Fractional Lotka-Volterra-Type Cooperation Models: Impulsive Control on Their Stability Behavior

We present a biological fractional n-species delayed cooperation model of Lotka-Volterra type. The considered fractional derivatives are in the Caputo sense. Impulsive control strategies are applied for several stability properties of the states, namely Mittag-Leffler stability, practical stability and stability with respect to sets. The proposed results extend the existing stability results for integer-order n−species delayed Lotka-Volterra cooperation models to the fractional-order case under impulsive control.

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Analogue Fractional-Order Generalized Memristive Devices

Volume 4: 7th International Conference on Multibody Systems, Nonlinear Dynamics, and Control, Parts A, B and C ◽

10.1115/detc2009-86861 ◽

2009 ◽

Cited By ~ 17

Author(s):

Calvin Coopmans ◽

Ivo Petra´sˇ ◽

YangQuan Chen

Keyword(s):

Fractional Order ◽

Functional Relation ◽

Theoretical Description ◽

Practical Realization ◽

Integer Order ◽

Order Case ◽

Memristive Systems ◽

First Time ◽

Fractional Order Integral ◽

Missing Element

Memristor is a new electrical element which has been predicted and described in 1971 by Leon O. Chua and for the first time realized by HP laboratory in 2008. Chua proved that memristor behavior could not be duplicated by any circuit built using only the other three elements (resistor, capacitor, inductor), which is why the memristor is truly fundamental. Memristor is a contraction of memory resistor, because that is exactly its function: to remember its history. The memristor is a two-terminal device whose resistance depends on the magnitude and polarity of the voltage applied to it and the length of time that voltage has been applied. The missing element—the memristor, with memristance M—provides a functional relation between charge and flux, dφ = Mdq. In this paper, for the first time, the concept of (integer-order) memristive systems is generalized to non-integer order case using fractional calculus. We also show that the memory effect of such devices can be also used for an analogue implementation of the fractional-order operator, namely fractional-order integral and fractional-order derivatives. This kind of operators are useful for realization of the fractional-order controllers. We present theoretical description of such implementation and we proposed the practical realization and did some experiments as well.

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Geometric interpretation of fractional-order derivative

Fractional Calculus and Applied Analysis ◽

10.1515/fca-2016-0062 ◽

2016 ◽

Vol 19 (5) ◽

Cited By ~ 6

Author(s):

Vasily E. Tarasov

Keyword(s):

Differential Geometry ◽

Fractional Order ◽

Infinite Series ◽

Fractional Derivatives ◽

Geometric Interpretation ◽

Order Derivative ◽

Jet Bundles ◽

Fractional Order Derivative ◽

Integer Order ◽

Derivatives Of

AbstractA new geometric interpretation of the Riemann-Liouville and Caputo derivatives of non-integer orders is proposed. The suggested geometric interpretation of the fractional derivatives is based on modern differential geometry and the geometry of jet bundles. We formulate a geometric interpretation of the fractional-order derivatives by using the concept of the infinite jets of functions. For this interpretation, we use a representation of the fractional-order derivatives by infinite series with integer-order derivatives. We demonstrate that the derivatives of non-integer orders connected with infinite jets of special type. The suggested infinite jets are considered as a reconstruction from standard jets with respect to order.

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Linear stationary fractional differential equations

Fractional Calculus and Applied Analysis ◽

10.1515/fca-2019-0025 ◽

2019 ◽

Vol 22 (2) ◽

pp. 412-423

Author(s):

Valeriy Nosov ◽

Jesús Alberto Meda-Campaña

Keyword(s):

Differential Equations ◽

Fractional Order ◽

Fractional Differential Equations ◽

Integer Order ◽

Fractional Differential ◽

Stability Results

Abstract In this paper, fractional-order derivatives satisfying conventional concepts, are considered in order to present some stability results on linear stationary differential equations of fractional-order. As expected, the obtained results are very close to the ones widely accepted in differential equations of integer order. Some examples are included in order to show how some restrictions of more sophisticated fractional-order derivatives are overcome.

