Document Zbl 1312.34094

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Calc. Appl. Anal.</a> <a href="/?q=in%3A310478" title="Articles in this Issue">15, No. 4, 639-668 (2012)</a>. </div> <div class="abstract">Summary: We investigate Cauchy problem for a class of time-fractional differential equation \[ \begin{aligned} D_t^\alpha u(t) + c_1 D_t^{\beta_{1}} u(t) + \cdots + c_d D_t^{\beta_{d}} u(t) = Au(t), \quad t > 0, \\ u^{(j)} (0) = x_j,\quad j = 0, \cdots, m-1, \end{aligned} \tag{0.1} \] where $A$ is a closed densely defined linear operator in a Banach space $X$, $\alpha > \beta_1 > \cdots > \beta_d > 0$, $c_j$ are constants and $m = \lceil\alpha\rceil$. A new type of resolvent family corresponding to well-posedness of (0.1) is introduced. We derive the generation theorems, algebraic equations and approximation theorems for such resolvent families. Moreover, we give the exact solution for a kind of generalized fractional telegraph equations. Some examples are given as illustrations.</div> <div class="clear"></div> <br> <div class="citations"><div class="clear"><a href="/?q=rf%3A6440190">Cited in <strong>21</strong> Documents</a></div></div> <div class="classification"> <h3>MSC:</h3> <table><tr> <td> <a class="mono" href="/classification/?q=cc%3A34G10" title="MSC2020">34G10</a> </td> <td class="space"> Linear differential equations in abstract spaces </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A34A08" title="MSC2020">34A08</a> </td> <td class="space"> Fractional ordinary differential equations </td> </tr></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3Atime-fractional+differential+equations">time-fractional differential equations</a>; <a href="/?q=ut%3Aresolvent+families">resolvent families</a>; <a href="/?q=ut%3A%5C%28%5Calpha%5C%29-times+resolvent+families">$\alpha$-times resolvent families</a>; <a href="/?q=ut%3Agenerators">generators</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 1312.34094" data-ciurl="/ci/06440190" data-biburl="/bibtex/06440190.bib" data-amsurl="/amsrefs/06440190.bib" data-xmlurl="/xml/06440190.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/06440190.pdf" title="Zbl 1312.34094 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/10.2478/s13540-012-0044-x" aria-label="DOI for “On a class of time-fractional differential equations”" title="10.2478/s13540-012-0044-x">DOI</a> </div> <div class="sfx" style="float: right;"> <a href="https://creativecommons.org/licenses/by-nc-nd/3.0/" 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"> <img src="https://static.zbmath.org/contrib/img/cc/svg/icons/nc.svg" alt="Creative Commons NC license icon" class="cc-license-icon"> <img src="https://static.zbmath.org/contrib/img/cc/svg/icons/nd.svg" alt="Creative Commons ND license icon" class="cc-license-icon"> </a> </div> </div> <div class="references"> <h3>References:</h3> <table><tr> <td>[1]</td> <td class="space">W. 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