Document Zbl 1253.39004

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Appl. Anal.</a> <a href="/?q=in%3A293529" title="Articles in this Issue">2012, Article ID 406757, 13 p. (2012)</a>. </div> <div class="abstract">Summary: We show that two recent definitions of discrete nabla fractional sum operators are related. Obtaining such a relation between two operators allows one to prove basic properties of the one operator by using the known properties of the other. We illustrate this idea with proving power rule and commutative property of discrete fractional sum operators. We also introduce and prove summation by parts formulas for the right and left fractional sum and difference operators, where we employ the Riemann-Liouville definition of the fractional difference. We formalize initial value problems for nonlinear fractional difference equations as an application of our findings. An alternative definition for the nabla right fractional difference operator is also introduced.</div> <div class="clear"></div> <br> <div class="citations"><div class="clear"><a href="/?q=rf%3A6116389">Cited in <strong>96</strong> Documents</a></div></div> <div class="classification"> <h3>MSC:</h3> <table><tr> <td> <a class="mono" href="/classification/?q=cc%3A39A12" title="MSC2020">39A12</a> </td> <td class="space"> Discrete version of topics in analysis </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></table> </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 1253.39004" data-ciurl="/ci/06116389" data-biburl="/bibtex/06116389.bib" data-amsurl="/amsrefs/06116389.bib" data-xmlurl="/xml/06116389.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/06116389.pdf" title="Zbl 1253.39004 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/10.1155/2012/406757" aria-label="DOI for “On the definitions of nabla fractional operators”" title="10.1155/2012/406757">DOI</a> </div> <div class="sfx" style="float: right;"> <a href="https://creativecommons.org/licenses/by/4.0/" target="_blank" title="Open Access License" class="cc-license-link no-new-tab-icon"> <img src="https://static.zbmath.org/contrib/img/cc/svg/icons/cc.svg" alt="Creative Commons CC license icon" class="cc-license-icon"> <img src="https://static.zbmath.org/contrib/img/cc/svg/icons/by.svg" alt="Creative Commons BY license icon" class="cc-license-icon"> </a> </div> </div> <div class="references"> <h3>References:</h3> <table><tr> <td>[1]</td> <td class="space">H. L. Gray and N. F. Zhang, “On a new definition of the fractional difference,” Mathematics of Computation, vol. 50, no. 182, pp. 513-529, 1988. · <a href="/0648.39002" class="nowrap">Zbl 0648.39002</a> · <a href="https://doi.org/10.2307/2008620" class="nowrap">doi:10.2307/2008620</a></td> </tr><tr> <td>[2]</td> <td class="space">F. M. Atıcı and P. W. Eloe, “Discrete fractional calculus with the nabla operator,” Electronic Journal of Qualitative Theory of Differential Equations, no. 3, pp. 1-12, 2009. · <a href="/1189.39004" class="nowrap">Zbl 1189.39004</a></td> </tr><tr> <td>[3]</td> <td class="space">K. S. Miller and B. Ross, “Fractional difference calculus,” in Proceedings of the International Symposium on Univalent Functions, Fractional Calculus and Their Applications, pp. 139-152, Nihon University, Koriyama, Japan, 1989. · <a href="/0693.39002" class="nowrap">Zbl 0693.39002</a></td> </tr><tr> <td>[4]</td> <td class="space">T. 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Trujillo, Theory and Application of Fractional Differential Equations, vol. 204 of North-Holland Mathematics Studies, 2006. · <a href="/1092.45003" class="nowrap">Zbl 1092.45003</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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