Wyniki wyszukiwania - BazTech

1

A study of issues in extreme cases in a numerical solver for nonlinear fractional circuits

Sowa Marcin

Przegląd Elektrotechniczny

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2023

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R. 99, nr 8

212--218

EN

The study deals with the computations of nonlinear fractional circuit analyses in transient states. A numerical solver for FDAE (Fractional Differential-Algebraic Equations) is recalled, which bases on SubIval (acronym for “the subinterval-based method”). In the study, a nonlinear fractional problem was deliberately selected and tuned so as to cause issues occuring only in extreme cases, but motivating the further development of the numerical method. A development idea arises after observations of the error estimation, the influence of the time step adaptivity and the polynomial order.

PL

Praca dotyczy obliczen w analizach nieliniowych obwodów ułamkowych w stanach nieustalonych. Przywołano solwer numeryczny dla ułamkowych równan różniczkowo-algebraicznych, który opiera się na metodzie numerycznej SubIval (akronim od anglojęzycznej nazwy “the subinterval-based method”, czyli “metody podprzedziałów”). W badaniu celowo wybrano i dostrojono nieliniowe zagadnienie ułamkowe tak, aby wywołac komplikacje występujące jedynie w skrajnych przypadkach, ale motywujące do dalszego rozwoju wspomnianej metody numerycznej. Pomysł rozwoju metody pojawia sie po obserwacjach oszacowania błędu, wpływu automatycznego dostosowania kroku czasowego i rzędu wielomianów aproksymujących.

2

Fractional stochastic differential equations driven by G-Brownian motion with delays

Saci Akram, Redjil Amel, Boutabia Hacene, Kebiri Omar

Probability and Mathematical Statistics

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2023

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Vol. 43, Fasc. 1

1--21

EN

This paper consists of two parts. In part I, existence and uniqueness of solution for fractional stochastic differential equations driven by G-Brownian motion with delays (G-FSDEs for short) is established. In part II, the averaging principle for this type of equations is given. We prove under some assumptions that the solution of G-FSDE can be approximated by solution of its averaged stochastic system in the sense of mean square.

3

Remarks on the Caputo fractional derivative

Sikora Beata

MINUT Matematyka i Informatyka na Uczelniach Technicznych

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2023

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Nr 5

76-84

EN

The purpose of the paper is to familiarise the reader with the concept of the Caputo fractional derivative. The definition and basic properties of the Caputo derivative are given. Formulas for the derivatives of selected functions are derived. Examples of calculating the derivatives of basic functions are presented. The paper also contains a number of self-solving exercises, with answers.

4

A New Class of Fractional Cumulative Residual Entropy - Some Theoretical Results

Benmahmoud Slimane

Journal of Telecommunications and Information Technology

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2023

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nr 1

25--29

EN

In this paper, by differentiating the entropy’s generating function (i.e., h(t) = R SX̄F tX (x)dx) using a Caputo fractional-order derivative, we derive a generalized non-logarithmic fractional cumulative residual entropy (FCRE). When the order of differentiation α → 1, the ordinary Rao CRE is recovered, which corresponds to the results from first-order ordinary differentiation. Some properties and examples of the proposed FCRE are also presented.

5

Processing characteristics correction of measuring systems by means a differintegral of variable order

Cioć Radosław

Journal of Automation, Electronics and Electrical Engineering

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2023

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Vol. 5, No. 3

7--14

EN

The paper presents new methods for correcting the processing characteristics of measurement systems based on a modified Grünwald-Letnikov fractional calculus definition. The presented methods are based on the determination of the fractional order as an estimation factor. Two methods are presented: a fractional order array and a fractional order function. Both methods can be used in DSP systems as methods to correct the processing characteristics of systems with measuring transducers and measurement systems in general.

