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The Motion of a Bead Sliding on a Wire in Fractional Sense

100%
Baleanu D., Jajarmi A., Asad J., Blaszczyk T.
Acta Physica Polonica A
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2017
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vol. 131
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issue 6
1561-1564
EN
In this study, we consider the motion of a bead sliding on a wire which is bent into a parabola form. We first introduce the classical Lagrangian from the system model under consideration and obtain the classical Euler-Lagrange equation of motion. As the second step, we generalize the classical Lagrangian to the fractional form and derive the fractional Euler-Lagrange equation in terms of the Caputo fractional derivatives. Finally, we provide numerical solution of the latter equation for some fractional orders and initial conditions. The method we used is based on a discretization scheme using a Grünwald-Letnikov approximation for the fractional derivatives. Numerical simulations verify that the proposed approach is efficient and easy to implement.
2
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Exact solution for the fractional cable equation with nonlocal boundary conditions

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Bazhlekova E., Dimovski I.
Open Physics
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2013
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vol. 11
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issue 10
1304-1313
EN
The fractional cable equation is studied on a bounded space domain. One of the prescribed boundary conditions is of Dirichlet type, the other is of a general form, which includes the case of nonlocal boundary conditions. In real problems nonlocal boundary conditions are prescribed when the data on the boundary can not be measured directly. We apply spectral projection operators to convert the problem to a system of integral equations in any generalized eigenspace. In this way we prove uniqueness of the solution and give an algorithm for constructing the solution in the form of an expansion in terms of the generalized eigenfunctions and three-parameter Mittag-Leffler functions. Explicit representation of the solution is given for the case of double eigenvalues. We consider some examples and as a particular case we recover a recent result. The asymptotic behavior of the solution is also studied.
3
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Numerical solution of fractionally damped beam by homotopy perturbation method

100%
Behera D., Chakraverty S.
Open Physics
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2013
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vol. 11
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issue 6
792-798
EN
This paper investigates the numerical solution of a viscoelastic continuous beam whose damping behaviours are defined in term of fractional derivatives of arbitrary order. The Homotopy Perturbation Method (HPM) is used to obtain the dynamic response. Unit step function response is considered for the analysis. The obtained results are depicted in various plots. From the results obtained it is interesting to note that by increasing the order of the fractional derivative the beam suffers less oscillation. Similar observations have also been made by keeping the order of the fractional derivative constant and varying the damping ratios. Comparisons are made with the analytic solutions obtained by Zu-feng and Xiao-yan [Appl. Math. Mech. 28, 219 (2007)] to show the effectiveness and validation of this method.
4
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Homotopy analysis method for solving Abel differential equation of fractional order

100%
Jafari H., Sayevand K., Tajadodi H., Baleanu D.
Open Physics
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2013
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vol. 11
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issue 10
1523-1527
EN
In this study, the homotopy analysis method is used for solving the Abel differential equation with fractional order within the Caputo sense. Stabilityand convergence of the proposed approach is investigated. The numerical results demonstrate that the homotopy analysis method is accurate and readily implemented.
5
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Fractional nonlinear systems with sequential operators

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Mozyrska D., Girejko E., Wyrwas M.
Open Physics
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2013
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vol. 11
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issue 10
1295-1303
EN
In the paper possible approximation of solutions to initial value problems stated for fractional nonlinear equations with sequential derivatives of Caputo type is presented. We proved that values of Caputo derivatives in continuous case can be approximated by corresponding values of h-difference operators with h being small enough. Numerical examples are presented.
6
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Two methods to solve a fractional single phase moving boundary problem

100%
Li X., Wang S., Zhao M.
Open Physics
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2013
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vol. 11
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issue 10
1387-1391
EN
A moving boundary problem of a melting problem is considered in this study. A mathematical model using the Caputo fractional derivative heat equation is proposed in the paper. Since moving boundary problems are difficult to solve for the exact solution, two methods are presented to approximate the evolution of the temperature. To simplify the computation, a similarity variable is adopted in order to reduce the partial differential equations to ordinary ones.
7
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Identification of the Thermal Conductivity Coefficient in the Heat Conduction Model with Fractional Derivative

