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1
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Sliding observer for synchronization of fractional order chaotic systems with mismatched parameter

100%
Delavari H., Senejohnny D., Baleanu D.
Open Physics
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2012
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vol. 10
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issue 5
1095-1101
EN
In this paper, we propose an observer-based fractional order chaotic synchronization scheme. Our method concerns fractional order chaotic systems in Brunovsky canonical form. Using sliding mode theory, we achieve synchronization of fractional order response with fractional order drive system using a classical Lyapunov function, and also by fractional order differentiation and integration, i.e. differintegration formulas, state synchronization proved to be established in a finite time. To demonstrate the efficiency of the proposed scheme, fractional order version of a well-known chaotic system; Arnodo-Coullet system is considered as illustrative examples.
2
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A discrete time method to the first variation of fractional order variational functionals

100%
Pooseh S., Almeida R., Torres D.
Open Physics
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2013
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vol. 11
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issue 10
1262-1267
EN
The fact that the first variation of a variational functional must vanish along an extremizer is the base of most effective solution schemes to solve problems of the calculus of variations. We generalize the method to variational problems involving fractional order derivatives. First order splines are used as variations, for which fractional derivatives are known. The Grünwald-Letnikov definition of fractional derivative is used, because of its intrinsic discrete nature that leads to straightforward approximations.
3
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Existence and uniqueness of a complex fractional system with delay

100%
Ibrahim R., Jalab H.
Open Physics
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2013
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vol. 11
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issue 10
1528-1535
EN
Chaotic complex systems are utilized to characterize thermal convection of liquid flows and emulate the physics of lasers. This paper deals with the time-delay of a complex fractional-order Liu system. We have examined its chaos, computed numerical solutions and established the existence and uniqueness of those solutions. Ultimately, we have presented some examples.
4
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Analysis of football player’s motion in view of fractional calculus

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Couceiro M., Clemente F., Martins F.
Open Physics
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2013
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vol. 11
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issue 6
714-723
EN
Accurately retrieving the position of football players over time may lay the foundations for a whole series of possible new performance metrics for coaches and assistants. Despite the recent developments of automatic tracking systems, the misclassification problem (i.e., misleading a given player by another) still exists and requires human operators as final evaluators. This paper proposes an adaptive fractional calculus (FC) approach to improve the accuracy of tracking methods by estimating the position of players based on their trajectory so far. One half-time of an official football match was used to evaluate the accuracy of the proposed approach under different sampling periods of 250, 500 and 1000 ms. Moreover, the performance of the FC approach was compared with position-based and velocity-based methods. The experimental evaluation shows that the FC method presents a high classification accuracy for small sampling periods. Such results suggest that fractional dynamics may fit the trajectory of football players, thus being useful to increase the autonomy of tracking systems.
5
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Diffusion problems on fractional nonlocal media

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Sapora A., Cornetti P., Carpinteri A.
Open Physics
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2013
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vol. 11
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issue 10
1255-1261
EN
In this paper, the nonlocal diffusion in one-dimensional continua is investigated by means of a fractional calculus approach. The problem is set on finite spatial domains and it is faced numerically by means of fractional finite differences, both for what concerns the transient and the steady-state regimes. Nonlinear deviations from classical solutions are observed. Furthermore, it is shown that fractional operators possess a clear physical-mechanical meaning, representing conductors, whose conductance decays as a power-law of the distance, connecting non-adjacent points of the body.
6
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Analysis on the time and frequency domain for the RC electric circuit of fractional order

100%
Guía M., Gómez F., Rosales J.
Open Physics
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2013
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vol. 11
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issue 10
1366-1371
EN
This paper provides an analysis in the time and frequency domain of an RC electrical circuit described by a fractional differential equation of the order 0 < α≤ 1. We use the Laplace transform of the fractional derivative in the Caputo sense. In the time domain we emphasize on the delay, rise and settling times, while in the frequency domain the interest is in the cutoff frequency, the bandwidth and the asymptotes in low and high frequencies. All these quantities depend on the order of differential equation.
7
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Fractional-order TV-L2 model for image denoising

