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1

Control of chaos in the burke-shaw system of fractal-fractional order in the sense of Caputo-Fabrizio

Saber Sayed

Journal of Applied Mathematics and Computational Mechanics

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2024

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Vol. 23, nr 1

83-96

EN

For the Burke-Shaw system, we propose a fractal-fractional order in the sense of the Caputo-Fabrizio derivative. The proposed system is solved by utilizing the fractal-fractional derivative operator with an exponential decay kernel. Time-fractional Caputo-Fabrizio fractal fractional derivatives are applied to the Burke-Shaw-type nonlinear chaotic systems.Based on fixed point theory, it has been demonstrated that a fractal-fractional-order model under the Caputo-Fabrizio operator exists and is unique. Using a numerical power series method, we solve the fractional Burke-Shaw model. Using Newton’s interpolation polynomial, we solve the equation numerically by implementing a novel numerical scheme based on an efficient polynomial.

2

Równanie Laplace’a w ujęciu pochodnych niecałkowitego rzędu

Zawadzki Andrzej, Włodarczyk Maciej, Różowicz Sebastian

Przegląd Elektrotechniczny

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2023

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R. 99, nr 4

107--110

PL

W pracy przedstawiono równanie Laplace’a w ujęciu pochodnych niecałkowitego rządu oraz podjęto próbę znalezienia analitycznego rozwiązania takiego równania. Do rozwiązania zastosowano metodę separacji zmiennej (metodę Fouriera).

EN

The paper attempts of Laplace type of a linear fractional order differential equation and find an analytical solution. To solve this equation the method of variable separation (Fourier method) was used.

3

Prediction of high-speed debris motion in the framework of time-fractional model: theory and validation

Malendowski Michał, Sumelka Wojciech, Gajewski Tomasz, Studziński Robert, Peksa Piotr, Sielicki Piotr W.

Archives of Civil and Mechanical Engineering

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2023

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Vol. 23, no. 1

art. no. e46, 2023

EN

In this work, a newly proposed fractional derivative framework is used for the prediction of high-speed debris motion. The paper focuses on the mathematical formulation of the equation of motion, in which the damping term is generalised using the fractional derivative. The capacity of the proposed approach to predict the motion of debris is justified by the experimental results. Furthermore, the mathematical formulation has been verified by extensive parametric studies on spherical projectiles. The general conclusion is that the elaborated formulation is more reliable compared to the classical approach or, in other words, the fractional viscous damping term (proportional to the fractional velocity of debris) provides a better description of the complexity of the real drag force.

4

Solution of the modified time fractional coupled burgers equations using laplace adomian decompostion method

Omame Andrew, Zaman Fiazud Din

Acta Mechanica et Automatica

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2023

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Vol. 17, no. 1

124--132

EN

In this work, a coupled system of time-fractional modified Burgers’ equations is considered. Three different fractional operators: Caputo, Caputo-Fabrizio and Atangana-Baleanu operators are implemented for the equations. Also, two different scenarios are examined for each fractional operator: when the initial conditions are u(x,y,0) = sin(xy), v(x,y,0) = sin(xy), and when they are u(x,y,0) = e{−kxy}, v(x,y,0) = e{−kxy}, where k,α are some positive constants. With the aid of computable Adomian polynomials, the solutions are obtained using Laplace Adomian decomposition method (LADM). The method does not need linearization, weak nonlinearity assumptions or perturbation theory. Simulations are also presented to support theoretical results, and the behaviour of the solutions under the three different fractional operators compared.

5

Exact solution for heat conduction inside a sphere with heat absorption using the regularized Hilfer-Prabhakar derivative

Elhadedy Hager, Latif Mohamed S. Abdel, Nour Hamed M., Kader Abbas H. Abdel

Journal of Applied Mathematics and Computational Mechanics

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2022

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Vol. 21, nr 2

27--37

EN

In this article, we utilize the finite Sine-Fourier transform and the Laplace transform for solving fractional partial differential equations with regularized Hilfer-Prabhakar derivative. These transforms are used to get analytical solutions for the time fractional heat conduction equation (TFHCE) with the regularized Hilfer-Prabhakar derivative associated with heat absorption in spherical coordinates. Two cases of Dirichlet boundary conditions are considered by obtaining an analytical solution in each case. The effect of the parameters of the regularized Hilfer-Prabhakar derivative on the heat transfer inside the sphere is discussed using some figures.

6

Stability of a class of entropies based on fractional calculus

Ferreira Rui A. C.

Journal of Applied Analysis

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2022

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Vol. 28, nr 1

105--107

EN

In this work, we argue about the Lesche stability of some systems that are motivated by the use of fractional derivatives.

