@article{ChaudharyDiethelmHashemishahraki, author = {Chaudhary, Renu and Diethelm, Kai and Hashemishahraki, Safoura}, title = {On the separation of solutions to fractional differential equations of order α ∈ (1, 2)}, series = {Applied Numerical Mathematics}, volume = {203}, journal = {Applied Numerical Mathematics}, doi = {10.1016/j.apnum.2024.05.020}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:863-opus-57338}, pages = {Article No. 38}, abstract = {Given a Caputo-type fractional differential equation with order between 1 and 2, we consider two distinct solutions to this equation subject to different sets of initial conditions. In this framework, we discuss nontrivial upper and lower bounds for the difference between these solutions. The main emphasis is on describing how such bounds are related to the differences of the associated initial values.}, language = {en} } @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}, doi = {10.1016/j.jmaa.2024.128642}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:863-opus-57547}, pages = {25}, abstract = {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.}, language = {en} } @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} }