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1

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.

2

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.

3

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.

4

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.

5

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.

6

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.

7

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.

8

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.

9

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.

10

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.

11

Laboratory investigations of dynamic properties of accelerometers with fractional orders for application in telematic equipment

Pietruszczak D.

Archives of Transport System Telematics

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2012

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Vol. 5, iss. 2

26-31

EN

The paper presents laboratory studies on measuring accelerometers, which were modelled in the classical differential equations, as well as the fractional calculus. Measurement errors were examined and the classical and fractional models in terms of dynamic properties were compared. The advantages of fractional calculus in modelling dynamic elements were also indicated.

12

Application of fractional calculus in modelling of the transducer and the measurement system

Luft M., Cioć R., Pietruszczak D.

Przegląd Elektrotechniczny

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2011

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R. 87, nr 7

110-114

EN

The paper outlines an example of modelling the measurement transducer and actual measurement system with the use of fractional calculus. The algorithm determining these models is presented in the form of a fractional calculus notation and then the models are compared to the ones described by means of classical differential equations. Tests are executed in the programming environment Matlab-Simulink.

PL

W artykule przedstawiono przykład modelowania przetwornika pomiarowego oraz rzeczywistego systemu pomiarowego za pomocą rachunku różniczkowo-całkowego ułamkowych rzędów (rachunku ułamkowego). Przedstawiono algorytm wyznaczania tych modeli zapisem ułamkowym oraz porównano ich z modelami opisanymi klasycznymi równaniami różniczkowymi. Badania symulacyjne wykonano w środowisku programistycznym Matlab-Simulink.

13

An operational Haar wavelet method for solving fractional Volterra integral equations

Saeedi H., Mollahasani N., Mohseni Moghadam M., Chuev G. N.

International Journal of Applied Mathematics and Computer Science

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2011

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Vol. 21, no 3

535-547

EN

A Haar wavelet operational matrix is applied to fractional integration, which has not been undertaken before. The Haar wavelet approximating method is used to reduce the fractional Volterra and Abel integral equations to a system of algebraic equations. A global error bound is estimated and some numerical examples with smooth, nonsmooth, and singular solutions are considered to demonstrate the validity and applicability of the developed method.

14

Positive and bounded below solutions for certain nonlinear fractional differential equations

Klimek M., Błasik M.

Prace Naukowe Instytutu Matematyki i Informatyki Politechniki Częstochowskiej

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2010

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Vol. 9, nr 2

91-102

EN

Two types of one-term nonlinear fractional differential equations are considered and the existence of solutions in the space of continuous, positive and bounded below functions is proved. We transform an equation containing the left- or right-sided Caputo derivative into a fixed point condition and apply the Banach theorem and extended Bielecki method of equivalent norms.

15

On differentiable solutions for one-term nonlinear fractional differential equations

Klimek M., Lupa M.

Prace Naukowe Instytutu Matematyki i Informatyki Politechniki Częstochowskiej

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2010

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Vol. 9, nr 2

103-113

EN

Two one-term nonlinear fractional differential equations with the left- or rightsided Caputo derivative are discussed. The existence and uniqueness of solutions, generated by the respective stationary function, is proved in the space of continuously differentiable function. The proof, based on the Banach theorem, includes the extension of the Bielecki method of equivalent norms.

16

Monotonic continuous solution for a mixed type integral inclusion of fractional order

El-Sayed A. M. A., Al-Issa Sh. M.

Journal of Mathematics and Applications

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2010

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Vol. 33

27-34

EN

In this paper we are concerned with the mixed type integral inclusion [formula/wzór]. The existence of monotonic continuous solution will be proved. As an application the initial-value problem of the arbitrary (fractional) orders differential inclusion [formula/wzór] will be studied.

17

The finite difference method for fractional Cattaneo-Vernotte equation

Ciesielski M.

Prace Naukowe Instytutu Matematyki i Informatyki Politechniki Częstochowskiej

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2009

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Vol. 8, nr 1

13-18

EN

In this work a numerical solution of modified Cattaneo-Vernotte equation is presented. This equation is obtained by replacing the second order time derivative by the fractional derivative in Caputo sense. In order to solve the problem with classical boundary-initial conditions, the finite difference method is applied. In the final part of the paper the examples of computations are shown.

18

Robust fractional adaptive control based on the strictly positive realness condition

Ladaci S., Charef A., Loiseau J. J.

International Journal of Applied Mathematics and Computer Science

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2009

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Vol. 19, no 1

69-76

EN

This paper presents a new approach to robust adaptive control, using fractional order systems as parallel feedforward in the adaptation loop. The problem is that adaptive control systems may diverge when confronted with finite sensor and actuator dynamics, or with parasitic disturbances. One of the classical robust adaptive control solutions to these problems makes use of parallel feedforward and simplified adaptive controllers based on the concept of positive realness. The proposed control scheme is based on the Almost Strictly Positive Realness (ASPR) property of the plant. We show that this condition implies also robust stability in the case of fractional order controllers. An application to Model Reference Adaptive Control (MRAC) with a fractional order adaptation rule is provided with an implementable algorithm. A simulation example of a SISO robust adaptive control system illustrates the advantages of the proposed method in the presence of disturbances and noise.

19

Certain classes of multivalent functions with negative coefficients defined by using a differential operator

Aouf M. K.

Journal of Mathematics and Applications

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2008

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Vol. 30

5-21

EN

In this paper, we investigate the various important prop-erties and characteristics of the subclasses Sn (p, q, α, β) and Cn (p, q, α, β) of multivalent functions with negative coefficients defined by using a differential operator. We also derive many results for the modified Hadamard products of functions belongingto the classes Sn(p, q, α, β) and Cn(p, q, α, β). Finally several applications involving an integral operator and certain fractional calculus operators are also considered.

20

Distortion and convolutional theorems for operators of generalized fractional calculus involving wright function

Aouf M. K., Dziok J.

Journal of Applied Analysis

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2008

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Vol. 14, nr 2

183-192

EN

Using the Wright's generalized hypergeornetric function, we investigate a class W(q,s: A, B, λ) of analytic functions with negative coefficients. We derive many results for the modified Hadamard product of functions belonging to the class W(q,s: A, B, λ) . Moreover, we generalize some of the distortion theorems to the classical fractional integrals and derivatives and the Saigo (hypergeornetric) operators of fractional calculus.