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Volume 11 Issue 6 - Topical issue: Fractional dynamics

June 2013

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Journal Overview

Open Access October 9, 2013

Fractionl dynamics

Richard Magin, Raoul Nigmatullin

Page range: 615-616

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Open Access October 9, 2013

On the derivative chain-rules in fractional calculus via fractional difference and their application to systems modelling

Guy Jumarie

Page range: 617-633

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Abstract

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.

Open Access October 9, 2013

Fractional dynamics of tracer transport in fractured media from local to regional scales

Yong Zhang, Donald Reeves, Karl Pohlmann, Jenny Chapman, Charles Russell

Page range: 634-645

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Abstract

Tracer transport through fractured media exhibits concurrent direction-dependent super-diffusive spreading along high-permeability fractures and sub-diffusion caused by mass transfer between fractures and the rock matrix. The resultant complex dynamics challenge the applicability of conventional physical models based on Fick’s law. This study proposes a multi-scaling tempered fractional-derivative (TFD) model to explore fractional dynamics for tracer transport in fractured media. Applications show that the TFD model can capture anomalous transport observed in small-scale single fractures, intermediate-scale fractured aquifers, and two-dimensional large-scale discrete fracture networks. Tracer transport in fractured media from local (0.255-meter long) to regional (400-meter long) scales therefore can be quantified by a general fractional-derivative model. Fractional dynamics in fractured media can be scale dependent, owning to 1) the finite length of fractures that constrains the large displacement of tracers, and 2) the increasing mass exchange capacity along the travel path that enhances sub-diffusion.

Open Access October 9, 2013

Numerical investigation of three types of space and time fractional Bloch-Torrey equations in 2D

Qiang Yu, Fawang Liu, Ian Turner, Kevin Burrage

Page range: 646-665

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Abstract

Recently, the fractional Bloch-Torrey model has been used to study anomalous diffusion in the human brain. In this paper, we consider three types of space and time fractional Bloch-Torrey equations in two dimensions: Model-1 with the Riesz fractional derivative; Model-2 with the one-dimensional fractional Laplacian operator; and Model-3 with the two-dimensional fractional Laplacian operator. Firstly, we propose a spatially second-order accurate implicit numerical method for Model-1 whereby we discretize the Riesz fractional derivative using a fractional centered difference. We consider a finite domain where the time and space derivatives are replaced by the Caputo and the sequential Riesz fractional derivatives, respectively. Secondly, we utilize the matrix transfer technique for solving Model-2 and Model-3. Finally, some numerical results are given to show the behaviours of these three models especially on varying domain sizes with zero Dirichlet boundary conditions.

Open Access October 9, 2013

Some properties of the fundamental solution to the signalling problem for the fractional diffusion-wave equation

Yuri Luchko, Francesco Mainardi

Page range: 666-675

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Abstract

In this paper, the one-dimensional time-fractional diffusion-wave equation with the Caputo fractional derivative of order α, 1 ≤ α ≤ 2 and with constant coefficients is revisited. It is known that the diffusion and the wave equations behave quite differently regarding their response to a localized disturbance. Whereas the diffusion equation describes a process where a disturbance spreads infinitely fast, the propagation speed of the disturbance is a constant for the wave equation. We show that the time-fractional diffusion-wave equation interpolates between these two different responses and investigate the behavior of its fundamental solution for the signalling problem in detail. In particular, the maximum location, the maximum value, and the propagation velocity of the maximum point of the fundamental solution for the signalling problem are described analytically and calculated numerically.

Open Access October 9, 2013

Electro-chemical manifestation of nanoplasmonics in fractal media

Emmanuel Baskin, Alexander Iomin

Page range: 676-684

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Abstract

Electrodynamics of composite materials with fractal geometry is studied in the framework of fractional calculus. This consideration establishes a link between fractal geometry of the media and fractional integrodifferentiation. The photoconductivity in the vicinity of the electrode-electrolyte fractal interface is studied. The methods of fractional calculus are employed to obtain an analytical expression for the giant local enhancement of the optical electric field inside the fractal composite structure at the condition of the surface plasmon excitation. This approach makes it possible to explain experimental data on photoconductivity in the nano-electrochemistry.

Open Access October 9, 2013

Thermoelasticity of thin shells based on the time-fractional heat conduction equation

Yuriy Povstenko

Page range: 685-690

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Abstract

The time-nonlocal generalizations of Fourier’s law are analyzed and the equations of the generalized thermoelasticity based on the time-fractional heat conduction equation with the Caputo fractional derivative of order 0 < α ≤ 2 are presented. The equations of thermoelasticity of thin shells are obtained under the assumption of linear dependence of temperature on the coordinate normal to the median surface of a shell. The conditions of Newton’s convective heat exchange between a shell and the environment have been assumed. In the particular case of classical heat conduction (α = 1) the obtained equations coincide with those known in the literature.