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Analysis of Fractional-Order Nonlinear Dynamic Systems with General Analytic Kernels: Lyapunov Stability and Inequalities

Mathematics ◽

10.3390/math9172084 ◽

2021 ◽

Vol 9 (17) ◽

pp. 2084

Author(s):

Oscar Martínez-Fuentes ◽

Fidel Meléndez-Vázquez ◽

Guillermo Fernández-Anaya ◽

José Francisco Gómez-Aguilar

Keyword(s):

Fractional Derivative ◽

Fractional Order ◽

Lyapunov Functions ◽

Fractional Derivatives ◽

Direct Method ◽

The Laplace Transform ◽

Uniformly Convergent ◽

The Stability ◽

Stability Results ◽

Stability Concept

In this paper, we study the recently proposed fractional-order operators with general analytic kernels. The kernel of these operators is a locally uniformly convergent power series that can be chosen adequately to obtain a family of fractional operators and, in particular, the main existing fractional derivatives. Based on the conditions for the Laplace transform of these operators, in this paper, some new results are obtained—for example, relationships between Riemann–Liouville and Caputo derivatives and inverse operators. Later, employing a representation for the product of two functions, we determine a form of calculating its fractional derivative; this result is essential due to its connection to the fractional derivative of Lyapunov functions. In addition, some other new results are developed, leading to Lyapunov-like theorems and a Lyapunov direct method that serves to prove asymptotic stability in the sense of the operators with general analytic kernels. The FOB-stability concept is introduced, which generalizes the classical Mittag–Leffler stability for a wide class of systems. Some inequalities are established for operators with general analytic kernels, which generalize others in the literature. Finally, some new stability results via convex Lyapunov functions are presented, whose importance lies in avoiding the calculation of fractional derivatives for the stability analysis of dynamical systems. Some illustrative examples are given.

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Epidemiological analysis of fractional order COVID-19 model with Mittag-Leffler kernel

AIMS Mathematics ◽

10.3934/math.2022046 ◽

2022 ◽

Vol 7 (1) ◽

pp. 756-783

Author(s):

Muhammad Farman ◽

◽

Ali Akgül ◽

Kottakkaran Sooppy Nisar ◽

Dilshad Ahmad ◽

...

Keyword(s):

Fractional Order ◽

Fractional Derivatives ◽

Fixed Point Theory ◽

Control Strategies ◽

Point Theory ◽

Fractional Operators ◽

Order Model ◽

Epidemiological Analysis ◽

Fractional Order Model ◽

Sumudu Transform

<abstract> <p>This paper derived fractional derivatives with Atangana-Baleanu, Atangana-Toufik scheme and fractal fractional Atangana-Baleanu sense for the COVID-19 model. These are advanced techniques that provide effective results to analyze the COVID-19 outbreak. Fixed point theory is used to derive the existence and uniqueness of the fractional-order model COVID-19 model. We also proved the property of boundedness and positivity for the fractional-order model. The Atangana-Baleanu technique and Fractal fractional operator are used with the Sumudu transform to find reliable results for fractional order COVID-19 Model. The generalized Mittag-Leffler law is also used to construct the solution with the different fractional operators. Numerical simulations are performed for the developed scheme in the range of fractional order values to explain the effects of COVID-19 at different fractional values and justify the theoretical outcomes, which will be helpful to understand the outbreak of COVID-19 and for control strategies.</p> </abstract>

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COMMENTS ON "RIEMANN–CHRISTOFFEL TENSOR IN DIFFERENTIAL GEOMETRY OF FRACTIONAL ORDER APPLICATION TO FRACTAL SPACE-TIME", [FRACTALS 21 (2013) 1350004]

Fractals ◽

10.1142/s0218348x15750018 ◽

2015 ◽

Vol 23 (02) ◽

pp. 1575001 ◽

Cited By ~ 4

Author(s):

VASILY E. TARASOV

Keyword(s):

Differential Geometry ◽

Fractional Derivative ◽

Power Function ◽

Fractional Order ◽

Fractional Derivatives ◽

Space Time ◽

Leibniz Rule ◽

Integer Order ◽

Fractal Space ◽

Derivatives Of

We prove that main properties represented by Eq. (4.2) for fractional derivative of power function and the non-fractional Leibniz rule in the form (4.3) of the considered paper, cannot hold together for derivatives of non-integer order. As a result, we prove that the usual Leibniz rule (4.3) cannot hold for fractional derivatives.

