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  <doc>
    <id>2025</id>
    <completedYear/>
    <publishedYear>2022</publishedYear>
    <thesisYearAccepted/>
    <language>eng</language>
    <pageFirst>166</pageFirst>
    <pageLast>180</pageLast>
    <pageNumber/>
    <edition/>
    <issue/>
    <volume>25</volume>
    <type>article</type>
    <publisherName/>
    <publisherPlace/>
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    <belongsToBibliography>1</belongsToBibliography>
    <completedDate>--</completedDate>
    <publishedDate>2022-02-07</publishedDate>
    <thesisDateAccepted>--</thesisDateAccepted>
    <title language="eng">Upper and lower estimates for the separation of solutions to fractional differential equations</title>
    <abstract language="eng">Given a fractional differential equation of order α∈(0,1] with Caputo derivatives, we investigate in a quantitative sense how the associated solutions depend on their respective initial conditions. Specifically, we look at two solutions x1 and x2, say, of the same differential equation, both of which are assumed to be defined on a common interval [0, T], and provide upper and lower bounds for the difference x1(t)−x2(t) for all t∈[0,T] that are stronger than the bounds previously described in the literature.</abstract>
    <parentTitle language="eng">Fractional Calculus and Applied Analysis</parentTitle>
    <identifier type="issn">1314-2224</identifier>
    <identifier type="doi">10.1007/s13540-021-00007-x</identifier>
    <enrichment key="opus.source">publish</enrichment>
    <licence>Creative Commons - CC BY - Namensnennung 4.0 International</licence>
    <author>Kai Diethelm</author>
    <author>Hoang The Tuan</author>
    <collection role="institutes" number="fang">Fakultät für angewandte Natur- und Geisteswissenschaften</collection>
    <collection role="oa-colour" number="">Gefördert (Hybrid/DEAL)</collection>
    <thesisPublisher>Hochschule für Angewandte Wissenschaften Würzburg-Schweinfurt</thesisPublisher>
    <file>https://opus4.kobv.de/opus4-fhws/files/2025/Diethelm_Estimates_separation_solutions.pdf</file>
  </doc>
  <doc>
    <id>5754</id>
    <completedYear/>
    <publishedYear>2024</publishedYear>
    <thesisYearAccepted/>
    <language>eng</language>
    <pageFirst/>
    <pageLast/>
    <pageNumber>25</pageNumber>
    <edition/>
    <issue/>
    <volume>540</volume>
    <type>article</type>
    <publisherName/>
    <publisherPlace/>
    <creatingCorporation/>
    <contributingCorporation/>
    <belongsToBibliography>1</belongsToBibliography>
    <completedDate>--</completedDate>
    <publishedDate>--</publishedDate>
    <thesisDateAccepted>--</thesisDateAccepted>
    <title language="eng">A constructive approach for investigating the stability of incommensurate fractional differential systems</title>
    <abstract language="eng">This paper is devoted to studying the asymptotic behaviour of solutions to generalized incommensurate fractional systems. To this end, we first consider fractional systems with rational orders and introduce a criterion that is necessary and sufficient to ensure the stability of such systems. Next, from the fractional order pseudospectrum definition proposed by Šanca et al., we formulate the concept of a rational approximation for the fractional spectrum of incommensurate fractional systems with general, not necessarily rational, orders. Our first important new contribution is to show the equivalence between the fractional spectrum of a incommensurate linear system and its rational approximation. With this result in hand, we use ideas developed in our earlier work to demonstrate the stability of an equilibrium point to nonlinear systems in arbitrary finite-dimensional spaces. A second novel aspect of our work is the fact that the approach is constructive. It is effective and widely applicable in studying the asymptotic behaviour of solutions to linear incommensurate fractional differential systems with constant coefficient matrices and linearized stability theory for nonlinear incommensurate fractional differential systems. Finally, we give numerical simulations to illustrate the merit of the proposed theoretical results.</abstract>
    <parentTitle language="eng">Journal of Mathematical Analysis and Applications</parentTitle>
    <identifier type="doi">10.1016/j.jmaa.2024.128642</identifier>
    <identifier type="urn">urn:nbn:de:bvb:863-opus-57547</identifier>
    <enrichment key="opus.doi.autoCreate">false</enrichment>
    <enrichment key="opus.urn.autoCreate">true</enrichment>
    <licence>Creative Commons - CC BY - Namensnennung 4.0 International</licence>
    <author>Kai Diethelm</author>
    <author>Safoura Hashemishahraki</author>
    <author>Ha Duc Thai</author>
    <author>Hoang The Tuan</author>
    <collection role="institutes" number="fang">Fakultät für angewandte Natur- und Geisteswissenschaften</collection>