Open Access October 9, 2013

Noether’s theorem for fractional variational problems of variable order

Tatiana Odzijewicz, Agnieszka Malinowska, Delfim Torres

Page range: 691-701

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Abstract

We prove a necessary optimality condition of Euler-Lagrange type for fractional variational problems with derivatives of incommensurate variable order. This allows us to state a version of Noether’s theorem without transformation of the independent (time) variable. Considered derivatives of variable order are defined in the sense of Caputo.

Open Access October 9, 2013

Tuning fractional PID controllers for a Steward platform based on frequency domain and artificial intelligence methods

Cosmin Copot, Yu Zhong, Clara Ionescu, Robin Keyser

Page range: 702-713

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Abstract

In this paper, two methods to tune a fractional-order PI λ D μ controller for a mechatronic system are presented. The first method is based on a genetic algorithm to obtain the parameter values for the fractionalorder PI λ D μ controller by global optimization. The second method used to design the fractional-order PI λ D μ controller relies on an auto-tuning approach by meeting some specifications in the frequency domain. The real-time experiments are conducted using a Steward platform which consists of a table tilted by six servo-motors with a ball on the top of the table. The considered system is a 6 degrees of freedom (d.o.f.) motion platform. The feedback on the position of the ball is obtained from images acquired by a visual sensor mounted above the platform. The fractional-order controllers were implemented and the performances of the steward platform are analyzed.

Open Access October 9, 2013

Analysis of football player’s motion in view of fractional calculus

Micael Couceiro, Filipe Clemente, Fernando Martins

Page range: 714-723

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Abstract

Accurately retrieving the position of football players over time may lay the foundations for a whole series of possible new performance metrics for coaches and assistants. Despite the recent developments of automatic tracking systems, the misclassification problem (i.e., misleading a given player by another) still exists and requires human operators as final evaluators. This paper proposes an adaptive fractional calculus (FC) approach to improve the accuracy of tracking methods by estimating the position of players based on their trajectory so far. One half-time of an official football match was used to evaluate the accuracy of the proposed approach under different sampling periods of 250, 500 and 1000 ms. Moreover, the performance of the FC approach was compared with position-based and velocity-based methods. The experimental evaluation shows that the FC method presents a high classification accuracy for small sampling periods. Such results suggest that fractional dynamics may fit the trajectory of football players, thus being useful to increase the autonomy of tracking systems.

Open Access October 9, 2013

Self-similarity principle: the reduced description of randomness

Raoul Nigmatullin, José Machado, Rui Menezes

Page range: 724-739

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Abstract

A new general fitting method based on the Self-Similar (SS) organization of random sequences is presented. The proposed analytical function helps to fit the response of many complex systems when their recorded data form a self-similar curve. The verified SS principle opens new possibilities for the fitting of economical, meteorological and other complex data when the mathematical model is absent but the reduced description in terms of some universal set of the fitting parameters is necessary. This fitting function is verified on economical (price of a commodity versus time) and weather (the Earth’s mean temperature surface data versus time) and for these nontrivial cases it becomes possible to receive a very good fit of initial data set. The general conditions of application of this fitting method describing the response of many complex systems and the forecast possibilities are discussed.

Open Access October 9, 2013

Time-fractional extensions of the Liouville and Zwanzig equations

Stanislav Lukashchuk

Page range: 740-749

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Abstract

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.

Open Access October 9, 2013

The variable, fractional-order discrete-time PD controller in the IISv1.3 robot arm control

Piotr Ostalczyk, Dariusz Brzezinski, Piotr Duch, Maciej Łaski, Dominik Sankowski

Page range: 750-759

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Abstract

In this paper, the discrete differentiation order functions of the variable, fractional-order PD controller (VFOPD) are considered. In the proposed VFOPD controller, a variable, fractional-order backward difference is applied to perform closed-loop, system error, discrete-time differentiation. The controller orders functions which may be related to the controller input or output signal or an input and output signal. An example of the VFOPD controller is applied to the robot arm closed-loop control due to system changes in moment of inertia. The close-loop system step responses are presented.