7

Fractional heat conduction in a rectangular plate with bending moments

Warbhe Shrikant

Journal of Applied Mathematics and Computational Mechanics

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2020

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Vol. 19, nr 4

115--126

EN

In this research work, we consider a thin, simply supported rectangular plate defined as 0 ≤ x ≤ a, 0 ≤ y ≤ b, 0 ≤ z ≤ c and determine the thermal stresses by using a thermal bending moment with the help of a time dependent fractional derivative. The constant temperature is prescribed on the surface y = 0 and other surfaces are maintained at zero temperature. A powerful technique of integral transform is used to find the analytical solution of initial-boundary value problem of a thin rectangular plate. The numerical result of temperature distribution, thermal deflection and thermal stress component are computed and represented graphically for a copper plate.

8

Optimal control problems with a fixed terminal time in linear fractional-order systems

Gomoyunov M. I.

Archives of Control Sciences

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2020

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Vol. 30, No. 4

721--744

EN

The paper deals with an optimal control problem in a dynamical system described by a linear differential equation with the Caputo fractional derivative. The goal of control is to minimize a Bolza-type cost functional, which consists of two terms: the first one evaluates the state of the system at a fixed terminal time, and the second one is an integral evaluation of the control on the whole time interval. In order to solve this problem, we propose to reduce it to some auxiliary optimal control problem in a dynamical system described by a first-order ordinary differential equation. The reduction is based on the representation formula for solutions to linear fractional differential equations and is performed by some linear transformation, which is called the informational image of a position of the original system and can be treated as a special prediction of a motion of this system at the terminal time. A connection between the original and auxiliary problems is established for both open-loop and feedback (closed-loop) controls. The results obtained in the paper are illustrated by examples.

9

Niejednoznaczności wyznaczania ułamkowej potęgi wektora wirującego

Zawadzki A., Włodarczyk M.

Przegląd Elektrotechniczny

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2017

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R. 93, nr 11

115--118

PL

W pracy przedstawiono problem niejednoznaczności wyznaczania pochodnych ułamkowego rzędu wektora wirującego. Jest to związane z określeniem ułamkowej potęgi liczby zespolonej.

EN

The paper presents problems of ambiguity of calculation of fractional derivatives for the rotating vector. This is related to the determining of fractional power of the imaginary number.

10

On the Critical Strip of the Riemann zeta Fractional Derivative

Cattani C., Guariglia E., Wang S.

Fundamenta Informaticae

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2017

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Vol. 151, nr 1/4

459--472

EN

The α-order fractional derivative of the Dirichlet η function is computed in order to investigate the behavior of the fractional derivative of the Riemann zeta function ζ(α) on the critical strip. The convergence of η(α) is studied. In particular, its half-plane of convergence gives the possibility to better understand the ζ(α) and its critical strip. As an application, two signal processing networks, corresponding to η(α) and to its Fourier transform respectively, are shortly described.

11

Transient states in quadripoles utilizing fractional order elements

Włodarczyk M., Szczapaniak J.

Prace Naukowe Politechniki Śląskiej. Elektryka

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2016

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z. 2

93--100

EN

The paper presents a mirror Γ-network containing fractional-order elements. Fractional calculus was employed for the transient analysis of network operation. Comparative analysis was performed for a classic two-port network and defined mirror Γ-network in no-load state, short-circuit state and under load conditions for unit step function input. The obtained results are presented in graphical form and compared to results obtained using classic methods.

PL

W pracy przedstawiono kątowy czwórnik typu Γ zawierający elementy ułamkowego rzędu. Do analizy takiego czwórnika w stanie nieustalonym zastosowano rachunek różniczkowy ułamkowego rzędu. Przeprowadzono analizę porównawczą dla czwórnika klasycznego i omawianego czwórnika dla stanu jałowego i zwarcia dla wymuszenia skokiem jednostkowym. Uzyskane wyniki przedstawiono na wykresach, w porównaniu z wynikami dla czwórnika klasycznego.

12

Explicit Finite-Difference Scheme for the Numerical Solution of the Model Equation of Nonlinear Hereditary Oscillator with Variable-Order Fractional Derivatives

Parovik R. I.

Archives of Control Sciences

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2016

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Vol. 26, no. 3

429--435

EN

The paper deals with the model of variable-order nonlinear hereditary oscillator based on a numerical finite-difference scheme. Numerical experiments have been carried out to evaluate the stability and convergence of the difference scheme. It is argued that the approximation, stability and convergence are of the first order, while the scheme is stable and converges to the exact solution

13

Axisymmetric thermal stresses in a half-space in the framework of fractional thermoelasticity

Povstenko Y.

Scientific Issues of Jan Długosz University in Częstochowa. Mathematics

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2014

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Vol. 19

207--216

EN

A theory of thermal stresses based on the time-fractional heat conduction equation is considered. The Caputo fractional derivative is used. The fundamental solution to the axisymmetric heat conduction equation in a half-space under the Dirichlet boundary condition and the associated thermal stresses are investigated.

