editorial new challenges in fractional systems...

EditorialNew Challenges in Fractional Systems 2014

Guido Maione,1 Raoul R. Nigmatullin,2 José A. Tenreiro Machado,3 and Jocelyn Sabatier4

1Department of Electrical and Information Engineering (DEI), Technical University of Bari, Via E. Orabona 4, 70125 Bari, Italy2Radio-Electronic and Information & Measuring Technology Department, Kazan National ResearchTechnical University (KNRTU-KAI), Karl Marx Street 10, Kazan, Tatarstan 420111, Russia3Department of Electrical Engineering, Institute of Engineering (ISEP), Polytechnic of Porto, Rua Dr. Antonio Bernardino deAlmeida 431, 4200-072 Porto, Portugal4IMS Laboratory, Bordeaux 1 University, 351 Cours de la Liberation, 33400 Talence, France

Correspondence should be addressed to Guido Maione; [email protected]

Received 17 November 2014; Accepted 17 November 2014

Copyright © 2015 Guido Maione et al. This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

In many fields, theoretical and applied researches havestrengthened the belief that fractional calculus is not onlya branch of the mathematical analysis, but also a usefuland powerful tool for engineers. Namely, fractional cal-culus allows both a better modeling of a wide class ofsystems with anomalous dynamic behavior and a betterunderstanding of the facets of both physical phenomena andartificial processes. Hence the mathematical models derivedfrom differential equations with noninteger/fractional orderderivatives or integrals are becoming a fundamental researchissue for scientists and engineers. In particular, fractionalmodels are successful when describing power-law long-termmemory or hereditary properties. They are also successfulwhen anomalous diffusion, transport phenomena, and wavespropagation in complex media require nonlocal operators, orwhen fractal-like properties are evident in some processes.These potentialities and the related benefits have attractedmany experts and practitioners of different fields to recon-sider mathematical models and engineering methods bothfor linear and for nonlinear systems.

To synthesize, many different areas of science and engi-neering invest on fractional order modeling and systems bynow [1–7]. Moreover, in the last years, many researchers arebecoming aware of the interdisciplinary aspects of complexengineering problems that require knowledge from differentfields. At the same time, inspiration is often taken fromtheories or applications of different areas and sometimes itis necessary to combine scientific methodologies that aredeveloped for different problems. To cite a recent example,

achieving good results in delivering a high-quality multime-dia service through wireless networks not only is a matterof advanced communication technologies and protocols, butalso requires to build a proper transmission scheduler thatemploys classical or innovative control systems. Namely, afeedback control techniquemay schedule the transmission inadvance [8, 9], but can also consider the fractional nature ofthe data traffic [10] and the fractionalmodeling of the streams[11].

The above remarks justify the interest in fractionalmodeling and fractional order systems that may represent acommunication language to integrate expertise, results, andnew ideas of researchers coming from different areas. Theconsequences could have a great impact on everyday life,services, technology, industrial processes, and environmentalissues. However, fractional calculus and fractional ordersystems are not a panacea. Namely, many potentialities offractional order systems are still unexplored or under inves-tigation so that there are many challenges. The contributionsgathered in this special issue offer a snapshot of differentinteresting researches, problems, and solutions.

In detail, the papers of this special issue cover topicsin the fields of control systems, electric circuits, and imageand signal processing, concern some aspects of mathemat-ical modeling for biological and physical phenomena, andconsider some applications of numerical and computationalmethods to engineering and finance. In the following, webriefly highlight these topics and synthesize the content ofeach paper.

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015, Article ID 870841, 3 pageshttp://dx.doi.org/10.1155/2015/870841

2 Mathematical Problems in Engineering

The paper “Fractional dynamics of computer virus prop-agation,” by C. M. A. Pinto and J. A. Tenreiro Machado,introduces a fractional order model to describe the prop-agation and spread of computer viruses due to dynamicalinteractions between computing and removable devices. Thefractional order system is inspired by mathematical modelsfor biological epidemics and extends an integer order modeltaken from the literature by using fractional order differen-tiation. Numerical results show that fractional modeling canserve to describe and capture very fast dynamics or long-termmemory effects that cannot be represented by classical integerorder models. The investigation is therefore interesting forantivirus development and for studying effective protectivemeasures for applications based on internet connections thatare exposed to computer viruses.