Open Access October 9, 2013

Analysis of two colliding fractionally damped spherical shells in modelling blunt human head impacts

Yury Rossikhin, Marina Shitikova

Page range: 760-778

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Abstract

The collision of two elastic or viscoelastic spherical shells is investigated as a model for the dynamic response of a human head impacted by another head or by some spherical object. Determination of the impact force that is actually being transmitted to bone will require the model for the shock interaction of the impactor and human head. This model is indended to be used in simulating crash scenarios in frontal impacts, and provide an effective tool to estimate the severity of effect on the human head and to estimate brain injury risks. The model developed here suggests that after the moment of impact quasi-longitudinal and quasi-transverse shock waves are generated, which then propagate along the spherical shells. The solution behind the wave fronts is constructed with the help of the theory of discontinuities. It is assumed that the viscoelastic features of the shells are exhibited only in the contact domain, while the remaining parts retain their elastic properties. In this case, the contact spot is assumed to be a plane disk with constant radius, and the viscoelastic features of the shells are described by the fractional derivative standard linear solid model. In the case under consideration, the governing differential equations are solved analytically by the Laplace transform technique. It is shown that the fractional parameter of the fractional derivative model plays very important role, since its variation allows one to take into account the age-related changes in the mechanical properties of bone.

Open Access October 9, 2013

Existence, uniqueness and limit property of solutions to quadratic Erdélyi-Kober type integral equations of fractional order

JinRong Wang, Chun Zhu, Michal Fečkan

Page range: 779-791

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Abstract

In this paper, we apply certain measure of noncompactness and fixed point theorem of Darbo type to derive the existence and limit property of solutions to quadratic Erdélyi-Kober type integral equations of fractional order with three parameters. Moreover, we also present the uniqueness and another existence results of the solutions to the above equations. Finally, two examples are given to illustrate the obtained results.

Open Access October 9, 2013

Numerical solution of fractionally damped beam by homotopy perturbation method

Diptiranjan Behera, Snehashish Chakraverty

Page range: 792-798

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Abstract

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.

Open Access October 9, 2013

Fractional diffusion equation in half-space with Robin boundary condition

Jun-Sheng Duan, Zhong Wang, Shou-Zhong Fu

Page range: 799-805

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Abstract

The initial and boundary value problem for the fractional diffusion equation in half-space with the Robin boundary condition is considered. The solution is comprised of two parts: the contribution of the initial value and the contribution of the boundary value, for which the respective fundamental solutions are given. Finally, the solution formula of the considered problem is obtained.

Open Access October 9, 2013

Consensus of compound-order multi-agent systems with communication delays

Hong-yong Yang, Lei Guo, Xun-lin Zhu, Ke-cai Cao, Hai-lin Zou

Page range: 806-812

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Abstract

In complex environments, many distributed networked systems can only be illustrated with fractional-order dynamics. When multi-agent systems show individual diversity with difference agents, heterogeneous (integer-order and fractional-order) dynamics are used to illustrate the agent systems and compose integerfractional compounded-order systems. Applying Laplace transform and frequency domain theory of the fractional-order operator, the consensus of delayed multi-agent systems with directed weighted topologies is studied. Since an integer-order model is the special case of a fractional-order model, the results in this paper can be extended to systems with integer-order models. Finally, numerical examples are used to verify our results.

Open Access October 9, 2013

Robust projective outer synchronization of coupled uncertain fractional-order complex networks

Junwei Wang, Yun Zhang

Page range: 813-823

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Abstract

In this work, we propose a novel projective outer synchronization (POS) between unidirectionally coupled uncertain fractional-order complex networks through scalar transmitted signals. Based on the state observer theory, a control law is designed and some criteria are given in terms of linear matrix inequalities which guarantee global robust POS between such networks. Interestingly, in the POS regime, we show that different choices of scaling factor give rise to different outer synchrony, with various special cases including complete outer synchrony, anti-outer synchrony and even a state of amplitude death. Furthermore, it is demonstrated that although stability of POS is irrelevant to the inner-coupling strength, it will affect the convergence speed of POS. In particular, stronger inner synchronization can induce faster POS. The effectiveness of our method is revealed by numerical simulations on fractional-order complex networks with small-world communication topology.

Open Access October 9, 2013

Synchronization of variable-order fractional financial system via active control method

Yufeng Xu, Zhimin He

Page range: 824-835

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Abstract

In this paper, we study the chaotic dynamics of a Variable-Order Fractional Financial System (VOFFS). The Variable-Order Fractional Derivative (VOFD) is defined in Caputo type. A necessary condition for occurrence of chaos in VOFFS is obtained. Numerical experiments on the dynamics of the VOFFS with various conditions are given. Based on them, it is shown that the VOFFS has complex dynamical behavior, and the occurrence of chaos depends on the choice of order function. Furthermore, the chaos synchronization of the VOFFS is studied via active control method. Numerical simulations demonstrate that the active control method is effective and simple for synchronizing the VOFFSs with commensurate or incommensurate order functions.

Open Access October 9, 2013

The effect of fractionality nature in differences between computer simulation and experimental results of a chaotic circuit

Salman Faraji, Mohammad Tavazoei

Page range: 836-844

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Abstract

In practice, some differences are usually observed between computer simulation and experimental results of a chaotic circuit. In this paper, it is tried to obtain computer simulation results having more correlation with those obtained in practice by using more realistic models for chaotic circuits. This goal is achieved by considering the fractionality nature of electrical capacitors in the model of a chaotic circuit.