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The properties of Bessel-type fractional derivatives and integrals

JOURNAL OF ADVANCES IN PHYSICS ◽

10.24297/jap.v11i8.203 ◽

2016 ◽

pp. 3973-3982

Author(s):

V. R. Lakshmi Gorty

Keyword(s):

Fractional Order ◽

Real Line ◽

Computer Algebra System ◽

Fractional Derivatives ◽

Graphical Representation ◽

Fractional Integrals ◽

The Real ◽

Fractional Derivatives And Integrals ◽

Half Axis ◽

Derivatives Of

The fractional integrals of Bessel-type Fractional Integrals from left-sided and right-sided integrals of fractional order is established on finite and infinite interval of the real-line, half axis and real axis. The Bessel-type fractional derivatives are also established. The properties of Fractional derivatives and integrals are studied. The fractional derivatives of Bessel-type of fractional order on finite of the real-line are studied by graphical representation. Results are direct output of the computer algebra system coded from MATLAB R2011b.

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Dynamic study of SIAISQVR−B fractional-order cholera model with control strategies in Cameroon Far North Region

Chaos Solitons & Fractals ◽

10.1016/j.chaos.2021.110702 ◽

2021 ◽

Vol 144 ◽

pp. 110702

Author(s):

Tchule Nguiwa ◽

Gabriel Guilsou Kolaye ◽

Mibaile Justin ◽

Djaouda Moussa ◽

Gambo Betchewe ◽

...

Keyword(s):

Fractional Order ◽

Control Strategies ◽

Dynamic Study ◽

Far North

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Fractional-Order Colour Image Processing

Mathematics ◽

10.3390/math9050457 ◽

2021 ◽

Vol 9 (5) ◽

pp. 457

Author(s):

Manuel Henriques ◽

Duarte Valério ◽

Paulo Gordo ◽

Rui Melicio

Keyword(s):

Image Processing ◽

Edge Detection ◽

Fractional Derivative ◽

Fractional Order ◽

Fractional Derivatives ◽

Grey Scale ◽

Colour Image ◽

Colour Image Processing ◽

Image Processing Algorithms ◽

Processing Algorithms

Many image processing algorithms make use of derivatives. In such cases, fractional derivatives allow an extra degree of freedom, which can be used to obtain better results in applications such as edge detection. Published literature concentrates on grey-scale images; in this paper, algorithms of six fractional detectors for colour images are implemented, and their performance is illustrated. The algorithms are: Canny, Sobel, Roberts, Laplacian of Gaussian, CRONE, and fractional derivative.

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Stability Analysis and Existence of Solutions for a Modified SIRD Model of COVID-19 with Fractional Derivatives

Symmetry ◽

10.3390/sym13081431 ◽

2021 ◽

Vol 13 (8) ◽

pp. 1431

Author(s):

Bilal Basti ◽

Nacereddine Hammami ◽

Imadeddine Berrabah ◽

Farid Nouioua ◽

Rabah Djemiat ◽

...

Keyword(s):

Mathematical Model ◽

Existence And Uniqueness ◽

Fractional Derivatives ◽

Fixed Point Theorems ◽

Existence Of Solutions ◽

Biological Models ◽

Uniqueness Of Solutions ◽

Caputo Fractional Derivatives ◽

Single Form ◽

Stability Results

This paper discusses and provides some analytical studies for a modified fractional-order SIRD mathematical model of the COVID-19 epidemic in the sense of the Caputo–Katugampola fractional derivative that allows treating of the biological models of infectious diseases and unifies the Hadamard and Caputo fractional derivatives into a single form. By considering the vaccine parameter of the suspected population, we compute and derive several stability results based on some symmetrical parameters that satisfy some conditions that prevent the pandemic. The paper also investigates the problem of the existence and uniqueness of solutions for the modified SIRD model. It does so by applying the properties of Schauder’s and Banach’s fixed point theorems.

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