89%
Brociek R., Słota D., Brociek R., Słota D.
Archives of Foundry Engineering
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2019
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vol. 19
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issue 3
8
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Mathematical modeling of traveling autosolitons in fractional-order activator-inhibitor systems

88%
Datsko B., Gafiychuk V.
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2018
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issue 4
9
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On the derivative chain-rules in fractional calculus via fractional difference and their application to systems modelling

88%
Jumarie G.
Open Physics
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2013
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vol. 11
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issue 6
617-633
EN
It has been pointed out that the derivative chains rules in fractional differential calculus via fractional calculus are not quite satisfactory as far as they can yield different results which depend upon how the formula is applied, that is to say depending upon where is the considered function and where is the function of function. The purpose of the present short note is to display some comments (which might be clarifying to some readers) on the matter. This feature is basically related to the non-commutativity of fractional derivative on the one hand, and furthermore, it is very close to the physical significance of the systems under consideration on the other hand, in such a manner that everything is right so. As an example, it is shown that the trivial first order system may have several fractional modelling depending upon the way by which it is observed. This suggests some rules to construct the fractional models of standard dynamical systems, in as meaningful a model as possible. It might happen that this pitfall comes from the feature that a function which is continuous everywhere, but is nowhere differentiable, exhibits random-like features.
10
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On optimal control problem subject to fractional order discrete time singular systems

88%
Chiranjeevi T., Biswas R. K., Devarapalli R., Babu N. R., García Márquez F. P., Chiranjeevi T., Biswas R. K., Devarapalli R., Babu N. R., García Márquez F. P.
Archives of Control Sciences
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2021
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vol. 31
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issue 4
11
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Time-fractional extensions of the Liouville and Zwanzig equations

88%
Lukashchuk S.
Open Physics
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2013
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vol. 11
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issue 6
740-749
EN
This paper presents extensions to the classical stochastic Liouville equation of motion that contain the Riemann-Liouville and Caputo time-fractional derivatives. At first, the dynamic equations with the time-fractional derivatives are formally obtained from the classical Liouville equation. A feature of these new equations is that they have the same common formal solution as the classical Liouville equation and therefore may be used for study of the Hamiltonian system dynamics. Two cases of the time-dependent and time-independent Hamiltonian are considered separately. Then, the time-fractional Liouville equations are deduced from the short- and long-time asymptotic expansions of the obtained dynamic equations. The physical meaning of the resulting equations is discussed. The statements of the Cauchy-type problems for the derived time-fractional Liouville equations are given, and the formal solutions of these problems are presented. At last, the projection operator formalism is employed to derive the time-fractional extensions of the Zwanzig kinetic equations and the corresponding formal statistical operators from the time-fractional Liouville equations.
12
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Oscillation of fractional order functional differential equations with nonlinear damping

88%
Bayram M., Adiguzel H., Ogrekci S.
Open Physics
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2015
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vol. 13
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issue 1
EN
In this paper, we are concerned with the oscillatory behavior of a class of fractional differential equations with functional terms. The fractional derivative is defined in the sense of the modified Riemann-Liouville derivative. Based on a certain variable transformation, by using a generalized Riccati transformation, generalized Philos type kernels, and averaging techniques we establish new interval oscillation criteria. Illustrative examples are also given.
13
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Applying a fractional coil model for power system ferroresonance analysis

63%
Majka Ł.
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2018
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issue 4
14
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The dynamics of multiparticle collisions in motion of a granular material

63%
Leszczyński J.
Open Physics
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2003
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vol. 1
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issue 4
596-605
EN
We consider the complex problem of how to calculate particle motions taking into account multiparticle collisions. Multiparticle contacts occur when a particle collides with neighbouring particles, so that those contacts have a direct influence on each other. We will focus on the molecular dynamics method. Particularly, we will analyse what happens in cohesive materials during multiparticle contacts. We investigated the expression of repulsive force formulated under fractional calculus which is able to control dynamically the transfer and dissipation of energy in granular media. Such approach allows to perform simulations of arbitrary multiparticle collisions and also granular cohesion dynamics.
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