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Chen D., Sun S., Zhang C., Chen Y., Xue D.
Open Physics
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2013
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vol. 11
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issue 10
1414-1422
EN
This paper proposes a new fractional order total variation (TV) denoising method, which provides a much more elegant and effective way of treating problems of the algorithm implementation, ill-posed inverse, regularization parameter selection and blocky effect. Two fractional order TV-L2 models are constructed for image denoising. The majorization-minimization (MM) algorithm is used to decompose these two complex fractional TV optimization problems into a set of linear optimization problems which can be solved by the conjugate gradient algorithm. The final adaptive numerical procedure is given. Finally, we report experimental results which show that the proposed methodology avoids the blocky effect and achieves state-of-the-art performance. In addition, two medical image processing experiments are presented to demonstrate the validity of the proposed methodology.
8
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Numerical solution of fractional differential equations via a Volterra integral equation approach

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Esmaeili S., Shamsi M., Dehghan M.
Open Physics
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2013
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vol. 11
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issue 10
1470-1481
EN
The main focus of this paper is to present a numerical method for the solution of fractional differential equations. In this method, the properties of the Caputo derivative are used to reduce the given fractional differential equation into a Volterra integral equation. The entire domain is divided into several small domains, and by collocating the integral equation at two adjacent points a system of two algebraic equations in two unknowns is obtained. The method is applied to solve linear and nonlinear fractional differential equations. Also the error analysis is presented. Some examples are given and the numerical simulations are also provided to illustrate the effectiveness of the new method.
9
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Posicast control of a class of fractional-order processes

100%
Gonzalez E., Hung J., Dorcak L., Terpak J., Petras I.
Open Physics
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2013
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vol. 11
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issue 6
868-880
EN
The number of studies on the control of fractional-order processes-processes having dynamics described by differential equations of arbitrary order-has been increasing in the past two decades and it is now ubiquitous. Various methods have emerged and have been proven to effectively control such processes-usually resulting in fractional-order controllers similar to their conventional integer-order counterparts, which include, but are not limited to fractional PID and fractional lead-lag controllers. However, such methods require a lot of computational effort and fractional-order controllers could be challenging when it comes to their synthesis and implementation. In this paper, we propose a simple yet effective delay-based controller with the use of the Posicast control methodology in controlling the overshoot of a fractional-order process of the class $$\mathcal{P}:\left\{ {P\left( s \right) = {1 \mathord{\left/ {\vphantom {1 {\left( {as^\alpha + b} \right)}}} \right. \kern-\nulldelimiterspace} {\left( {as^\alpha + b} \right)}}} \right\}$$ having orders 1 < α < 2. Such controllers have proven to be easy to implement because they only require delays and summers. In this paper, the Posicast control methodology introduced in the past few years is modified to minimize the overshoot of the processes step response to a level that is acceptable in control engineering and automation practices. Furthermore, proof of the existence of overshoot for such class of processes, as well as the determination of the peak-time of the open-loop response of a fractional-order process of the class P is presented. Validation through numerical simulations for a class of fractional-order processes are presented in this paper.
10
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Conciliating the nonadditive entropy approach and the fractional model formulation when describing subdiffusion

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Kosztołowicz T., Lewandowska K.
Open Physics
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2012
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vol. 10
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issue 3
645-651
EN
We consider here two different models describing subdiffusion. One of them is derived from Continuous Time Random Walk formalism and utilizes a subdiffusion equation with a fractional time derivative. The second model is based on Sharma-Mittal nonadditive entropy formalism where the subdiffusive process is described by a nonlinear equation with ordinary derivatives. Using these two models we describe the process of a substance released from a thick membrane and we find functions which determine the time evolution of the amount of substance remaining inside this membrane. We then find ‘the agreement conditions’ under which these two models provide the same relation defining subdiffusion and give the same function characterizing the process of the released substance. These agreement conditions enable us to determine the relation between the parameters occuring in both models.
11
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Content available

Economic growth in the European Union modelled with fractional derivatives: first results

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Tejado I., Valério D., Pérez E.
Bulletin of the Polish Academy of Sciences: Technical Sciences
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2018
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issue 4
12
Publication available in full text mode
Content available

Stability conditions for fractional-order linear equations with delays

100%
Mozyrska D., Ostalczyk P., Wyrwas M.
Bulletin of the Polish Academy of Sciences: Technical Sciences
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2018
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issue 4
13
Content available remote