14

Zastosowanie pochodnych ułamkowych do matematycznego opisu kinetyki sorpcji dla układu sorbent roślinny - jony metali ciężkich

Tomczak E., Kamiński W., Szczerkowska D.

Proceedings of ECOpole

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2013

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Vol. 7, No. 1

267--272

PL

W pracy wykorzystano łuskę gryki do procesu sorpcji jonów metali ciężkich Cu(II), Ni(II), Zn(II), Co(II) i Cd(II) z roztworów wodnych. Wyznaczone zostały maksymalne pojemności sorpcyjne oraz stałe kinetyczne i równowagowe. Obliczone wartości pozwoliły na zastosowanie równań różniczkowych ułamkowych do opisu kinetyki sorpcji oraz uzyskania uogólnionego równania kinetyki sorpcji. Opracowanie wyników według tej koncepcji wymaga napisania procedury obliczeniowej wykorzystującej funkcje gamma oraz szeregi nieskończone. Równania kinetyki z wykorzystaniem pochodnych ułamkowych są równaniami o dwóch parametrach. Są to ułamek pochodnej i stała kinetyczna zależne od analizowanego układu sorbent - adsorbat.

EN

Buckwheat husk for sorption of heavy metal ions Cu(II), Ni(II), Zn(II), Co(II) and Cd(II) from aqueous solutions was presented in the paper. The maximum sorption capacity and kinetics and equilibrium constants were determined. Calculated values were used to describe sorption kinetics by means of fractional derivatives. The calculations were carried out using gamma functions and the infinite series. Two parameters equation was obtained. These parameters are a fraction of a derivative and a kinetics constant depend on the analyzed system.

15

Fractional Derivatives for Description of Sorption Kinetics in the Plant Sorbent - Metal Ions System

Tomczak E., Kamiński W., Szczerkowska D.

Ecological Chemistry and Engineering. S

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2013

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Vol. 20, nr 3

499--506

EN

It was examined if buckwheat hull has a potential to be used to adsorb heavy metal ions Zn(II), Cd(II), Co(II), Cu(II), Ni(II) from water. The research involved experiments aimed at the determination of sorption kinetics taking into consideration changes of concentration in a solution and sorbent over time. According to the literature data, kinetics is described with the use of pseudo first-order equations. Application of fractional derivatives for the description of sorption kinetics enables the development of the generalised sorption kinetics equation. Result analysis with this concept requires making a computational procedure using gamma functions and infinite series. Kinetics description using fractional derivatives will be equations with two parameters ie fraction of derivative α and the kinetics constant K dependent on the analysed sorbent-adsorbate system.

PL

W pracy wykorzystano łuskę gryki do procesu sorpcji jonów metali ciężkich Cu(II), Ni(II), Zn(II), Co(II) i Cd(II) z roztworów wodnych. Wyznaczone zostały maksymalne pojemności sorpcyjne oraz stałe kinetyczne i równowagowe. Obliczone wartości pozwoliły na zastosowanie równań różniczkowych ułamkowych do opisu kinetyki sorpcji oraz uzyskania uogólnionego równania kinetyki sorpcji. Opracowanie wyników według tej koncepcji wymaga napisania procedury obliczeniowej wykorzystującej funkcje gamma oraz szeregi nieskończone. Równania kinetyki z wykorzystaniem pochodnych ułamkowych są równaniami o dwóch parametrach. Są to ułamek pochodnej i stała kinetyczna zależne od analizowanego układu sorbent - adsorbat.

16

Mathematical description of rheological properties of asphalt-aggregate mixes

Zbiciak A.

Bulletin of the Polish Academy of Sciences. Technical Sciences

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2013

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Vol. 61, nr 1

65-72

EN

The procedure of the formulation of constitutive equations for asphalt-aggregate mixes is based very often on rheological schemes composed of classical elastic, plastic and viscous elements. The parameters of these schemes can be obtained based on laboratory experiments. In order to obtain better curve fitting results one can use non-classical viscoelastic elements described by fractional derivatives. In this paper we present the characteristics of the fractional viscoelastic Huet-Sayegh model as well as the characteristics of an original simplified fractional model. The results have been obtained using algorithms of numerical calculation of inverse Laplace transforms. Then the proposal of an original rheological model including plasticity has been given. The non-linear differential constitutive relationships of such a model are presented in the paper. The results of computer simulations are also visualized. Finally, 3D viscoelasticplastic models of asphalt aggregatemixes are proposed. The models are based on a generalized macroscopic theory taking into account the effect of pressure-dependency on yielding.

17

Reflection symmetry properties of generalized fractional derivatives

Klimek M., Lupa M.

