Wyniki wyszukiwania - Biblioteka Nauki

1

A note on some elementary properties and applications of certain operators to certain functions analytic in the unit disk

100%

Irmak H.

Annales Universitatis Paedagogicae Cracoviensis. Studia Mathematica

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2020

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tom 19

193-201

EN

This scientific note relates to introducing certain elementary operators defined in the unit disk in the complex plane, then determining various applications (specified by those operators) to certain analytic functions, and also revealing a number of possible implications of them.

2

Set-valued fractional order differential equations in the space of summable functions

63%

Salem H.

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

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2008

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tom 28

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nr 1

83-93

EN

In this paper, we study the existence of integrable solutions for the set-valued differential equation of fractional type $(D^{αₙ} - ∑_{i=1}^{n-1} a_i D^{α_i})x(t) ∈ F(t,x(φ(t)))$, a.e. on (0,1), $I^{1 - αₙ} x(0) = c$, αₙ ∈ (0,1), where F(t,·) is lower semicontinuous from ℝ into ℝ and F(·,·) is measurable. The corresponding single-valued problem will be considered first.

3

Fractional continua for linear elasticity

63%

Sumelka W. , Blaszczyk T.

Archives of Mechanics

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2014

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tom Vol. 66, nr 3

147--172

EN

Fractional continua is a generalisation of the classical continuum body. This new concept shows the application of fractional calculus in continuum mechanics. The advantage is that the obtained description is non-local. This natural non-locality is inherently a consequence of fractional derivative definition which is based on the interval, thus variates from the classical approach where the definition is given in a point. In the paper, the application of fractional continua to one-dimensional problem of linear elasticity under small deformation assumption is presented.

4

On reflection symmetry in fractional mechanics

51%

Klimek M. , Lupa M.

Prace Naukowe Instytutu Matematyki i Informatyki Politechniki Częstochowskiej

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2011

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tom Vol. 10, nr 1

109-121

EN

We study the properties of fractional differentiation with respect to reflection mapping in a finite interval. The symmetric and anti-symmetric fractional derivatives in a full interval are expressed as fractional differential operators in left or right subintervals obtained by subsequent partitions. These representation properties and the reflection symmetry of the action and variation are applied to derive Euler-Lagrange equations of fractional free motion. Then the localization phenomenon for these equations is discussed.

5

Description of selected pneumatic elements and systems using fractional calculus

51%

Pietruszczak D. , Nowocień A.

Journal of Automation, Electronics and Electrical Engineering

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2019

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tom Vol. 1, No. 1

43--49

EN

The paper presents the application of fractional calculus to describe the dynamics of selected pneumatic elements and systems. In the construction of mathematical models of the analysed dynamic systems, the Riemann-Liouville definition of differintegral of non- integer order was used. For the analysed model, transfer function of integer and non-integer order was determined. Functions describing characteristics in frequency domains were determined, whereas the characteristics of the elements and systems were obtained by means of computer simulation. MATLAB programme were used for the simulation research.

PL

W artykule przedstawiono zastosowanie rachunku różniczkowego niecałkowitych rzędów (ang. fractional calculus) do opisu dynamiki zjawisk układów pneumatycznych wybranych elementów i układów. W budowie modeli matematycznych, analizowanych układów dynamicznych, wykorzystano definicję Riemanna–Liouville’a pochodno–całki niecałkowitego rzędu. Dla analizowanego modelu, wyznaczono transmitancję operatorową całkowitego i niecałkowitego rzędu. Wyznaczono zależności opisujące charakterystyki częstotliwościowe, na drodze symulacji komputerowej uzyskano charakterystyki analizowanych układów. Do badań symulacyjnych wykorzystano oprogramowanie MATLAB.

6

Methods of parameters identification of fractional systems

51%

Janiczek T. , Janiczek J.

Przegląd Elektrotechniczny

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2011

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tom R. 87, nr 3

123-125

EN

This paper presents the parameters of identification method generalized for systems that for obtaining estimates of parameters of models use fractional differential equations i.e. systems containing elements of fractional degree. These elements introduce fractional derivatives to differential equations and in the operator equations they are presented as operators of fractional degree. The discussed method based on the operators wchich affect the defined/ measured incoming and outgoing signals allows to obtain equations or a set of equations with unknown parameters. This method generalized to systems in which there are elements of the fractional degree. The way how to transform the algorithm of the model as to avoid large scale mathematical complications and with alittle loss ot generality is demonstrated.

PL

W niniejszym artykule przedstawiono uogólnienie metody różniczkowej i zbadano ją pod kątem przydatności w identyfikacji systemów zawierających elementy stopnia ułamkowego. Elementy te w równaniach różniczkowych wprowadzają pochodne ułamkowe, zaś w równaniach operatorowych pojawiają się w postaci operatorów z potęgami ułamkowymi. Rozpatrywana poniżej metoda wykorzystująca operatory działające na zadane/mierzone sygnały wejściowe i wyjściowe umożliwia takie przekształcenie sygnałów, aby uzyskać równania bądź układy równań z niewiadomymi parametrami . Metodę tą uogólniono do układów, w których występują elementy stopnia ułamkowego, obrazując sposób przekształceń i poszukiwanie parametrów wybranego modelu procesu.

