Special Issue "Fractional Order Systems"

A special issue of Mathematics (ISSN 2227-7390).

Deadline for manuscript submissions: closed (30 April 2019)

Special Issue Editor

Guest Editor
Prof. Dr. Ivo Petráš

Institute of Control and Informatization of Production Processes, BERG Faculty, Technical University of Košice, B. Němcovej 3, 042 00 Košice, Slovakia
Website | E-Mail
Interests: fractional calculus and its applications; dynamical systems; chaos theory; control theory; mathematical modelling; simulations; process control; automation; signal processing

Special Issue Information

Dear Colleagues,

It is well known that fractional calculus is recognized since the regular calculus with the first written reference dated in September 1695 in letter from Leibniz to L’Hospital. Nowadays, the fractional calculus has a wide area of applications, for example, physics, chemistry, bioengineering, chaos theory, control systems engineering, and many others. In all those applications we deal, in general, with fractional order systems. Moreover, the fractional calculus plays an important role even in the complex systems and therefore allows us to use a better description of some real-world phenomena. Based on this fact, the fractional order systems are ubiquitous as well as whole real world around us is fractional, not integer one. Due to this reason it is so urgent consider almost all systems as the fractional order systems. 

This Special Issue is focused on the theory and multidisciplinary applications of fractional order systems in science and engineering, and will accept only high-quality survey, and/or original research papers.

Prof. Dr. Ivo Petráš
Guest Editor

Manuscript Submission Information

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Keywords

  • Fractional calculus and its applications
  • Fractional differential equations
  • Fractional order dynamical systems
  • Fractional order control
  • Fractional nonlinear and chaotic systems

Published Papers (3 papers)

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Research

Open AccessArticle
Time-Fractional Diffusion-Wave Equation with Mass Absorption in a Sphere under Harmonic Impact
Mathematics 2019, 7(5), 433; https://doi.org/10.3390/math7050433
Received: 24 April 2019 / Revised: 10 May 2019 / Accepted: 12 May 2019 / Published: 16 May 2019
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Abstract
The time-fractional diffusion equation with mass absorption in a sphere is considered under harmonic impact on the surface of a sphere. The Caputo time-fractional derivative is used. The Laplace transform with respect to time and the finite sin-Fourier transform with respect to the [...] Read more.
The time-fractional diffusion equation with mass absorption in a sphere is considered under harmonic impact on the surface of a sphere. The Caputo time-fractional derivative is used. The Laplace transform with respect to time and the finite sin-Fourier transform with respect to the spatial coordinate are employed. A graphical representation of the obtained analytical solution for different sets of the parameters including the order of fractional derivative is given. Full article
(This article belongs to the Special Issue Fractional Order Systems)
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Open AccessArticle
Adaptive Pinning Synchronization of Fractional Complex Networks with Impulses and Reaction–Diffusion Terms
Mathematics 2019, 7(5), 405; https://doi.org/10.3390/math7050405
Received: 7 March 2019 / Revised: 23 April 2019 / Accepted: 24 April 2019 / Published: 7 May 2019
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Abstract
In this paper, a class of fractional complex networks with impulses and reaction–diffusion terms is introduced and studied. Meanwhile, a class of more general network structures is considered, which consists of an instant communication topology and a delayed communication topology. Based on the [...] Read more.
In this paper, a class of fractional complex networks with impulses and reaction–diffusion terms is introduced and studied. Meanwhile, a class of more general network structures is considered, which consists of an instant communication topology and a delayed communication topology. Based on the Lyapunov method and linear matrix inequality techniques, some sufficient criteria are obtained, ensuring adaptive pinning synchronization of the network under a designed adaptive control strategy. In addition, a pinning scheme is proposed, which shows that the nodes with delayed communication are good candidates for applying controllers. Finally, a numerical example is given to verify the validity of the main results. Full article
(This article belongs to the Special Issue Fractional Order Systems)
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Open AccessArticle
Fractional Order Complexity Model of the Diffusion Signal Decay in MRI
Mathematics 2019, 7(4), 348; https://doi.org/10.3390/math7040348
Received: 11 March 2019 / Revised: 1 April 2019 / Accepted: 8 April 2019 / Published: 12 April 2019
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Abstract
Fractional calculus models are steadily being incorporated into descriptions of diffusion in complex, heterogeneous materials. Biological tissues, when viewed using diffusion-weighted, magnetic resonance imaging (MRI), hinder and restrict the diffusion of water at the molecular, sub-cellular, and cellular scales. Thus, tissue features can [...] Read more.
Fractional calculus models are steadily being incorporated into descriptions of diffusion in complex, heterogeneous materials. Biological tissues, when viewed using diffusion-weighted, magnetic resonance imaging (MRI), hinder and restrict the diffusion of water at the molecular, sub-cellular, and cellular scales. Thus, tissue features can be encoded in the attenuation of the observed MRI signal through the fractional order of the time- and space-derivatives. Specifically, in solving the Bloch-Torrey equation, fractional order imaging biomarkers are identified that connect the continuous time random walk model of Brownian motion to the structure and composition of cells, cell membranes, proteins, and lipids. In this way, the decay of the induced magnetization is influenced by the micro- and meso-structure of tissues, such as the white and gray matter of the brain or the cortex and medulla of the kidney. Fractional calculus provides new functions (Mittag-Leffler and Kilbas-Saigo) that characterize tissue in a concise way. In this paper, we describe the exponential, stretched exponential, and fractional order models that have been proposed and applied in MRI, examine the connection between the model parameters and the underlying tissue structure, and explore the potential for using diffusion-weighted MRI to extract biomarkers associated with normal growth, aging, and the onset of disease. Full article
(This article belongs to the Special Issue Fractional Order Systems)
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