Prace Naukowe Instytutu Matematyki i Informatyki Politechniki Częstochowskiej

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2012

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Vol. 11, nr 3

71-80

EN

In this paper we study the properties of generalized fractional derivatives (GFDs) with respect to the reflection mapping in finite intervals. We introduce symmetric and antisymmetric derivatives in a given interval and a split of arbitrary function into [J]- projections - parts with well-defined reflection symmetry properties. The main result are representation formulas for the symmetric and anti-symmetric GFDs of order α ∈ (0,1) which allow us to reduce the operators defined in the interval [a,b] to the ones given in arbitrarily short subintervals.

18

Zastosowanie pochodnej ułamkowego rzędu do modelowania mieszanek mineralno-asfaltowych

Grzesikiewicz W., Zbiciak A.

Pomiary Automatyka Kontrola

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2011

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R. 57, nr 9

1048-1051

EN

Asphalt-aggregate mixtures constitute basic component for the road pavements construction. The fundamental problem in the procedure of pavement design is to elaborate an appropriate constitutive model suited for the structural behaviour modelling within a wide range of mechanical and environmental loadings. The paper deals with the identification problem for a model representing constitutive properties of asphalt-aggregate mixtures. The constants characterising such a model should be evaluated based on expe-riments. There were analysed two viscoelastic rheological schemes. The classical Burgers model (Fig. 3a) contains linear elastic and viscous elements, while the Huet-Sayegh model (Fig. 3b) is composed of non-classical viscoelastic elements and its constitutive properties can be described using fractional derivatives. The results of cyclic tests of an asphalt-aggregate mixture were taken from the literature [8]. The experiments were carried out when assuming one-dimensional state of stress within the range of frequencies between 0.5 Hz and 35 Hz. The curve fitting procedure for both models was executed based on its complex moduli evaluation (Eqs. 11a and 11b). The obtained results show that by applying the fractional Huet-Sayegh model one can better fit the experiment in comparison with the classical Burgers model (see Figs. 4 and 5). Further investigations should be focused on description of non-linear effects using fractional and plastic elements. The plasticity phenomenon is important within the range of large intensities of loadings. The work on this area is currently underway by the authors.

19

Characteristics of fractional rheological models of asphalt-aggregate mixtures

Zbiciak A., Grzesikiewicz W.

Logistyka

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2011

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nr 6

EN

The procedure of the formulation of constitutive equations for asphalt-aggregate mixtures is based very often on linear rheological schemes composed of classical elastic and viscous elements. The parameters of these schemes can be obtained based on laboratory experiments. It was proved in the literature that it is possible to obtain better curve fitting results using non-classical viscoelastic elements described by fractional derivatives. In this paper we will present the characteristics of the fractional Huet-Sayegh model. The problem of estimation of its parameters was analyzed in our previous paper [2]. We will show the results of calculations in the form of creep and relaxation curves as well as hysteretic loops. The characteristics of the fractal model will be compared with the characteristics of the classical viscoelastic Burgers model. The results were obtained using algorithms of numerical calculation of inverse Laplace transforms.

PL

W procedurze formułowania relacji konstytutywnych mieszanek mineralno-asfaltowych (MMA) są wykorzystywane najczęściej liniowe struktury reologiczne zawierające klasyczne elementy sprężyste i lepkie. Wartości parametrów takich struktur wyznacza się na podstawie wyników badań doświadczalnych. W wielu pracach wykazano, iż zastosowanie nieklasycznych elementów lepkosprężystych, opisywanych za pomocą pochodnej ułamkowego rzędu (pochodnej fraktalnej), umożliwia lepsze dopasowanie wyników badań. W niniejszym opracowaniu zajmujemy się charakterystykami fraktalnego modelu Hueta-Sayegha. Zagadnienie estymacji jego parametrów przedstawiliśmy we wcześniejszej pracy [2]. Podamy wyniki obliczeń w postaci krzywych pełzania, relaksacji i pętli histerezy. Charakterystyki modelu fraktalnego zostaną przedstawione na tle charakterystyk klasycznego, lepkosprężystego modelu Burgersa. Rozwiązania uzyskano przy zastosowaniu algorytmów numerycznego wyznaczania odwrotnych transformat Laplace'a.

20

Properties of resonance-type and dynamic vibration eliminators

Golec Z.

Vibrations in Physical Systems

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2010

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Vol. 24

125--130

EN

The paper presents a rheological model of a body, which properties are described by means of a fractional derivative of its deformation. Such a model of a body was used to describe the coupling between a protected object and vibration eliminator. Then differential equations of motion were solved and effectiveness of vibration elimination was determined.

PL

W pracy przedstawiono model reologiczny ciała, którego własności opisano niecałkowitą pochodną jego deformacji. Taki model ciała wykorzystano w opisie sprzężenia obiektu chronionego z eliminatorem drgań. Rozwiązano różniczkowe równania ruchu i określono skuteczność eliminacji drgań mechanicznych.