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  1. 5 Journal of Applied Mathematics and Computational Mechanics
  2. 4 Bulletin of the Polish Academy of Sciences. Technical Sciences
  3. 3 Archives of Control Sciences
  4. 3 Poznan University of Technology Academic Journals. Electrical Engineering
  5. 2 Archives of Mechanics
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  1. 2 Babu Naladi Ram
  2. 2 Chiranjeevi Tirumalasetty
  3. 2 Devarapalli Ramesh
  4. 2 Dinzart F.
  5. 2 Lipiński P.
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  1. 1 2024
  2. 2 2023
  3. 1 2022
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1
Content available On the Darboux vector of a new fractional frame
Toplama Aykut, Dede Mustafa
Journal of Applied Mathematics and Computational Mechanics
|
2024
|
Vol. 23, nr 1
109-120
EN
Fractional derivatives are useful tools for many applications in different branch of science such as optics and engineering. In this paper, the ∧-fractional frame that is obtained along a space curve by using the ∧-fractional derivative is being examined in Euclidean E3 space. In addition, the Darboux vector of the ∧-fractional Frenet frame is constructed. Then the curvatures of the standard Frenet frame, the ∧-fractional Frenet frame and the ∧-fractional Darboux vector are compared geometrically.
2
Content available Remarks on the Caputo fractional derivative
Sikora Beata
MINUT Matematyka i Informatyka na Uczelniach Technicznych
|
2023
|
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.
3
Content available Necessary optimality conditions for quasi-singular controls for systems with Caputo fractional derivatives
Yusubov Shakir Sh., Mahmudov Elimhan N.
Archives of Control Sciences
|
2023
|
Vol. 33, No. 3
463--496
EN
In this paper, we consider an optimal control problem in which a dynamical system is controlled by a nonlinear Caputo fractional state equation. First we get the linearized maximum principle. Further, the concept of a quasi-singular control is introduced and, on this basis, an analogue of the Legendre-Clebsch conditions is obtained. When the analogue of Legendre-Clebsch condition degenerates, a necessary high-order optimality condition is derived. An illustrative example is considered.
4
Content available Fixed terminal time fractional optimal control problem for discrete time singular system
Chiranjeevi Tirumalasetty, Devarapalli Ramesh, Babu Naladi Ram, Vakkapatla Kiran Babu, Rao R. Gowri Sankara, Garcìa Màrquez Fausto Pedro
Archives of Control Sciences
|
2022
|
Vol. 32, No. 3
489--506
EN
This paper presents the formulation and numerical simulation for linear quadratic optimal control problem (LQOCP) of free terminal state and fixed terminal time fractional order discrete time singular system (FODSS). System dynamics is expressed in terms of Riemann-Liouville fractional derivative (RLFD), and performance index (PI) in terms of state and costate. Because of its complexity, finding analytical and numerical solutions to singular system (SS) is difficult. As a result, we use coordinate transformation to convert FODSS to its corresponding fractional order discrete time nonsingular system (FODNSS). After that, we obtain the necessary conditions by employing a Hamiltonian approach. The relevant conditions are solved using the general solution approach. For the analysis of formulation and solution algorithm, a numerical example is illustrated. Results are obtained for various 𝛼 values. According to state of the art, this is the first time that a formulation and numerical simulation of free terminal state and fixed terminal time optimal control problem (OCP) of FODSS is presented.
5
Content available The implicit numerical method for the one-dimensional anomalous subdiffusion equation with a nonlinear source term
Błasik Marek
Bulletin of the Polish Academy of Sciences. Technical Sciences
|
2021
|
Vol. 69, nr 6
art. no. e138240
EN
In the paper, the numerical method of solving the one-dimensional subdiffusion equation with the source term is presented. In the approach used, the key role is played by transforming of the partial differential equation into an equivalent integro-differential equation. As a result of the discretization of the integro-differential equation obtained an implicit numerical scheme which is the generalized Crank-Nicolson method. The implicit numerical schemes based on the finite difference method, such as the Carnk-Nicolson method or the Laasonen method, as a rule are unconditionally stable, which is their undoubted advantage. The discretization of the integro-differential equation is performed in two stages. First, the left-sided Riemann-Liouville integrals are approximated in such a way that the integrands are linear functions between successive grid nodes with respect to the time variable. This allows us to find the discrete values of the integral kernel of the left-sided Riemann-Liouville integral and assign them to the appropriate nodes. In the second step, second order derivative with respect to the spatial variable is approximated by the difference quotient. The obtained numerical scheme is verified on three examples for which closed analytical solutions are known.