6

Numerical approximation of the Riemann-Liouville fractional integrals using the akima spline interpolation

Grodzki Grzegorz

Journal of Applied Mathematics and Computational Mechanics

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2023

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Vol. 22, nr 4

30-43

EN

This paper presents the numerical algorithms for evaluating the values of the left- and right-sided Riemann-Liouville fractional integrals using the linear and Akima cubic spline interpolations. Sample numerical calculations have been performed based on the derived algorithms. The results are presented in two tables. Knowledge of the closed analytical expressions for sample fractional integrals makes it possible to determine the numerical errors and the experimental rates of convergence for each derived algorithm.

7

A numerical scheme for time-fractional fourth-order reaction-diffusion model

Koç Dilara Altan

Journal of Applied Mathematics and Computational Mechanics

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2023

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Vol. 22, nr 2

15--25

EN

In fractional calculus, the fractional differential equation is physically and theoretically important. In this article an efficient numerical process has been developed. Numerical solutions of the time fractional fourth order reaction diffusion equation in the sense of Caputo derivative is obtained by using the implicit method, which is a finite difference method and is developed by increasing the number of iterations. The advantage of the implicit difference scheme is unconditionally stable. The stability analysis and convergency have been proven. A numerical example has been presented, and the validity of the method is supported by tables and graphics.

8

Applications of the fractional Sturm-Liouville difference problem to the fractional diffusion difference equation

Malinowska Agnieszka B., Odzijewicz Tatiana, Poskrobko Anna

International Journal of Applied Mathematics and Computer Science

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2023

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Vol. 33, no. 3

349--359

EN

This paper deals with homogeneous and non-homogeneous fractional diffusion difference equations. The fractional operators in space and time are defined in the sense of Grünwald and Letnikov. Applying results on the existence of eigenvalues and corresponding eigenfunctions of the Sturm-Liouville problem, we show that solutions of fractional diffusion difference equations exist and are given by a finite series.

9

New directions in electric arc furnace modeling

Grabowski Dariusz, Klimas Maciej

Archives of Electrical Engineering

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2023

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Vol. 72, nr 1

157--172

EN

This paper presents new directions in the modeling of electric arc furnaces. This work is devoted to an overview of new approaches based on random differential equations, artificial neural networks, chaos theory, and fractional calculus. The foundation of proposed solutions consists of an instantaneous power balance equation related to the electric arc phenomenon. The emphasis is mostly placed on the conclusions that come from a novel interpretation of the equation coefficients.

10

Probing 3D chaotic Thomas’ cyclically attractor with multimedia encryption and electronic circuitry

Khan NajeebAlam, Qureshi Muhammad Ali, Akbar Saeed, Ara Asmat

Archives of Control Sciences

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2023

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Vol. 33, No. 1

239--271

EN

This study investigates Thomas’ cyclically symmetric attractor dynamics with mathematical and electronic simulations using a proportional fractional derivative to comprehend the dynamics of a given chaotic system. The three-dimensional chaotic flow was examined in detail with Riemann-Liouville derivative for different values of the fractional index to highlight the sensitivity of chaotic systems with initial conditions. Thus, the dynamics of the fractional index system were investigated with Eigenvalues, Kaplan-Yorke dimension, Lyapunov exponent, and NIST testing, and their corresponding trajectories were visualized with phase portraits, 2D density plot, and Poincaré maps. After obtaining the results, we found that the integer index dynamics are more complex than the fractional index dynamics. Furthermore, the chaotic system circuit is simulated with operational amplifiers for different fractional indices to generate analog signals of the symmetric attractor, making it an important aspect of engineering. The qualitative application of our nonlinear chaotic system is then applied to encrypt different data types such as voice, image, and video, to ensure that the developed nonlinear chaotic system can widely applied in the field of cyber security.

