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1

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.

2

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.

3

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.

4

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.

5

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.

6

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.

7

Fox H-functions as exact solutions for Caputo type mass spring damper system under Sumudu transform

Qureshi Sania

Journal of Applied Mathematics and Computational Mechanics

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2021

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Vol. 20, nr 1

83--89

EN

Closed form solutions for mathematical systems are not easy to find in many cases. In particular, linear systems such as the population growth/decay model, RLC circuit, mixing problems in chemistry, first-order kinetic reactions, and mass spring damper system in mechanical and mechatronic engineering can be handled with tools available in theoretical study of linear systems. One such linear system has been investigated in the present research study. The second order linear ordinary differential equation called the mass spring damper system is explored under the Caputo type differential operator while using the Sumudu integral transform. The closed form solution has been found in terms of the Fox H-function wherein different aspects of the solution can be obtained with variation in α ∈ 2 (1;2] and β ∈ 2 (0;1]: The classical mass spring damper model is retrieved for α = β = 1:

8

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.

9

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.

10

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.

11

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.

12

Fractional order based velocity control system for a nanorobot in non-Newtonian fluids

Muresan C. I., Birs I. R., Folea S., Ionescu C.

Bulletin of the Polish Academy of Sciences. Technical Sciences

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2018

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Vol. 66, nr 6

991--997

EN

Customized patient drug delivery overcomes classic medicine setbacks such as side effects, improper drug absorption or slow action. Nanorobots can be successfully used for targeted patient-specific drug administration, but they must be reliable in the entire circulatory system environment. This paper analyzes the possibility of fractional order control applied to the nanomedicine field. The parameters of a fractional order proportional integral controller are determined with the purpose of controlling the velocity of the nanorobot in non-Newtonian fluids envisioning the blood flow in the circulatory system.

13

Behaviour of fractional discrete-time consensus models with delays for summator dynamics

Girejko E., Mozyrska D., Wyrwas M.

Bulletin of the Polish Academy of Sciences. Technical Sciences

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2018

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Vol. 66, nr 4

403--410

EN

The leader-following consensus problem of fractional-order multi-agent discrete-time systems with delays is considered. In the systems, interactions between agents are defined like in Krause and Cucker-Smale models, but the memory is included by taking both the fractional-order discrete-time operator on the left hand side of the nonlinear systems and the delays. Since in practical problems only bounded number of delays can be considered, we study the fractional order discrete-time models with a finite number of delays. The models of opinions under consideration are investigated for single- and double-summator dynamics of discrete-time by means of analytical methods as well as computer simulations.

14

Fractional calculus for continuum mechanics – anisotropic non-locality

Sumelka W.

Bulletin of the Polish Academy of Sciences. Technical Sciences

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2016

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Vol. 64, nr 2

361--372

EN

In this paper, a generalisation of previous author’s formulation of fractional continuum mechanics for the case of anisotropic non-locality is presented. The discussion includes a review of competitive formulations available in literature. The overall concept is based on the fractional deformation gradient which is non-local due to fractional derivative definition. The main advantage of the proposed formulation is its structure, analogous to the general framework of classical continuum mechanics. In this sense, it allows to define similar physical and geometrical meaning of introduced objects. The theoretical discussion is illustrated by numerical examples assuming anisotropy limited to single direction.

15

Non-local Kirchhoff–Love plates in terms of fractional calculus

Sumelka W.

Archives of Civil and Mechanical Engineering

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2015

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Vol. 15, no. 1

231--242

EN

Modern continuum mechanics needs new mathematical techniques to describe the complexity of real physical processes. Recently fractional calculus, a branch of mathematical analysis that studies differential operators of an arbitrary (real or complex) order, emerged as a powerful tool for modelling complex systems. It is due to the fact that fractional differential operators introduce non-locality to the description considered in a natural way. In this sense they generalize classical (local) formulations and make the description more realistic. This paper deals with the generalisation of the Kirchhoff–Love plates theory using fractional calculus. This new formulation in non-local, thus all common fields like e.g. internal forces or displacements at a specific point contain somehow information from its finite surroundings, which is in agreement with experimental observations.

16

Analiza modelu ułamkowego rzędu procesów szybkich reaktora jądrowego

Nowak T. K., Duzinkiewicz K.

Informatyka, Automatyka, Pomiary w Gospodarce i Ochronie Środowiska

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2014

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nr 1

44--47

PL

W artykule przedstawiono wyniki badań dotyczące rozwiązań numerycznych punktowego modelu ułamkowego rzędu kinetyki neutronów oraz wymiany ciepła w reaktorze jądrowym. Zbudowano model ułamkowego rzędu z sześcioma grupami neutronów opóźnionych wraz równaniami wymiany ciepła. Model matematyczny został zaimplementowany w środowisku Matlab i zbadany symulacyjnie dla skoków reaktywności. Przeprowadzono analizę wpływu wybranych parametrów modelu na uzyskiwane rozwiązania.

EN

The paper presents the results concerning numerical solutions of the fractional point kinetics and heat exchange model for nuclear reactor. The fractional neutron point kinetics model with six groups of delayed neutron precursors was developed and numerical solutions were proposed. Mathematical model has been implemented in the Matlab environment and tested using typical step input change. The analysis of the impact of chosen parameters was conducted.

17

Analysis of selected dynamic properties of quasi-fractional-order measuring transducer used in transportation facilities

Luft M., Cioć R., Pietruszczak D.

Archives of Transport System Telematics

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2014

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Vol. 7, iss. 3

17--21

EN

The paper presents possibilities of using fractional calculus in dynamic measurements used in telematic equipment in cars and railway vehicles diagnostic systems. It describes a laboratory measurement system for investigating dynamic properties of accelerometers. Tests are executed in the MATLAB&Simulink programme. Properties of the examined transducers of integral and quasi-fractional-orders are compared. The authors indicate the fractional calculus advantages from the point of view of their dynamics description.

18

Tuning and digital implementation of a fractional-order PD controller for a position servo

Tepljakov A., Petlenkov E., Belikov J., Astapov S

International Journal of Microelectronics and Computer Science

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2013

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Vol. 4, nr 3

116--123

EN

Fractional-order calculus offers ﬂexible computational possibilities that can be applied to control design thereby improving industrial control loop performance. However, before theoretical results can be carried over to an industrial setting it is important to study the effects of fractional-order control by means of laboratory experiments. In this paper, we study the practical aspects of tuning and implementing a fractional-order PD controller for position control of a laboratory modular servo system using FOMCON (“Fractional-order Modeling and Control”) toolbox for MATLAB. We provide an overview of the tools used to model, analyze, and design the control system. The procedure of tuning and implementation of a suitable digital fractional-order controller is described. The results of the real-time experiments conﬁrm the effectiveness of used methods.

19

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.

20

Equivalent descriptions of a discrete-time fractional-order linear system and its stability domains

Ostalczyk P.

International Journal of Applied Mathematics and Computer Science

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2012

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Vol. 22, no. 3

533-538

EN

Two description forms of a linear fractional-order discrete system are considered. The first one is by a fractional-order difference equation, whereas the second by a fractional-order state-space equation. In relation to the two above-mentioned description forms, stability domains are evaluated. Several simulations of stable, marginally stable and unstable unit step responses of fractional-order systems due to different values of system parameters are presented.