Open Access October 9, 2013

Fractional-order linear time invariant swarm systems: asymptotic swarm stability and time response analysis

Mojtaba Soorki, Mohammad Tavazoei

Page range: 845-854

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Abstract

This paper deals with fractional-order linear time invariant swarm systems. Necessary and sufficient conditions for asymptotic swarm stability of these systems are presented. Also, based on a time response analysis the speed of convergence in an asymptotically swarm stable fractional-order linear time invariant swarm system is investigated and compared with that of its integer-order counterpart. Numerical simulation results are presented to show the effectiveness of the paper results.

Open Access October 9, 2013

General formula for stability testing of linear systems with fractional-delay characteristic equation

Farshad Merrikh-Bayat

Page range: 855-862

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Abstract

In some applications (especially in the filed of control theory) the characteristic equation of system contains fractional powers of the Laplace variable s possibly in combination with exponentials of fractional powers of s. The aim of this paper is to propose an easy-to-use and effective formula for bounded-input boundedoutput (BIBO) stability testing of a linear time-invariant system with fractional-delay characteristic equation in the general form of $$\Delta \left( s \right) = P_0 \left( s \right) + \sum\nolimits_{i = 1}^N {P_i \left( s \right)\exp ( - \zeta _i s^{\beta _i } ) = 0}$$, where P i(s) (i = 0,...,N) are the so-called fractional-order polynomials and ξ i and β i are positive real constants. The proposed formula determines the number of the roots of such a characteristic equation in the right half-plane of the first Riemann sheet by applying Rouche’s theorem. Numerical simulations are also presented to confirm the efficiency of the proposed formula.

Open Access October 9, 2013

About Maxwell’s equations on fractal subsets of ℝ3

Alireza Golmankhaneh, Ali Golmankhaneh, Dumitru Baleanu

Page range: 863-867

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Abstract

In this paper we have generalized $$F^{\bar \xi }$$-calculus for fractals embedding in ℝ3. $$F^{\bar \xi }$$-calculus is a fractional local derivative on fractals. It is an algorithm which may be used for computer programs and is more applicable than using measure theory. In this Calculus staircase functions for fractals has important role. $$F^{\bar \xi }$$-fractional differential form is introduced such that it can help us to derive the physical equation. Furthermore, using the $$F^{\bar \xi }$$-fractional differential form of Maxwell’s equations on fractals has been suggested.

Open Access October 9, 2013

Posicast control of a class of fractional-order processes

Emmanuel Gonzalez, John Hung, Lubomir Dorcak, Jan Terpak, Ivo Petras

Page range: 868-880

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Abstract

The number of studies on the control of fractional-order processes—processes having dynamics described by differential equations of arbitrary order—has been increasing in the past two decades and it is now ubiquitous. Various methods have emerged and have been proven to effectively control such processes—usually resulting in fractional-order controllers similar to their conventional integer-order counterparts, which include, but are not limited to fractional PID and fractional lead-lag controllers. However, such methods require a lot of computational effort and fractional-order controllers could be challenging when it comes to their synthesis and implementation. In this paper, we propose a simple yet effective delay-based controller with the use of the Posicast control methodology in controlling the overshoot of a fractional-order process of the class $$\mathcal{P}:\left\{ {P\left( s \right) = {1 \mathord{\left/ {\vphantom {1 {\left( {as^\alpha + b} \right)}}} \right. \kern-\nulldelimiterspace} {\left( {as^\alpha + b} \right)}}} \right\}$$ having orders 1 < α < 2. Such controllers have proven to be easy to implement because they only require delays and summers. In this paper, the Posicast control methodology introduced in the past few years is modified to minimize the overshoot of the processes step response to a level that is acceptable in control engineering and automation practices. Furthermore, proof of the existence of overshoot for such class of processes, as well as the determination of the peak-time of the open-loop response of a fractional-order process of the class P is presented. Validation through numerical simulations for a class of fractional-order processes are presented in this paper.

Open Access October 9, 2013

Fractional order modelling of zero length column desorption response for adsorbents with variable particle sizes

J. Machado, Sharif Zaman, Dumitru Baleanu

Page range: 881-885

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Abstract

This manuscript analyses the data generated by a Zero Length Column (ZLC) diffusion experimental set-up, for 1,3 Di-isopropyl benzene in a 100% alumina matrix with variable particle size. The time evolution of the phenomena resembles those of fractional order systems, namely those with a fast initial transient followed by long and slow tails. The experimental measurements are best fitted with the Harris model revealing a power law behavior.

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