    <collection role="oa-colour" number="">Gefördert (Hybrid/DEAL)</collection>
    <file>https://opus4.kobv.de/opus4-fhws/files/5754/Diethelm_Constructive_approach_incommensurate.pdf</file>
  </doc>
  <doc>
    <id>5195</id>
    <completedYear/>
    <publishedYear>2024</publishedYear>
    <thesisYearAccepted/>
    <language>eng</language>
    <pageFirst/>
    <pageLast/>
    <pageNumber>21</pageNumber>
    <edition/>
    <issue>2</issue>
    <volume>540</volume>
    <type>article</type>
    <publisherName/>
    <publisherPlace/>
    <creatingCorporation/>
    <contributingCorporation/>
    <belongsToBibliography>1</belongsToBibliography>
    <completedDate>--</completedDate>
    <publishedDate>--</publishedDate>
    <thesisDateAccepted>--</thesisDateAccepted>
    <title language="eng">A constructive approach for investigating the stability of incommensurate fractional differential systems</title>
    <abstract language="eng">This paper is devoted to studying the asymptotic behaviour of solutions to generalized noncommensurate fractional systems. To this end, we first consider fractional systems with rational orders and introduce a criterion that is necessary and sufficient to ensure the stability of such systems. Next, from the fractional-order pseudospectrum definition proposed by Sanca et al., we formulate the concept of a rational approximation for the fractional spectrum of a noncommensurate fractional systems with general, not necessarily rational, orders. Our first important new contribution is to&#13;
show the equivalence between the fractional spectrum of a noncommensurate linear system and its rational approximation. With this result in hand, we use ideas developed in our earlier work to demonstrate the stability of an equilibrium point to nonlinear systems in arbitrary finite-dimensional spaces. A second novel aspect of our work is the fact that the approach is constructive. Finally, we give numerical simulations to illustrate the merit of the proposed theoretical results.</abstract>
    <parentTitle language="eng">Journal of Mathematical Analysis and Applications</parentTitle>
    <identifier type="urn">urn:nbn:de:bvb:863-opus-51954</identifier>
    <identifier type="doi">10.1016/j.jmaa.2024.128642</identifier>
    <enrichment key="opus.source">publish</enrichment>
    <licence>Creative Commons - CC BY-NC-SA - Namensnennung - Nicht kommerziell -  Weitergabe unter gleichen Bedingungen 4.0 International</licence>
    <author>Kai Diethelm</author>
    <author>Safoura Hashemishahraki</author>
    <author>Ha Duc Thai</author>
    <author>Hoang The Tuan</author>
    <collection role="institutes" number="fang">Fakultät für angewandte Natur- und Geisteswissenschaften</collection>
    <collection role="oa-colour" number="">Zweitveröffentlichung (Grün)</collection>
    <thesisPublisher>Technische Hochschule Würzburg-Schweinfurt</thesisPublisher>
    <file>https://opus4.kobv.de/opus4-fhws/files/5195/Pseudospectra_arxiv.pdf</file>
  </doc>
  <doc>
    <id>5196</id>
    <completedYear/>
    <publishedYear>2024</publishedYear>
    <thesisYearAccepted/>
    <language>eng</language>
    <pageFirst>231</pageFirst>
    <pageLast>236</pageLast>
    <pageNumber>6</pageNumber>
    <edition/>
    <issue>12</issue>
    <volume>58</volume>
    <type>article</type>
    <publisherName/>
    <publisherPlace/>
    <creatingCorporation/>
    <contributingCorporation/>
    <belongsToBibliography>1</belongsToBibliography>
    <completedDate>--</completedDate>
    <publishedDate>--</publishedDate>
    <thesisDateAccepted>--</thesisDateAccepted>
    <title language="eng">Stability properties of multi-order fractional differential systems in 3D</title>
    <abstract language="eng">This paper is devoted to studying three-dimensional non-commensurate fractional order differential equation systems with Caputo derivatives. Necessary and sufficient conditions are for the asymptotic stability of such systems are obtained.</abstract>
    <parentTitle language="eng">IFAC-PapersOnLine</parentTitle>
    <identifier type="urn">urn:nbn:de:bvb:863-opus-51960</identifier>
    <identifier type="doi">10.1016/j.ifacol.2024.08.195</identifier>
    <enrichment key="opus.source">publish</enrichment>
    <licence>Creative Commons - CC BY-NC-SA - Namensnennung - Nicht kommerziell -  Weitergabe unter gleichen Bedingungen 4.0 International</licence>
    <author>Kai Diethelm</author>
    <author>Safoura Hashemishahraki</author>
    <author>Ha Duc Thai</author>
    <author>Hoang The Tuan</author>
    <collection role="institutes" number="fang">Fakultät für angewandte Natur- und Geisteswissenschaften</collection>
    <thesisPublisher>Technische Hochschule Würzburg-Schweinfurt</thesisPublisher>
    <file>https://opus4.kobv.de/opus4-fhws/files/5196/ICFDAPaper.pdf</file>
  </doc>
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