@article{DiethelmHashemishahrakiThaietal., author = {Diethelm, Kai and Hashemishahraki, Safoura and Thai, Ha Duc and Tuan, Hoang The}, title = {A constructive approach for investigating the stability of incommensurate fractional differential systems}, series = {Journal of Mathematical Analysis and Applications}, volume = {540}, journal = {Journal of Mathematical Analysis and Applications}, number = {2}, doi = {10.1016/j.jmaa.2024.128642}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:863-opus-51954}, pages = {21}, abstract = {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 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.}, language = {en} } @article{DiethelmHashemishahrakiThaietal., author = {Diethelm, Kai and Hashemishahraki, Safoura and Thai, Ha Duc and Tuan, Hoang The}, title = {Stability properties of multi-order fractional differential systems in 3D}, series = {IFAC-PapersOnLine}, volume = {58}, journal = {IFAC-PapersOnLine}, number = {12}, doi = {10.1016/j.ifacol.2024.08.195}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:863-opus-51960}, pages = {231 -- 236}, abstract = {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.}, language = {en} } @article{ChaudharyDiethelm, author = {Chaudhary, Renu and Diethelm, Kai}, title = {Novel variants of diffusive representation of fractional integrals: Construction and numerical computation}, series = {IFAC-PapersOnLine}, volume = {58}, journal = {IFAC-PapersOnLine}, number = {12}, doi = {10.1016/j.ifacol.2024.08.226}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:863-opus-51975}, pages = {412 -- 417}, abstract = {In this paper, we revisit the diffusive representations of fractional integrals established in Diethelm (2023a) to explore novel variants of such representations which provide highly efficient numerical algorithms for the approximate numerical evaluation of fractional integrals.}, language = {en} } @inproceedings{Diethelm, author = {Diethelm, Kai}, title = {Shooting methods for fractional Dirichlet-type boundary value problems of order α ∈ (1, 2) with Caputo derivatives}, series = {Recent Developments in Operator Theory, Mathematical Physics and Complex Analysis}, booktitle = {Recent Developments in Operator Theory, Mathematical Physics and Complex Analysis}, editor = {Ball, Joseph and Tylli, Hans-Olav and Virtanen, Jani A.}, publisher = {Birkh{\"a}user}, address = {Cham}, isbn = {978-3-032-00154-2}, doi = {10.1007/978-3-032-00155-9_8}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:863-opus-59665}, abstract = {For the numerical solution of Dirichlet-type boundary value problems associated to nonlinear fractional differential equations of order 𝛼 ∈ (1, 2) that use Caputo derivatives, we suggest to employ shooting methods. In particular, we demonstrate that the so-called proportional secting technique for selecting the required initial values leads to numerical schemes that converge to high accuracy in a very small number of shooting iterations, and we provide an explanation of the analytical background for this favourable numerical behaviour.}, language = {en} }