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This book is a result of the contributions of scientists involved in a Special Issue entitled “The Craft of Fractional Modelling in Science and Engineering” published by the journal Fractal and Fractional (MDPI). Most of the articles were published at the end of 2017 and the beginning 2018. In accordance with the initial aim of the Special Issue, the best published have now been consolidated into this book. The articles included span a broad area of applications of fractional calculus and demonstrate the feasibility of the noninteger differentiation and integration approach in modeling, directly related to pertinent problems in science and engineering.This a good beginning and it would be beneficial to continue with this collection under the same title and potentially provide a second volume of this book in the future.
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Historically, the notion of entropy emerged in conceptually very distinct contexts. This book deals with the connection between entropy, probability, and fractional dynamics as they appeared, for example, in solar neutrino astrophysics since the 1970's (Mathai and Rathie 1975, Mathai and Pederzoli 1977, Mathai and Saxena 1978, Mathai, Saxena, and Haubold 2010).The original solar neutrino problem, experimentally and theoretically, was resolved through the discovery of neutrino oscillations and was recently enriched by neutrino entanglement entropy. To reconsider possible new physics of solar neutrinos, diffusion entropy analysis, utilizing Boltzmann entropy, and standard deviation analysis was undertaken with SuperKamiokande solar neutrino data. This analysis revealed a nonGaussian signal with harmonic content. The Hurst exponent is different from the scaling exponent of the probability density function and both Hurst exponent and scaling exponent of the SuperKamiokande data deviate considerably from the value of ½, which indicates that the statistics of the underlying phenomenon is anomalous. Here experiment may provide guidance about the generalization of theory of Boltzmann statistical mechanics. Arguments in the socalled BoltzmannPlanckEinstein discussion related to Planck's discovery of the blackbody radiation law are recapitulated mathematically and statistically and emphasize from this discussion is pursued that a meaningful implementation of the complex ‘entropyprobabilitydynamics’ may offer two ways for explaining the results of diffusion entropy analysis and standard deviation analysis. One way is to consider an anomalous diffusion process that needs to use the fractional spacetime diffusion equation (Gorenflo and Mainardi) and the other way is to consider a generalized Boltzmann entropy by assuming a power law probability density function. Here new mathematical framework, invented by sheer thought, may provide guidance for the generalization of Boltzmann statistical mechanics. In this book Boltzmann entropy, generalized by Tsallis and Mathai, is considered. The second one contains a varying parameter that is used to construct an entropic pathway covering generalized type1 beta, type2 beta, and gamma families of densities. Similarly, pathways for respective distributions and differential equations can be developed. Mathai's entropy is optimized under various conditions reproducing the wellknown Boltzmann distribution, Raleigh distribution, and other distributions used in physics. Properties of the entropy measure for the generalized entropy are examined. In this process the role of special functions of mathematical physics, particularly the Hfunction, is highlighted.
special functions  fractional calculus  entropic functional  mathematical physics  applied analysis  statistical distributions  geometrical probabilities  multivariate analysis
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Fractional calculus is allowing integrals and derivatives of any positive order (the term fractional is kept only for historical reasons). It can be considered a branch of mathematical physics that deals with integrodifferential equations, where integrals are of convolution type and exhibit mainly singular kernels of power law or logarithm type.It is a subject that has gained considerably popularity and importance in the past few decades in diverse fields of science and engineering. Efficient analytical and numerical methods have been developed but still need particular attention.The purpose of this Special Issue is to establish a collection of articles that reflect the latest mathematical and conceptual developments in the field of fractional calculus and explore the scope for applications in applied sciences.
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Complex systems are pervasive in many areas of science integrated in our daily lives. Examples include financial markets, highway transportation networks, telecommunication networks, world and country economies, social networks, immunological systems, living organisms, computational systems and electrical and mechanical structures. Complex systems are often composed of a large number of interconnected and interacting entities, exhibiting much richer global scale dynamics than the properties and behavior of individual entities. Complex systems are studied in many areas of natural sciences, social sciences, engineering and mathematical sciences. This special issue therefore intends to contribute towards the dissemination of the multifaceted concepts in accepted use by the scientific community. We hope readers enjoy this pertinent selection of papers which represents relevant examples of the state of the art in present day research. [...]