The paper “On fractional order dengue epidemic model,”by H. Al-Sulamia et al., applies fractional calculus to modelthe spread of the tropical fever and considers memoryeffects. The model is very sensitive to the fractional orderof differentiation. Moreover, the authors study the stabilityof the equilibrium points of the fractional order systemdescribing the epidemic model. Numerical simulation of thefractional order system is based on the generalized Adams-Bashforth-Moulton method that provides an approximatesolution, which converges to the fixed point after a longertime than with an integer order model.

The paper “Algorithms of finite difference for pricingamerican options under fractional diffusion models,” by J. Xiet al., regards the application of fractional calculus tools toa financial topic, namely, developing accurate and effectivealgorithms for the price of American options. The authorsstart from fractional order partial differential equations usedunder fractional diffusion models to develop a first-orderapproximation.This is obtained by an iterative algorithm thatavoids singularities in the integral part of partial integrod-ifferential equations and computes the numerical estimatesby combining a fractional difference approach and a penaltymethod.Then the authors employ a spatial extrapolation for asecond-order accurate estimate. The numerical results in thepaper are intended to show the effectiveness and feasibility ofthe approach.

The paper “Adaptive sliding control for a class of fractionalcommensurate order chaotic systems,” by J. Yuan and B. Shi,is in the framework of control methodologies developedfor nonlinear fractional dynamic systems. The authors takeadvantage of fractional calculus tools and nonlinear controltheory. They propose adaptive sliding mode control designfor a class of commensurate fractional order chaotic systems.The authors first introduce a fractional integral sliding man-ifold for the nominal systems. Next, they prove the stabilityof the corresponding fractional sliding dynamics. Then, theyobtain the desired sliding control law by using a Lyapunovcandidate function and the Mittag-Leffler stability theory.The proposed sliding manifold is also adapted by a fractionaladaptation law for perturbed systems because of uncertaintiesand external disturbances. Simulation tests are provided toshow the performance of the designed controllers.

The paper “Approximated fractional-order chebyshev low-pass filters,” by T. Freeborn et al., is in the field of research

studies devoted to design of fractional order filters foranalog signal processing. The authors use a nonlinear leastsquares optimization method to determine the coefficientsof the fractional order transfer function that approximatesthe passband ripple characteristics of traditional Chebyshevlowpass filters. Matlab and SPICE simulation tests are used toverify the implementation of the fractional order filters.

The paper “Computational challenge of fractional differen-tial equations and the potential solutions: a survey,” byC.Gonget al., surveys computational costs of numerical methods tosolve fractional differential equations. The authors analyzethe computational complexity to solve time fractional, spacefractional, and space-time fractional equations, with respectto the case of integer order partial differential equationssolved by finite difference methods. The investigation aimsat giving a useful guide for solving problems that are mathe-matically represented by complex fractional differential equa-tions, also with variable fractional orders, and then requiresnumerical techniques with many time steps and many spacegrid points.

The paper “On a time-fractional integrodifferential equa-tion via three-point boundary value conditions,” by D. Baleanuet al., considers fractional differential equations useful tomodel physical processes that exhibit a fractional orderbehavior that varies with time and space. The complex,nonlinear fractional differential equations can be solvednumerically but it is important to prove existence andunique-ness of the solution. To this aim, the authors investigateand prove the existence of solutions for a time-fractionalintegrodifferential equation via three-point boundary valueconditions.

