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1

On the Darboux vector of a new fractional frame

Toplama Aykut, Dede Mustafa

Journal of Applied Mathematics and Computational Mechanics

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2024

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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

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.

3

Solution of hybrid time fractional diffusion problem via weighted inner product

Cetinkaya Suleyman, Demir Ali

Journal of Applied Mathematics and Computational Mechanics

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2021

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Vol. 20, nr 2

17--27

EN

In this research, we discuss the construction of the analytic solution of homogenous initial boundary value problem including partial differential equations of fractional order. Since the homogenous initial boundary value problem involves the Hybrid fractional order derivative with various coefficients functions, it has classical initial and boundary conditions. By means of separation of the variables method and the inner product defined on L 2 [0,l], the solution is constructed in the form of a Fourier series including the bivariate Mittag-Leffler function. An illustrative example presents the applicability and influence of the separation of variables method on time fractional diffusion problems. Moreover, as the fractional order α tends to 1, the solution of the fractional diffusion problem tends to the solution of the diffusion problem which proves the accuracy of the solution.

4

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.

5

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

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2021

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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

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

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2020

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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.

7

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

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2020

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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.

8

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.

9

Existence and Ulam-Hyers stability of the implicit fractional boundary value problem with ψ-Caputo fractional derivative

Wahash Hanan A., Abdo Mohammed S, Panchal Satish K.

Journal of Applied Mathematics and Computational Mechanics

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2020

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Vol. 19, nr 1

89--101

EN

In this paper, we investigate the existence, uniqueness and Ulam-Hyers stability of solutions for nonlinear implicit fractional differential equations with boundary conditions involving a ψ-Caputo fractional derivative. The obtained results for the proposed problem are proved under a new approach and minimal assumptions on the function ƒ. The analysis is based upon the reduction of the problem considered to the equivalent integral equation, while some fixed point theorems of Banach and Schauder and generalized Gronwall inequality are employed to obtain our results for the problem at hand. Finally, the investigation is illustrated by providing a suitable example.

10

Effects of the porous boundary and inclined magnetic field on MHD flow in a rectangular duct

Chutia Muhim

Journal of Applied Mathematics and Computational Mechanics

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2020

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Vol. 19, nr 4

33--44

EN

In this work, a steady two dimensional MHD flow of a viscous incompressible fluid through a rectangular duct under the action of an inclined magnetic field with a porous boundary has been investigated. The coupled partial differential equations are transformed into a system of algebraic equations using the finite difference method and are then solved simultaneously using the Gauss Seidal iteration method by programming in Matlab software. Numerical solutions for velocity, induced magnetic field and current density lines are obtained and analyzed for different values of dimensionless parameters namely suction/injection parameter (S), Hartmann number (M) and inclination angle (θ) and are presented graphically.

11

On existence and uniqueness results for iterative mixed integrodifferential equation of fractional order

Unhale S. I., Kendre Subhash D.

Journal of Applied Analysis

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2020

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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

The analytical solution of Van der Pol and Lienard differential equations within conformable fractional operator by retarded integral inequalities

Usta Fuat, Sarıkaya Mehmet Zeki

Demonstratio Mathematica

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2019

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Vol. 52, nr 1

204--212

EN

In this study we introduced and tested retarded conformable fractional integral inequalities utilizing non-integer order derivatives and integrals. In line with this purpose, we used the Katugampola type conformable fractional calculus which has several practical properties. These inequalities generalize some famous integral inequalities which provide explicit bounds on unknown functions. The results provided here had been implemented to the global existence of solutions to the conformable fractional differential equations with time delay.

13

Identification of the Thermal Conductivity Coefficient in the Heat Conduction Model with Fractional Derivative

Brociek R., Słota D.

Archives of Foundry Engineering

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2019

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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.

14

Axial vibration of bars using fractional viscoelastic material models

Łabędzki P., Pawlikowski R., Radowicz A.

Vibrations in Physical Systems

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2018

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Vol. 29

art. no. 2018009

EN

This paper presents dynamic analysis of a bar with one end fixed and other free, loaded with force at its free end. The viscoelastic material of the bar is described by fractional models (Scot-Blair, Voigt, Maxwell and Zener models). Rayleigh-Ritz and Laplace transform methods were applied to obtain closed-form solution of the considered problem.

15

O pewnych aspektach stosowania pochodnych ułamkowych w elektrodynamice

Sikora R., Pawłowski S.

Przegląd Elektrotechniczny

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2018

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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

Positive solution of a fractional differential equation with integral boundary conditions

Abdo M. S., Wahash H. A., Panchal S. K.

Journal of Applied Mathematics and Computational Mechanics

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2018

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Vol. 17, nr 3

5--15

EN

In this paper, we prove the existence and uniqueness of a positive solution for a boundary value problem of nonlinear fractional differential equations involving a Caputo fractional operator with integral boundary conditions. The technique used to prove our results depends on the upper and lower solution, the Schauder fixed point theorem and the Banach contraction principle. The result of existence obtained through constructing the upper and lower control functions of the nonlinear term without any monotone requirement.

17

Analysis of positive linear continuous-time systems using the conformable derivative

Kaczorek T.

International Journal of Applied Mathematics and Computer Science

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2018

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Vol. 28, no. 2

335--340

EN

Positive linear continuous-time systems are analyzed via conformable fractional calculus. A solution to a fractional linear system is derived. Necessary and sufficient conditions for the positivity of linear systems are established. Necessary and sufficient conditions for the asymptotic stability of positive linear systems are also given. The solutions of positive fractional linear systems based on the Caputo and conformable definitions are compared.

18

Mathematical modeling of traveling autosolitons in fractional-order activator-inhibitor systems

Datsko B., Gafiychuk V.

Bulletin of the Polish Academy of Sciences. Technical Sciences

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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.

19

Applying a fractional coil model for power system ferroresonance analysis

Majka Ł.

Bulletin of the Polish Academy of Sciences. Technical Sciences

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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.

20

Derivatives of a polynomial of best approximation and modulus of smoothness in generalized Lebesgue spaces with variable exponent

Jafarov S. Z.

Demonstratio Mathematica

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2017

|

Vol. 50, nr 1

245--251

EN

The relation between derivatives of a polynomial of best approximation and the best approximation of the function is investigated in generalized Lebesgue spaces with variable exponent. In addition, the relationship between the fractional modulus of smoothness of the function and its de la Vallée-Poussin sums is studied.