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This book is focused on fractional order systems. Historically, fractional calculus has been recognized since the inception of regular calculus, with the first written reference dated in September 1695 in a letter from Leibniz to L’Hospital. Nowadays, fractional calculus has a wide area of applications in areas such as physics, chemistry, bioengineering, chaos theory, control systems engineering, and many others. In all those applications, we deal with fractional order systems in general. Moreover, fractional calculus plays an important role even in complex systems and therefore allows us to develop better descriptions of realworld phenomena. On that basis, fractional order systems are ubiquitous, as the whole real world around us is fractional. Due to this reason, it is urgent to consider almost all systems as fractional order systems.
anomalous diffusion  complexity  magnetic resonance imaging  fractional calculus  fractional complex networks  adaptive control  pinning synchronization  timevarying delays  impulses  reaction–diffusion terms  fractional calculus  mass absorption  diffusionwave equation  Caputo derivative  harmonic impact  Laplace transform  Fourier transform  MittagLeffler function  fractional calculus  fractionalorder system  long memory  time series  Hurst exponent  fractional  control  PID  parameter  meaning  audio signal processing  linear prediction  fractional derivative  musical signal  optimal randomness  swarmbased search  cuckoo search  heavytailed distribution  global optimization
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Investigations involving the theory and applications of mathematical analytic tools and techniques are remarkably widespread in many diverse areas of the mathematical, physical, chemical, engineering and statistical sciences. In this Special Issue, we invite and welcome review, expository and original research articles dealing with the recent advances in mathematical analysis and its multidisciplinary applications.
Mathematical (or Higher Transcendental) Functions and Their Applications  Fractional Calculus and Its Applications  qSeries and qPolynomials  Analytic Number Theory  Special Functions of Mathematical Physics and Applied Mathematics  Geometric Function Theory of Complex Analysis
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During the past four decades or so, various operators of fractional calculus, such as those named after Riemann–Liouville, Weyl, Hadamard, Grunwald–Letnikov, Riesz, Erdelyi–Kober, Liouville–Caputo, and so on, have been found to be remarkably popular and important due mainly to their demonstrated applications in numerous diverse and widespread fields of the mathematical, physical, chemical, engineering, and statistical sciences. Many of these fractional calculus operators provide several potentially useful tools for solving differential, integral, differintegral, and integrodifferential equations, together with the fractionalcalculus analogues and extensions of each of these equations, and various other problems involving special functions of mathematical physics, as well as their extensions and generalizations in one and more variables. In this Special Issue, we invite and welcome review, expository, and original research articles dealing with the recent advances in the theory of fractional calculus and its multidisciplinary applications.
operators of fractional calculus  chaos and fractional dynamics  fractional differential  fractional differintegral equations  fractional integrodifferential equations  fractional integrals  fractional derivatives associated with special functions of mathematical physics  applied mathematics  identities and inequalities involving fractional integrals  fractional derivatives
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This Special Issue contains a series of excellent research works on telecommunications and signal processing, selected from the 2018 41st International Conference on Telecommunications and Signal Processing (TSP) which was held on July 4–6, 2018, in Athens, Greece. The conference was organized in cooperation with the IEEE Region 8 (Europe, Middle East, and Africa), IEEE Greece Section, IEEE Czechoslovakia Section, and IEEE Czechoslovakia Section SP/CAS/COM Joint Chapter by seventeen universities from the Czech Republic, Hungary, Turkey, Taiwan, Japan, Slovak Republic, Spain, Bulgaria, France, Slovenia, Croatia, and Poland, for academics, researchers, and developers, and serves as a premier international forum for the annual exchange and promotion of the latest advances in telecommunication technology and signal processing. The aim of the conference is to bring together both novice and experienced scientists, developers, and specialists, to meet new colleagues, collect new ideas, and establish new cooperation between research groups from universities, research centers, and private sectors worldwide. This collection of 10 papers is highly recommended for researchers, and believed to be interesting, inspiring, and motivating for readers in their further research.
