Stability Analysis of Linear Distributed Order Fractional Systems With Distributed Delays

Boyadzhiev, Doychin; Kiskinov, Hristo; Veselinova, Magdalena; Zahariev, Andrey

doi:10.1515/fca-2017-0048

Cited by 25 publications

(16 citation statements)

References 27 publications

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“…Neutral fractional systems with distributed delays are essentially studied less (see [19][20][21]). Stability properties of retarded fractional systems with derivatives of distributed order are studied in [22]. One of the existing best applications of fractional order equations with delays is modeling human manual control, in which perceptual and neuromuscular delays introduce a delay term.…”

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confidence: 99%

Integral Representation for the Solutions of Autonomous Linear Neutral Fractional Systems with Distributed Delay

et al. 2020

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The aim of this work is to obtain an integral representation formula for the solutions of initial value problems for autonomous linear fractional neutral systems with Caputo type derivatives and distributed delays. The results obtained improve and extend the corresponding results in the particular case of fractional systems with constant delays and will be a useful tool for studying different kinds of stability properties. The proposed results coincide with the corresponding ones for first order neutral linear differential systems with integer order derivatives. MSC: 34A08; 34A12 Introduction and NotationsFractional Calculus has a long history, but it has attracted considerable attention recently as an important tool for modeling of various real problems, such as viscoelastic systems, diffusion processes, signal and control processing, and seismic processes. Detailed information about the fractional calculus theory and its applications can be found in the monographs [1][2][3][4]. Some results for fractional linear systems with delays are in given in the book [5]. The monograph [6] is devoted to the impulsive differential and functional differential equations with fractional derivatives, as well as to some of their applications.It is well known that the study of linear fractional equations (integral representation, several types of stability, etc.) is an evergreen theme for research. Concerning these fields of fundamental and qualitative investigations for linear fractional ordinary differential equations and systems we refer to [2,4,7] and the references therein. Using the Laplace transform method, several interesting results in this direction are obtained in [8,9] as well. Regarding works concerning fractional differential systems with constant delays, we point out [10][11][12][13]. Concerning the retarded differential systems with variable or distributed delays-fundamental theory and application (stability properties)-we refer to [11,[14][15][16][17][18]. Neutral fractional systems with distributed delays are essentially studied less (see [19][20][21]). Stability properties of retarded fractional systems with derivatives of distributed order are studied in [22]. One of the existing best applications of fractional order equations with delays is modeling human manual control, in which perceptual and neuromuscular delays introduce a delay term. As interesting studies, we refer to [23,24].

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Integral Representation for the Solutions of Autonomous Linear Neutral Fractional Systems with Distributed Delay

et al. 2020

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“…is the fundamental solution for the IVP, (21) and (22). If we first calculate the inverse Laplace transform for Equation 25, using the formula of the Laplace transform on the Mittag-Leffler functions…”

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confidence: 99%

“…For such equations and systems, existence and uniqueness of solutions, and stability analysis were established. 17,21,22 In this paper, we propose a new model of the fractional Black-Scholes equation by using the right fractional derivatives to model the terminal value problem. The pricing of options is a backward problem, and so the option pricing at time t depends on the terminal value at time T. The right fractional derivatives of a function f(t) is determined by the value of the future of f(t).…”

Section: Introductionmentioning

confidence: 99%

Fractional model and solution for the Black‐Scholes equation

Duan

Lü

Chen

et al. 2017

Math Methods in App Sciences

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This work presents a new model of the fractional Black-Scholes equation by using the right fractional derivatives to model the terminal value problem. Through nondimensionalization and variable replacements, we convert the terminal value problem into an initial value problem for a fractional convection diffusion equation. Then the problem is solved by using the Fourier-Laplace transform. The fundamental solutions of the derived initial value problem are given and simulated and display a slow anomalous diffusion in the fractional case. KEYWORDS asset pricing models, Black-Scholes equation, fractional derivative, initial value problem, mathematical finance, terminal value problem Math Meth Appl Sci. 2018;41 697-704.wileyonlinelibrary.com/journal/mma

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“…Furthermore, in [19,20], the asymptotical stability of fractional systems has been analyzed by using Lyapunov functional method. Some new results were derived for the stability of fractional differential systems with distributed delays in [21,22].…”

Section: Introductionmentioning

confidence: 99%

Global Asymptotical Stability Analysis for Fractional Neural Networks with Time-Varying Delays

Zhang

Liu

et al. 2019

Mathematics

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In this paper, the global asymptotical stability of Riemann-Liouville fractional-order neural networks with time-varying delays is studied. By combining the Lyapunov functional function and LMI approach, some sufficient criteria that guarantee the global asymptotical stability of such fractional-order neural networks with both discrete time-varying delay and distributed time-varying delay are derived. The stability criteria is suitable for application and easy to be verified by software. Lastly, some numerical examples are presented to check the validity of the obtained results.

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Stability Analysis of Linear Distributed Order Fractional Systems With Distributed Delays

Cited by 25 publications

References 27 publications

Integral Representation for the Solutions of Autonomous Linear Neutral Fractional Systems with Distributed Delay

Integral Representation for the Solutions of Autonomous Linear Neutral Fractional Systems with Distributed Delay

Fractional model and solution for the Black‐Scholes equation

Global Asymptotical Stability Analysis for Fractional Neural Networks with Time-Varying Delays

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