6
Content available On optimal control problem subject to fractional order discrete time singular systems
Chiranjeevi Tirumalasetty, Biswas Raj Kumar, Devarapalli Ramesh, Babu Naladi Ram, García Márquez Fausto Pedro
Archives of Control Sciences
|
2021
|
Vol. 31, No. 4
849--863
EN
In this work, we present optimal control formulation and numerical algorithm for fractional order discrete time singular system (DTSS) for fixed terminal state and fixed terminal time endpoint condition. The performance index (PI) is in quadratic form, and the system dynamics is in the sense of Riemann-Liouville fractional derivative (RLFD). A coordinate transformation is used to convert the fractional-order DTSS into its equivalent non-singular form, and then the optimal control problem (OCP) is formulated. The Hamiltonian technique is used to derive the necessary conditions. A solution algorithm is presented for solving the OCP. To validate the formulation and the solution algorithm, an example for fixed terminal state and fixed terminal time case is presented.
7
Content available Dynamic characteristics of the structure with viscoelastic dampers combined with inerters and subjected to temperature changes
Pawlak Zdzisław M., Lewandowski Roman
Vibrations in Physical Systems
|
2020
|
Vol. 31, nr 2
art. no. 2020222
EN
The purpose of the work is dynamic analysis of passive dampers used in structural systems to reduce excessive vibrations caused by wind or earthquakes. Special systems are considered that contain inerter, i.e. device using rotational inertia, in combination with a viscoelastic damper. The so-called fractional models of viscoelastic dampers describe their dynamic behavior in a wide frequency range using a small number of model parameters. To describe material behavior over a wider frequency range, the time-temperature superposition principle is used. The shifting factor is calculated from the well-known William-Landel-Ferry formula. This allows for determination of damper parameters at any temperature based on the parameters obtained at the reference temperature. Laplace transformation of the derived equations of motion leads to the non-linear eigenproblem, which could be solved using the continuation method. The influence of temperature on the dynamic characteristics of the system is examined.
8
Content available Exciter fractional model and its susceptibility on parameter changes
Majka Łukasz, Sowa Marcin
Poznan University of Technology Academic Journals. Electrical Engineering
|
2020
|
No. 104
87--98
EN
The paper concerns the application of fractional calculus in the modeling of a selected part of a power system generating unit, which is the high frequency AC exciter. The model’s fractional derivative-based generalization is recalled. The basis of the estimation process for the model consists of two sets of measurement waveforms. In order to solve the fractional and nonlinear problem – a numerical solver is applied. The solver and the estimation procedure have been both implemented in GNU Octave. The model parameter susceptibility is examined. The changes of each model parameter value is studied in a way that the influence on the model output is observed.
9
Content available On the fractional-order dynamics of a double pendulum with a forcing constraint using the nonsingular fractional derivative approach
Rangaig Norodin A., Pido Alvanh Alem G., Pada-Dulpina Caironesa T.
Journal of Applied Mathematics and Computational Mechanics
|
2020
|
Vol. 19, nr 2
95-106
EN
In this paper, we presented the fractional-order dynamics of a double pendulum, at a small oscillation, with a non-singular derivative kernel. The equation of motion has been derived from the fractional Lagrangian of the system and the considered fractional Euler-Lagrange equation. The generalized force has also been presented in studying the different cases of force, such as horizontal and vertical forcing. The source term is described by the imposed periodic force, and the memory effect gives an additional damping factor described by the fractional order. The integer and fractional orders of the sample phase diagrams were obtained and presented to visualize the effect of the presented fractional order on the system. Also, since the motion of the system dissipates in the fractional regime, the applied force will drive the system out of equilibrium.