7

Continuous solutions of some fractional order integral equations

51%

Salem H. A. H. , El-Sayed A. M. A. , Moustafa O. L.

Annales Societatis Mathematicae Polonae. Seria 1: Commentationes Mathematicae

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2002

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tom [Z] 42(2)

209-220

EN

In this paper, Schauder fixed point theorem is used to prove an existence of positive continuous solutions for the nonlinear fractional order integral equation x(t) = h(t) + λ Iα (ƒ(x(t)) + g(x(t))), t ϵ [0, 1], α > 0 (E), where ƒ and g are nonlinear continuous functions and ƒ is nondecreasing while g is nonincreasing. Also the existence of maximal and minimal solutions of (E) will be proved. Some fractional order differential equations will be considered.

8

Fractional-order TV-L2 model for image denoising

51%

Chen D. , Sun S. , Zhang C. , Chen Y. , Xue D.

Open Physics

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2013

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tom 11

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nr 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.

9

Analysis on the time and frequency domain for the RC electric circuit of fractional order

51%

Guía M. , Gómez F. , Rosales J.

Open Physics

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2013

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tom 11

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nr 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.

10

Nonlinear Degenerate Fractional Evolution Equations with Nonlocal Conditions

51%

Derdar N. , Debbouche A.

Fundamenta Informaticae

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2017

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tom Vol. 151, nr 1/4

473--485

EN

We investigate the unique solvability of a class of nonlinear nonlocal differential equations associated with degenerate linear operator at the fractional Caputo derivative. For the main results, we use the theory of fractional calculus and (L, p)-boundedness technique that based on the analysis of both strongly (L, p)-sectorial operators and strongly (L, p)-radial operators. The obtained results are applicable to degenerate fractional Cauchy and Showalter–Sidorov problems in Banach spaces. Finally, we give an application described by time-fractional order Oskolkov system.

11

Application of fractional order theory of thermoelasticity to a 1D problem for a cylindrical cavity

51%

Raslan W. E.

Archives of Mechanics

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2014

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tom Vol. 66, nr 4

257--267

EN

In this work, we apply the fractional order theory of thermoelasticity to a 1D problem of an infinitely long cylindrical cavity. Laplace transform techniques are used to solve the problem. Numerical results are computed and represented graphically for the temperature, displacement and stress distributions.

12

Posicast control of a class of fractional-order processes

51%

Gonzalez E. , Hung J. , Dorcak L. , Terpak J. , Petras I.

Open Physics

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2013

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tom 11

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nr 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.

13

A discrete time method to the first variation of fractional order variational functionals

51%

Pooseh S. , Almeida R. , Torres D.

Open Physics

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2013

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tom 11

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nr 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.

14

Existence and uniqueness of a complex fractional system with delay

51%

Ibrahim R. , Jalab H.

Open Physics

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2013

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tom 11

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nr 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.

15

Analysis of football player’s motion in view of fractional calculus

51%

Couceiro M. , Clemente F. , Martins F.

Open Physics

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2013

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tom 11

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nr 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.

16

Existence of a solution for the fractional forced pendulum

51%

Torres C.

Journal of Applied Mathematics and Computational Mechanics

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2014

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tom Vol. 13, nr 1

125--142

EN

In this work we study the fractional forced pendulum equation with combined fractional derivatives - tDαT 0Dαt u( t ) + g ( u ( t )) = f ( t ), t ∈ ( 0, T ) ( 0. 1 ) u ( 0 ) = u ( T ) = 0 where ½ < α < 1, g ∈ C ( R, R ), bounded f ∈ C [ 0, T ]. Using minimization techniques form variational calculus we show that ( 0. 1 ) has a nontrivial solution.

17

Fractional order model of measured quantity errors

51%

Cioć R. , Chrzan M.

Bulletin of the Polish Academy of Sciences. Technical Sciences

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2019

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tom Vol. 67, nr 6

1023--1030

EN

The paper presents an interpretation of fractional calculus for positive and negative orders of functions based on sampled measured quantities and their errors connected with digital signal processing. The derivatives as a function limit and the Grunwald-Letnikov differintegral are shown in chapter 1 due to the similarity of the presented definition. Notation of fractional calculus based on the gradient vector of measured quantities and its geometrical and physical interpretation of positive and negative orders are shown in chapter 2 and 3.

18

The use of a non-integer order PI controller with an active queue management mechanism

51%

Domański A. , Domańska J. , Czachórski T. , Klamka J.

International Journal of Applied Mathematics and Computer Science

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2016

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tom 26

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nr 4

777-789

EN

In this paper the performance of a fractional order PI controller is compared with that of RED, a well-known active queue management (AQM) mechanism. The article uses fluid flow approximation and discrete-event simulation to investigate the influence of the AQM policy on the packet loss probability, the queue length and its variability. The impact of self-similar traffic is also considered.

19

Numerical solution of fractional differential equations via a Volterra integral equation approach

51%

Esmaeili S. , Shamsi M. , Dehghan M.

Open Physics

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2013

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tom 11

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nr 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.

20

RLC electrical circuit of non-integer order

51%

Gómez F. , Rosales J. , Guía M.

Open Physics

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2013

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tom 11

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nr 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.