RLC electrical circuit of non-integer order

100%
Gómez F., Rosales J., Guía M.
Open Physics
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2013
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vol. 11
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issue 10
1361-1365
EN
In this work a fractional differential equation for the electrical RLC circuit is studied. The order of the derivative being considered is 0 < γ ≤ 1. To keep the dimensionality of the physical quantities R, L and C an auxiliary parameter γ is introduced. This parameter characterizes the existence of fractional components in the system. It is shown that there is a relation between and σ through the physical parameters RLC of the circuit. Due to this relation, the analytical solution is given in terms of the Mittag-Leffler function depending on the order of the fractional differential equation.
14
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Response of a fractional nonlinear system to harmonic excitation by the averaging method

100%
Duan J.-S., Huang C., Liu L.-L.
Open Physics
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2015
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vol. 13
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issue 1
EN
In this work, we consider a fractional nonlinear vibration system of Duffing type with harmonic excitation by using the fractional derivative operator -∞−1Dαt and the averaging method. We derive the steady-state periodic response and the amplitude-frequency and phase-frequency relations. Jumping phenomena caused by the nonlinear term and resonance peaks are displayed, which is similar to the integer-order case. It is possible that a minimum of the amplitude exists before the resonance appears for some values of the modelling parameters, which is a feature for the fractional case. The effects of the parameters in the fractional derivative term on the amplitude-frequency curve are discussed.
15
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Problems in solving fractional differential equations in a microcontroller implementation of an FOPID controller

88%
Matusiak M., Ostalczyk P., Matusiak M., Ostalczyk P.
Archives of Electrical Engineering
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2019
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vol. 68
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issue 3
16
Publication available in full text mode
Content available

Stability and robustness analysis of discrete-time fractional-piecewise-constant-order PID controller

88%
Oziablo P., Mozyrska D., Wyrwas M., Oziablo P., Mozyrska D., Wyrwas M.
Bulletin of the Polish Academy of Sciences: Technical Sciences
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2021
17
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On the derivative chain-rules in fractional calculus via fractional difference and their application to systems modelling

88%
Jumarie G.
Open Physics
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2013
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vol. 11
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issue 6
617-633
EN
It has been pointed out that the derivative chains rules in fractional differential calculus via fractional calculus are not quite satisfactory as far as they can yield different results which depend upon how the formula is applied, that is to say depending upon where is the considered function and where is the function of function. The purpose of the present short note is to display some comments (which might be clarifying to some readers) on the matter. This feature is basically related to the non-commutativity of fractional derivative on the one hand, and furthermore, it is very close to the physical significance of the systems under consideration on the other hand, in such a manner that everything is right so. As an example, it is shown that the trivial first order system may have several fractional modelling depending upon the way by which it is observed. This suggests some rules to construct the fractional models of standard dynamical systems, in as meaningful a model as possible. It might happen that this pitfall comes from the feature that a function which is continuous everywhere, but is nowhere differentiable, exhibits random-like features.
18
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Content available

Optimal input signal design for fractional-order system identification

88%
Jakowluk W.
Bulletin of the Polish Academy of Sciences: Technical Sciences
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2019
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issue 1
19
Content available remote

Thermoelasticity of thin shells based on the time-fractional heat conduction equation

88%
Povstenko Y.
Open Physics
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2013
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vol. 11
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issue 6
685-690
EN
The time-nonlocal generalizations of Fourier’s law are analyzed and the equations of the generalized thermoelasticity based on the time-fractional heat conduction equation with the Caputo fractional derivative of order 0 < α ≤ 2 are presented. The equations of thermoelasticity of thin shells are obtained under the assumption of linear dependence of temperature on the coordinate normal to the median surface of a shell. The conditions of Newton’s convective heat exchange between a shell and the environment have been assumed. In the particular case of classical heat conduction (α = 1) the obtained equations coincide with those known in the literature.
20
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Fractional diffusion equation in half-space with Robin boundary condition

88%
Duan J.-S., Wang Z., Fu S.-Z.
Open Physics
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2013
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vol. 11
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issue 6
799-805
EN
The initial and boundary value problem for the fractional diffusion equation in half-space with the Robin boundary condition is considered. The solution is comprised of two parts: the contribution of the initial value and the contribution of the boundary value, for which the respective fundamental solutions are given. Finally, the solution formula of the considered problem is obtained.
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