10
Content available Analytical and numerical study for a fractional boundary value problem with a conformable fractional derivative of Caputo and its fractional integral
Bekkouche M. Moumen, Guebbai H.
Journal of Applied Mathematics and Computational Mechanics
|
2020
|
Vol. 19, nr 2
31-42
EN
We study the existence and uniqueness of the solution of a fractional boundary value problem with conformable fractional derivation of the Caputo type, which increases the interest of this study. In order to study this problem we have introduced a new definition of fractional integral as an inverse of the conformable fractional derivative of Caputo, therefore, the proofs are based upon the reduction of the problem to a equivalent linear Volterra-Fredholm integral equations of the second kind, and we have built the minimum conditions to obtain the existence and uniqueness of this solution. The analytical study is followed by a complete numerical study.
11
On existence and uniqueness results for iterative mixed integrodifferential equation of fractional order
Unhale S. I., Kendre Subhash D.
Journal of Applied Analysis
|
2020
|
Vol. 26, nr 2
263--272
EN
The objective of this work is to study the local existence, uniqueness, stability and other properties of solutions of iterative mixed integrodifferential equations of fractional order. The Successive Approximation Method is applied for the numerical solution of iterative mixed integrodifferential equations of fractional order.
12
Content available remote Occupation time problem for multifractional Brownian motion
Ouahra Mohamed Ait, Guerbaz Raby, Ouahhabi Hanae, Sghir Aissa
Probability and Mathematical Statistics
|
2019
|
Vol. 39, Fasc. 1
99--113
EN
In this paper, by using a Fourier analytic approach, we investigate sample path properties of the fractional derivatives of multifractional Brownian motion local times. We also show that those additive functionals satisfy a property of local asymptotic self-similarity. As a consequence, we derive some local limit theorems for the occupation time of multifractional Brownian motion in the space of continuous functions.
13
Content available DAQ-based measurements and study of supercapacitor frequency characteristics
Sowa Marcin, Jakubowska-Ciszek Agnieszka
Poznan University of Technology Academic Journals. Electrical Engineering
|
2019
|
No. 97
203--213
EN
The paper concerns the beginning of a modeling study for supercapacitors. A data acquisition (DAQ) based system is presented, where an automated procedure has been implemented for the measurement of frequency characteristics. For a typical range of the supply voltage the characteristics are obtained. A model basing on fractional calculus is recalled and parameters for the model are obtained. The frequency characteristics of the model are compared with those obtained from measurements. Later the tested supercapacitor has its characteristics taken for various amplitudes and offsets of the source voltage. A few remarks are given for a possible expansion of the model when nonlinearity should be considered.
14
Content available Identification of the Thermal Conductivity Coefficient in the Heat Conduction Model with Fractional Derivative
Brociek R., Słota D.
Archives of Foundry Engineering
|
2019
|
iss. 3
38--42
EN
Main goal of the paper is to present the algorithm serving to solve the heat conduction inverse problem. Authors consider the heat conduction equation with the Riemann-Liouville fractional derivative and with the second and third kind boundary conditions. This type of model with fractional derivative can be used for modelling the heat conduction in porous media. Authors deal with the heat conduction inverse problem, which, in this case, consists in identifying an unknown thermal conductivity coefficient. Measurements of temperature, in selected point of the region, are the input data for investigated inverse problem. Basing on this information, a functional describing the error of approximate solution is created. Minimizing of this functional is necessary to solve the inverse problem. In the presented approach the Ant Colony Optimization (ACO) algorithm is used for minimization.
15
Content available remote O pewnych aspektach stosowania pochodnych ułamkowych w elektrodynamice
Sikora R., Pawłowski S.
Przegląd Elektrotechniczny
|
2018
|
R. 94, nr 1
101--104
PL
W artykule zwrócono uwagę na pewne problemy związane z zastosowaniem pochodnych ułamkowych w opisie zjawisk elektromagnetycznych oraz często popełniane przy tym błędy.