activation process  EPON  GPON  MATLAB  NGPON2  timing  transmission convergence layer  XGPON  authentication  identification  security  wireless communication  machine learning  phonation  acoustic analysis  followup study  hypokinetic dysarthria  Parkinson’s disease  pattern recognition  dynamic time warping  maximum likelihood criterion  similarity measure  multitemporal data  multispectral information  cognitive radio  interference alignment  interference leakage  multihop relay network  Parkinson’s disease dysgraphia  micrographia  online handwriting  kinematic analysis  fractionalorder derivative  fractional calculus  electrocardiogram (ECG)  Least Absolute Shrinkage and Selection Operator (LASSO)  overcomplete multiscale dictionary construction  signal representation  sparse inference  fractionalorder filters  fractional calculus  Chebyshev filters  lowpass filters  magnitude responses  timeinterleaved analogtodigital converter (TIADC)  sampleandhold (S/H) mismatch  modulo M quasistationary  Taylor series  Richardson iteration  3D segmentation  brain  deep learning  neural network  opensource  semantic segmentation  spine  unet  n/a
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In order to measure and quantify the complex behavior of realworld systems, either novel mathematical approaches or modifications of classical ones are required to precisely predict, monitor, and control complicated chaotic and stochastic processes. Though the term of entropy comes from Greek and emphasizes its analogy to energy, today, it has wandered to different branches of pure and applied sciences and is understood in a rather rough way, with emphasis placed on the transition from regular to chaotic states, stochastic and deterministic disorder, and uniform and nonuniform distribution or decay of diversity. This collection of papers addresses the notion of entropy in a very broad sense. The presented manuscripts follow from different branches of mathematical/physical sciences, natural/social sciences, and engineeringoriented sciences with emphasis placed on the complexity of dynamical systems. Topics like timing chaos and spatiotemporal chaos, bifurcation, synchronization and antisynchronization, stability, lumped mass and continuous mechanical systems modeling, novel nonlinear phenomena, and resonances are discussed.
multitime scale fractional stochastic differential equations  fractional Brownian motion  fractional stochastic partial differential equation  analytical solution  nonautonomous (autonomous) dynamical system  topological entropy  (asymptotical) focal entropy point  disturbation  mdimensional manifold  geometric nonlinearity  Bernoulli–Euler beam  colored noise  noise induced transitions  true chaos  Lyapunov exponents  wavelets  Lyapunov exponents  Wolf method  Rosenstein method  Kantz method  neural network method  method of synchronization  Benettin method  Fourier spectrum  Gauss wavelets  fractional calculus  Adomian decomposition  Mittag–Leffler function  descriptor fractional linear systems  regular pencils  Schur factorization  hyperchaotic system  selfsynchronous stream cipher  permutation entropy  image encryption  wavelet transform  product MValgebra  partition  Tsallis entropy  conditional Tsallis entropy  dynamical system  discrete chaos  discrete fractional calculus  hidden attractors  approximate entropy  stabilization  Information transfer  continuous flow  discrete mapping  Lorenz system  Chua’s system  deterministic chaos  random number generator  unbounded chaos  bounded chaos  phaselocked loop  Gaussian white noise  n/a
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The many technical and computational problems that appear to be constantly emerging in various branches of physics and engineering beg for a more detailed understanding of the fundamental mathematics that serves as the cornerstone of our way of understanding natural phenomena. The purpose of this Special Issue was to establish a brief collection of carefully selected articles authored by promising young scientists and the world's leading experts in pure and applied mathematics, highlighting the stateoftheart of the various research lines focusing on the study of analytical and numerical mathematical methods for pure and applied sciences.
ultraparabolic equation  ultradiffusion process  probabilistic representation  mathematical finance  linear elastostatics  layer potentials  fredholmian operators  fractional differential equations  fractional derivative  Abeltype integral  time delay  distributed lag  gamma distribution  macroeconomics  Keynesian model  integral transforms  Laplace integral transform  transmutation operator  generating operator  integral equations  differential equations  operational calculus of Mikusinski type  Mellin integral transform  fractional derivative  fractional integral  Mittag–Leffler function  Riemann–Liouville derivative  Caputo derivative  Grünwald–Letnikov derivative  spacetime fractional diffusion equation  fractional Laplacian  subordination principle  MittagLeffler function  Bessel function  exterior calculus  exterior algebra  electromagnetism  Maxwell equations  differential forms  tensor calculus  Fourier Theory  DFT in polar coordinates  polar coordinates  multidimensional DFT  discrete Hankel Transform  discrete Fourier Transform  Orthogonality  multispecies biofilm  biosorption  free boundary value problem  heavy metals toxicity  method of characteristics  relativistic diffusion equation  Caputo fractional derivatives of a function with respect to another function  BesselRiesz motion  Mittag–Leffler function  matrix function  Schur decomposition  Laplace transform  fractional calculus  central limit theorem  anomalous diffusion  stable distribution  fractional calculus  power law  n/a
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