11

Formation control of nonholonomic wheeled mobile robots using adaptive distributed fractional order fast terminal sliding mode control

Damani Allaeddine Yahia, Benselama Zoubir Abdeslem, Hedjar Ramdan

Archive of Mechanical Engineering

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2023

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LXX, nr 4

567–-587

EN

In this paper, an adaptive distributed formation controller for wheeled nonholonomic mobile robots is developed. The dynamical model of the robots is first derived by employing the Euler-Lagrange equation while taking into consideration the presence of disturbances and uncertainties in practical applications. Then, by incorporating fractional calculus in conjunction with fast terminal sliding mode control and consensus protocol, a robust distributed formation controller is designed to assure a fast and finite-time convergence of the robots towards the required formation pattern. Additionally, an adaptive mechanism is integrated to effectively counteract the effects of disturbances and uncertain dynamics. Moreover, the suggested control scheme’s stability is theoretically proven through the Lyapunov theorem. Finally, simulation outcomes are given in order to show the enhanced performance and efficiency of the suggested control technique.

12

Finite length triple estimation algorithm and its application to gyroscope mems noise identification

Macias Michał, Sierociuk Dominik

Acta Mechanica et Automatica

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2023

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Vol. 17, no. 2

219--229

EN

The noises associated with MEMS measurements can significantly impact their accuracy. The noises characterised by random walk and bias instability errors strictly depend on temperature effects that are difficult to specify during direct measurements. Therefore, the paper aims to estimate the fractional noise dynamics of the stationary MEMS gyroscope based on finite length triple estimation algorithm (FLTEA). The paper deals with the state, order and parameter estimation of fractional order noises originating from the MEMS gyroscope, being part of the popular Inertial Measurement Unit denoted as SparkFun MPU9250. The noise measurements from 𝑥,𝑦 and 𝑧 gyroscope axes are identified using a modified triple estimation algorithm (TEA) with finite approximation length. The TEA allows a simultaneous estimation of the state, order and parameter of fractional order systems. Moreover, as it is well-known that the number of samples in fractional difference approximations plays a key role, we try to show the influence of applying the TEA with various approximation length constraints on final estimation results. The validation of finite length TEA in the noise estimation process coming from MEMS gyroscope has been conducted for implementation length reduction achieving 50% of samples needed to estimate the noise with no implementation losses. Additionally, the capabilities of modified TEA in the analysis of fractional constant and variable order systems are confirmed in several numerical examples.

13

On Opial-type inequality for a generalized fractional integral operator

Vivas-Cortez Miguel, Martínez Francisco, Valdes Juan E. Nápoles, Hernández Jorge E.

Demonstratio Mathematica

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2022

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Vol. 55, nr 1

695--709

EN

This article is aimed at establishing some results concerning integral inequalities of the Opial type in the fractional calculus scenario. Specifically, a generalized definition of a fractional integral operator is introduced from a new Raina-type special function, and with certain results proposed in previous publications and the choice of the parameters involved, the established results in the work are obtained. In addition, some criteria are established to obtain the aforementioned inequalities based on other integral operators. Finally, a more generalized definition is suggested, with which interesting results can be obtained in the field of fractional integral inequalities.

14

Analysis of dynamic characteristics of selected pneumatic systems with rractional calculus. simulation and laboratory research

Luft M., Krzysztoszek K., Pietruszczak D., Nowocień A.

TransNav : International Journal on Marine Navigation and Safety of Sea Transportation

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2021

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Vol. 15 No. 4

835--843

EN

The section of the paper on simulation studies presents the application of fractional calculus to describe the dynamics of pneumatic systems. In the construction of mathematical models of the analysed dynamic systems, the Riemann-Liouville definition of differ-integral of non-integer order was used. For the analysed model, transfer function of integer and non-integer order was determined. Functions describing characteristics in time and frequency domains were determined, whereas the characteristics of the analysed systems were obtained by means of computer simulation. MATLAB were used for the simulation research. The section of the paper on laboratory research presents the results of the laboratory tests of the injection system of the internal combustion engine with special attention to the verification of simulated tests of selected pneumatic systems described with the use of fractional calculus.

15

A new numerical scheme for solving the two dimensional fractional diffusion equation

Altan Koç Dilara, Gülsu Mustafa

Journal of Applied Mathematics and Computational Mechanics

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2021

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Vol. 20, nr 2

5--16

EN

In this study, the locally one dimensional (LOD) method is used to solve the two dimensional time fractional diffusion equation. The fractional derivative is the Caputo fractional derivative of order α. The rate of convergence of the finite difference method is presented. It is seen that this method is in agreement with the obtained numerical solutions with acceptable central processing unit time (CPU time). Error estimates, numerical and exact results are tabulated. The graphics of errors are given.

