Finite time stability of tempered fractional systems with time delays

We investigate the notion of finite time stability for tempered fractional systems (TFSs) with time delays and variable coefficients. Then, we examine some sufficient conditions that allow concluding the TFSs stability in a finite time interval, which include the nonhomogeneous and the homogeneous delayed cases. We present two different approaches. The first one is based on H\"older's and Jensen's inequalities, while the second one concerns the Bellman--Gr\"onwall method using the tempered Gr\"onwall inequality. Finally, we provide two numerical examples to show the practicability of the developed procedures.Comment: This is a preprint version of the paper published open access in 'Chaos Solitons Fractals

The stability and stabilization of infinite dimensional Caputo-time fractional differential linear systems

We investigate the stability and stabilization concepts for infinite dimensional time fractional differential linear systems in Hilbert spaces with Caputo derivatives. Firstly, based on a family of operators generated by strongly continuous semigroups and on a probability density function, we provide sufficient and necessary conditions for the exponential stability of the considered class of systems. Then, by assuming that the system dynamics is symmetric and uniformly elliptic and by using the properties of the Mittag-Leffler function, we provide sufficient conditions that ensure strong stability. Finally, we characterize an explicit feedback control that guarantees the strong stabilization of a controlled Caputo time fractional linear system through a decomposition approach. Some examples are presented that illustrate the effectiveness of our results.publishe

Generalized Taylor’s formula for power fractional derivatives

We establish a new generalized Taylor’s formula for power fractional derivatives with nonsingular and nonlocal kernels, which includes many known Taylor’s formulas in the literature. Moreover, as a consequence, we obtain a general version of the classical mean value theorem. We apply our main result to approximate functions in Taylor’s expansions at a given point. The explicit interpolation error is also obtained. The new results are illustrated through examples and numerical simulations.This work was supported by CIDMA and is funded by the Fundação para a Ciência e a Tecnologia, I.P. (FCT, Funder ID 50110000187) under Grants UIDB/04106/2020 and UIDP/04106/2020. Open access funding provided by FCT|FCCN (b-on).publishe

A class of fractional differential equations via power non-local and non-singular kernels: Existence, uniqueness and numerical approximations

We prove a useful formula and new properties for the recently introduced power fractional calculus with non-local and non-singular kernels. In particular, we prove a new version of Gronwall’s inequality involving the power fractional integral; and we establish existence and uniqueness results for nonlinear power fractional differential equations using fixed point techniques. Moreover, based on Lagrange polynomial interpolation, we develop a new explicit numerical method in order to approximate the solutions of a rich class of fractional differential equations. The approximation error of the proposed numerical scheme is analyzed. For illustrative purposes, we apply our method to a fractional differential equation for which the exact solution is computed, as well as to a nonlinear problem for which no exact solution is known. The numerical simulations show that the proposed method is very efficient, highly accurate and converges quickly.The Center for Research and Development in Mathematics and Applications (CIDMA) through the Portuguese Foundation for Science and Technology (FCT – Fundação para a Ciência e a Tecnologia ), project UIDB/04106/2020. Zitane is also grateful to the post-doc fellowship at CIDMA-DMat-UA, reference UIDP/04106/2020.publishe

Pharmacokinetic/Pharmacodynamic Anesthesia Model Incorporating psi-Caputo Fractional Derivatives

We present a novel Pharmacokinetic/Pharmacodynamic (PK/PD) model for the induction phase of anesthesia, incorporating the $\psi$-Caputo fractional derivative. By employing the Picard iterative process, we derive a solution for a nonhomogeneous $\psi$-Caputo fractional system to characterize the dynamical behavior of the drugs distribution within a patient's body during the anesthesia process. To explore the dynamics of the fractional anesthesia model, we perform numerical analysis on solutions involving various functions of $\psi$ and fractional orders. All numerical simulations are conducted using the MATLAB computing environment. Our results suggest that the $\psi$ functions and the fractional order of differentiation have an important role in the modeling of individual-specific characteristics, taking into account the complex interplay between drug concentration and its effect on the human body. This innovative model serves to advance the understanding of personalized drug responses during anesthesia, paving the way for more precise and tailored approaches to anesthetic drug administration.This research was developed within the project “Mathematical Modelling of Multi-scale Control Systems: applications to human diseases (CoSysM3)”, 2022.03091.PTDC, financially supported by national funds (OE), through FCT/MCTES. The authors are also supported by FCT and CIDMA via projects UIDB/04106/2020 and UIDP/04106/ 2020.publishe