TY - JOUR A1 - Diethelm, Kai A1 - Hashemishahraki, Safoura A1 - Thai, Ha Duc A1 - Tuan, Hoang The T1 - A constructive approach for investigating the stability of incommensurate fractional differential systems JF - Journal of Mathematical Analysis and Applications N2 - 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. Y1 - 2024 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:bvb:863-opus-51954 VL - 540 IS - 2 ER - TY - JOUR A1 - Diethelm, Kai A1 - Hashemishahraki, Safoura A1 - Thai, Ha Duc A1 - Tuan, Hoang The T1 - Stability properties of multi-order fractional differential systems in 3D JF - IFAC-PapersOnLine N2 - 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. Y1 - 2024 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:bvb:863-opus-51960 VL - 58 IS - 12 SP - 231 EP - 236 ER - TY - JOUR A1 - Chaudhary, Renu A1 - Diethelm, Kai T1 - Novel variants of diffusive representation of fractional integrals: Construction and numerical computation JF - IFAC-PapersOnLine N2 - 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. Y1 - 2024 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:bvb:863-opus-51975 VL - 58 IS - 12 SP - 412 EP - 417 ER - TY - CHAP A1 - Diethelm, Kai ED - Ball, Joseph ED - Tylli, Hans-Olav ED - Virtanen, Jani A. T1 - Shooting methods for fractional Dirichlet-type boundary value problems of order α ∈ (1, 2) with Caputo derivatives T2 - Recent Developments in Operator Theory, Mathematical Physics and Complex Analysis N2 - 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. KW - fractional differential equation KW - Caputo derivative KW - boundary condition KW - boundary value problem KW - shooting method KW - proportional secting KW - secant method Y1 - 2025 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:bvb:863-opus-59665 SN - 978-3-032-00154-2 PB - Birkhäuser CY - Cham ER -