, B., An Introduction to the Fractional Calculus and Fractional Differential Equations (1993), John Wiley and Sons: John Wiley and Sons New York · <a href="/0789.26002" class="nowrap">Zbl 0789.26002</a></td> </tr><tr> <td>[23]</td> <td class="space">Mainardi, F., Some basic problems in continuum and statistical mechanics, (Carpinteri, A.; Mainardi, F., Fractals and Fractional Calculus in Continuum Mechanics (1997), Springer: Springer Wien), 291-348 · <a href="/0917.73004" class="nowrap">Zbl 0917.73004</a></td> </tr><tr> <td>[24]</td> <td class="space">Oldham, K. B.; Spanier, J., The fractional Calculus (1974), Academic Press: Academic Press New York and London · <a href="/0428.26004" class="nowrap">Zbl 0428.26004</a></td> </tr><tr> <td>[25]</td> <td class="space">I. Podlubny, A.M.A. El-Sayed, On two definitions of fractional calculus, Solvak Academy of Science-Institute of Experimental Phys. UEF-03-96 ISBN 80-7099-252-2, 1996; I. Podlubny, A.M.A. El-Sayed, On two definitions of fractional calculus, Solvak Academy of Science-Institute of Experimental Phys. UEF-03-96 ISBN 80-7099-252-2, 1996</td> </tr><tr> <td>[26]</td> <td class="space">Palczewski, A., Ordinary Differential Equations (1999), WNT: WNT Warsaw, (in Polish)</td> </tr><tr> <td>[27]</td> <td class="space">Podlubny, I., Fractional Differential Equations (1999), Academic Press · <a href="/0918.34010" class="nowrap">Zbl 0918.34010</a></td> </tr><tr> <td>[28]</td> <td class="space">S.G. Samko, A.A. Kilbas, O.I. Marichev, Integrals and Derivatives of the Fractional Order and Some of Their Applications, Nauka i Tehnika, Minsk, 1987 (in Russian), (English trans. 1993); S.G. Samko, A.A. Kilbas, O.I. Marichev, Integrals and Derivatives of the Fractional Order and Some of Their Applications, Nauka i Tehnika, Minsk, 1987 (in Russian), (English trans. 1993) · <a href="/0617.26004" class="nowrap">Zbl 0617.26004</a></td> </tr></table> <div class="reference_disclaimer"> This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. 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