The paper “Hybrid prediction and fractal hyperspectralimage compression,” by S. Zhu et al., presents a method forhyperspectral image compression based on the combinationof a prediction and a modified fractal coding technique.Hyperspectral images are important in remote sensing appli-cations. The key point is to take advantage of the local self-similarity that exist between adjacent bands in the consideredimages to obtain a high compression ratio at low bitrate,resolution independence, and a fast decoding speed. Theauthors implement a hybrid algorithm that first carriesout an intraband prediction and then applies an interbandfractal encoding. The authors show the reduced encodingcomplexity, enhanced decoding quality, and higher peaksignal-to-noise performance.

The paper “A novel high efficiency fractal multiview videocodec,” by S. Zhu et al., proposes a fractal multiview videocodec for compressing three-dimensional video signals. Thecontribution can be considered in the context of researchefforts devoted to improving transmission and reproductionof high-quality multimedia over broadband communicationchannels, for example, wireless networks of last generation.These processes are very important in streaming, videocon-ferencing, interactive videos, and other similar applicationsin delivery of multimedia digital services. In particular,the authors propose a compression algorithm that exploitstemporal and spatial correlations and a disparity estimationto increase the codec efficiency for encoding and/or decodinga digital data stream or signal. Improvements are shown

Mathematical Problems in Engineering 3

in encoding performance (lower time and bitrate, highercompression ratio) and in quality of decoding.

Guido MaioneJose A. Tenreiro Machado

Raoul R. NigmatullinJocelyn Sabatier

References

[1] R. Caponetto, G. Dongola, L. Fortuna, and I. Petras, FractionalOrder Systems: Modeling and Control Application, vol. 72 ofWorld Scientific Series on Nonlinear Science Series A, WorldScientific Publishing, Singapore, 2010.

[2] R. Hilfer, Applications of Fractional Calculus in Physics, Aca-demic Press, Orlando, Fla, USA, 1999.

[3] F. Mainardi, Fractional Calculus and Waves in Linear Viscoelas-ticity: An Introduction toMathematicalModels, Imperial CollegePress, London, UK, 2010.

[4] C. A. Monje, Y. Q. Chen, B. M. Vinagre, D. Xue, and V.Feliu, Fractional-Order Systems and Controls: Fundamentalsand Applications, Springer, London, UK, 2010.

[5] I. Podlubny, Fractional Differential Equations. An Introduc-tion to Fractional Derivatives, Fractional Differential Equations,Some Methods of Their Solution and Some of Their Applications,vol. 198 of Mathematics in Science and Engineering, AcademicPress, San Diego, Calif, USA, 1999.

[6] J. Sabatier, O. P. Agrawal, and J. A. Tenreiro Machado, Advancesin Fractional Calculus: Theoretical Developments and Applica-tions in Physics and Engineering, Springer, 2007.

[7] S. G. Samko, A. A. Kilbas, and O. I. Marichev, FractionalIntegrals and Derivatives: Theory and Applications, Gordon andBreach Science, Amsterdam, The Netherlands, 1993.

[8] C. Ceglie, G. Maione, and D. Striccoli, “Periodic feedbackcontrol for streaming 3D videos in last-generation cellular net-works,” inProceedings of the 5th IFAC InternationalWorkshop onPeriodic Control Systems (PSYCO '13), F. Giri and V. van Assche,Eds., vol. 5, part 1, pp. 23–28, IFAC Proceedings Online, Caen,France, July 2013.

[9] G. Maione and D. Striccoli, “Transmission control of Variable-Bit-Rate video streaming inUMTSnetworks,”Control Engineer-ing Practice, vol. 20, no. 12, pp. 1366–1373, 2012.

[10] C. Ceglie, G. Maione, and D. Striccoli, “Statistical analysisof long-range dependence in three-dimensional video traffic,”in Proceedings of the Mathematical Methods in EngineeringInternational Conference (MME ’13), Porto, Portugal, July 2013.

[11] R. R. Nigmatullin, C. Ceglie, G. Maione, and D. Striccoli,“Reduced fractional modeling of 3-D video streams: theFERMA approach,” Nonlinear Dynamics, 2014.

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