16

Time-optimal control of linear fractional systems with variable coefficients

Matychyn Ivan, Onyshchenko Viktoriia

International Journal of Applied Mathematics and Computer Science

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2021

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Vol. 31, no. 3

375--386

EN

Linear systems described by fractional differential equations (FDEs) with variable coefficients involving Riemann–Liouville and Caputo derivatives are examined in the paper. For these systems, a solution of the initial-value problem is derived in terms of the generalized Peano–Baker series and a time-optimal control problem is formulated. The optimal control problem is treated from the convex-analytical viewpoint. Necessary and sufficient conditions for time-optimal control similar to that of Pontryagin’s maximum principle are obtained. Theoretical results are supported by examples.

17

The IoT gateway with active queue management

Domański Adam, Domańska Joanna, Czachórski Tadeusz, Klamka Jerzy, Szyguła Jakub, Marek Dariusz

International Journal of Applied Mathematics and Computer Science

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2021

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Vol. 31, no. 1

165--178

EN

As the traffic volume from various Internet of things (IoT) networks increases significantly, the need for adapting the quality of service (QoS) mechanisms to the new Internet conditions becomes essential. We propose a QoS mechanism for the IoT gateway based on packet classification and active queue management (AQM). End devices label packets with a special packet field (type of service (ToS) for IPv4 or traffic class (TC) for IPv6) and thus classify them as priority for real-time IoT traffic and non-priority for standard IP traffic. Our AQM mechanism drops only non-priority packets and thus ensures that real-time traffic packets for critical IoT systems are not removed if the priority traffic does not exceed the maximum queue capacity. This AQM mechanism is based on the PIα controller with non-integer integration order. We use fluid flow approximation and discrete event simulation to determine the influence of the AQM policy on the packet loss probability, queue length and its variability. The impact of the long-range dependent (LRD) traffic is also considered. The obtained results show the properties of the proposed mechanism and the merits of the PIα controller.

18

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

Oziablo Piotr, Mozyrska Dorota, Wyrwas Malgorzata

Bulletin of the Polish Academy of Sciences. Technical Sciences

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2021

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Vol. 69, nr 5

art. no. e137937

EN

In the paper we propose a fractional-piecewise-constant-order PID controller and discuss the stability and robustness of a closed loop system. In stability analysis we use the transform method and include the Nyquist-like criteria. Simulations for designed controllers are performed for the second-order plant with a delay.

19

Mathematical models of a pneumatic cascade and pneumatic membrane actuator described with a fractional calculus

Luft M., Łukasik Z., Pietruszczak D.

TransNav : International Journal on Marine Navigation and Safety of Sea Transportation

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2020

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Vol. 14. no. 3

643--648

EN

The paper presents the analysis of dynamic properties of pneumatic systems such as: cascade connection of membrane pressure transmitters and a pneumatic membrane actuator by means of differential equations of integer and non-integer order. The analyzed systems were described from the time perspective by means of step response, and in terms of frequency with the help of the Bode plot, i.e. logarithmic magnitude and phase responses. Each response was determined using differential equations of non-integer order. To determine the responses, the interactive Simulink package was an irreplaceable programming tool built on the basis of the MATLAB program, which enables the analysis and synthesis of continuous dynamic systems.

20

The fourth-order ordinary differential equation with the fractional initial/boundary conditions

Siedlecki Jarosław

Journal of Applied Mathematics and Computational Mechanics

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2020

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Vol. 19, nr 1

79--87

EN

The initial/boundary value problem for the fourth-order homogeneous ordinary differential equation with constant coefficients is considered. In this paper, the particular solutions an ordinary differential equation with respect to the set of boundary conditions are studied. At least one of the boundary conditions is described by a fractional derivative. Finally, a few illustrative examples of particular solutions to the considered problem are shown.