EN
The article focuses on some of the problems associated with the use of fractional derivatives in the description of electromagnetic phenomena, and common errors.
16
Content available “gcdAlpha” – a semi-analytical method for solving fractional state equations
Sowa M.
Poznan University of Technology Academic Journals. Electrical Engineering
|
2018
|
No. 96
231--242
EN
The paper discusses a semi-analytical method for solving systems of fractional state equations with various orders. The method bases on an expansion of the system into one where there is a single derivative order. The formulation of the matrices of the new system is explained in detail. Another characteristic feature of the method is also introduced – a consideration of forms, in which the time functions appear and the terms appearing in the solution as a result. A fractional circuit example is presented in order to test the method. The computation time for the method is also studied.
17
Content available Mathematical modeling of traveling autosolitons in fractional-order activator-inhibitor systems
Datsko B., Gafiychuk V.
Bulletin of the Polish Academy of Sciences. Technical Sciences
|
2018
|
Vol. 66, nr 4
411--418
EN
In the article, basic properties of traveling spatially nonhomogeneous auto-wave solutions in nonlinear fractional-order reactiondiffusion systems are investigated. Such solutions, called autosolitons, arise in a stability region of the system and can coexist with the spatially homogeneous states. By a linear stability analysis and computer simulation, it is shown that the order of the fractional derivative can substantially change the properties of such auto-wave solutions and significantly enrich nonlinear system dynamics. The results of the linear stability analysis are confirmed by computer simulations of the generalized fractional van der Pol-FitzHugh-Nagumo model. A common picture of traveling auto-waves including series in time-fractional two-component activator-inhibitor systems is presented. The results obtained in the article for the distributed system have also been of interest for nonlinear dynamical systems described by fractional ordinary differential equations.
18
Content available Applying a fractional coil model for power system ferroresonance analysis
Majka Ł.
Bulletin of the Polish Academy of Sciences. Technical Sciences
|
2018
|
Vol. 66, nr 4
467--474
EN
This paper addresses the problem of modeling the nonlinear coil used for ferroresonant circuit analysis. The effect of ferroresonance is described and a general modeling approach is presented. The hysteresis modeling problem is also shortly discussed, on the example of a ferromagnetic coil. A brief overview of available literature and contributors to this area are provided. A series RLC circuit supplied from an AC source is discussed. The application of the fractional derivative in the modeling of an iron core coil is presented and suggestions of model implementations are given. The computations performed are illustrated by means of waveform data obtained from computer simulations and compared with those obtained from measurements performed in a specially prepared laboratory setup.
19
Content available remote CFE method – quality analysis of the approximation of reverse Laplace transform of fractional order
Zawadzki A., Włodarczyk M.
Prace Naukowe Politechniki Śląskiej. Elektryka
|
2017
|
z. 3-4
21--29
EN
The paper presents the CFE (Continued fraction expansion) method, which allows determination of inverse transform of expression containing element sα by expanding it into a continued fraction. On the basis of the presented method, analysis of selected electrical circuits containing quasi-elements described with fractional derivatives was performed. Calculations and numerical simulations were also carried out.
PL
W pracy przedstawiono metodę CFE (continued fraction expansion) umożliwiającą wyznaczanie transformaty odwrotnej wyrażenia zawierającego czynnik sα dzięki rozwinięciu w ułamek łańcuchowy. W oparciu o prezentowaną metodę przeprowadzono analizę wybranych obwodów elektrycznych zawierających quasielementy opisane pochodnymi niecałkowitego rzędu. Przeprowadzono obliczenia i wykonano symulacje numeryczne.
20
Content available remote Interpretation of Fractional Derivatives as Reconstruction from Sequence of Integer Derivatives
Tarasov V. E.
Fundamenta Informaticae
|
2017
|
Vol. 151, nr 1/4
431--442
EN
In this paper, we propose an “informatic” interpretation of the Riemann-Liouville and Caputo derivatives of non-integer orders as reconstruction from infinite sequence of standard derivatives of integer orders. The reconstruction is considered with respect to orders of derivatives.
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