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This eBook is a collection of articles from a Frontiers Research Topic. Frontiers Research Topics are very popular trademarks of the Frontiers Journals Series: they are collections of at least ten articles, all centered on a particular subject. With their unique mix of varied contributions from Original Research to Review Articles, Frontiers Research Topics unify the most influential researchers, the latest key findings and historical advances in a hot research area! Find out more on how to host your own Frontiers Research Topic or contribute to one as an author by contacting the Frontiers Editorial Office: frontiersin.org/about/contact

Science: general issues --- Physics --- Fractional Differential Equations --- Real-World Applications --- Physics --- fractional calculus --- computational techniques --- Fractional order models --- Fractional Differential Equations --- Real-World Applications --- Physics --- fractional calculus --- computational techniques --- Fractional order models

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In recent research on natural processes, mathematical modeling has become a very useful tool. It is often the case that, in fields such as economics and biology, a temporal lag between cause and effect must often be taken into consideration. In modeling, a natural and practical implementation of this phenomenon is through the use of distributed delays. This is because they illustrate the situation where temporal lags arise in certain ranges of values for certain related probability distributions, taking into account the variables’ entire history of behavior. Another mathematical tool that allows for the memory and inherited properties of systems to be encompassed in a model is the replacement of integer-order derivatives with fractional derivatives. To address realistic conditions, stochastic perturbation framed by a stochastic differential delay system can be used to explain the ambiguity about the context in which the system operates. The present book comprises all the 16 articles accepted and published in the Special Issue “Advances in Differential Dynamical Systems with Applications to Economics and Biology” of the MDPI journal Mathematics, with focuses on the dynamical analysis of mathematical models, arising from economy and biology, and innovative developments in mathematical techniques for their applications. We expect that the international scientific community will find this collection of research papers influential and that they will spur additional investigations on diverse applications with respect to dynamical systems in all scientific areas.

Research & information: general --- Mathematics & science --- dynamical systems --- time delay --- stability --- bifurcation analysis in economic and biological systems --- chaotic behaviors --- fractional order systems --- numerical methods --- economic modeling --- infection modeling --- epidemic spreading --- blood flow models --- optimal control --- reaction&ndash --- diffusion systems

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A wide range of ongoing research in the areas of controller design and information engineering reveals that the pace of technological change in this domain seems to be accelerating. The primary focus is on futuristic segments ranging from robotics, genomics to chemical feedstock and electrical storage. The main objective of this special issue was to provide a forum for researchers and practitioners to exchange their latest theoretical and technological achievements and to identify critical issues and challenges for future investigation on topics regarding advanced automation and control techniques based on information system technologies.

Technology: general issues --- fractional calculus --- fractional-order control --- experimental tuning --- smart beam --- vibration suppression --- industrial internet of things --- prediction strategy --- proactive historian --- water industry --- lithium-ion batteries --- fast charging --- battery ageing --- sample and hold --- hierarchical clustering --- multidimensional scaling --- distances --- moving horizon estimation --- Kalman filter --- parameter estimation --- tumor growth estimation --- vibration --- control --- linear quadratic regulator (LQR) --- algebraic Riccati equation --- iteration --- state observer --- viscoelasticity --- mechatronic system --- 6DOF platform --- co-simulation platform --- robust control --- fractional order PID control --- linear parameter varying system --- fractional calculus --- fractional-order control --- experimental tuning --- smart beam --- vibration suppression --- industrial internet of things --- prediction strategy --- proactive historian --- water industry --- lithium-ion batteries --- fast charging --- battery ageing --- sample and hold --- hierarchical clustering --- multidimensional scaling --- distances --- moving horizon estimation --- Kalman filter --- parameter estimation --- tumor growth estimation --- vibration --- control --- linear quadratic regulator (LQR) --- algebraic Riccati equation --- iteration --- state observer --- viscoelasticity --- mechatronic system --- 6DOF platform --- co-simulation platform --- robust control --- fractional order PID control --- linear parameter varying system

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A wide range of ongoing research in the areas of controller design and information engineering reveals that the pace of technological change in this domain seems to be accelerating. The primary focus is on futuristic segments ranging from robotics, genomics to chemical feedstock and electrical storage. The main objective of this special issue was to provide a forum for researchers and practitioners to exchange their latest theoretical and technological achievements and to identify critical issues and challenges for future investigation on topics regarding advanced automation and control techniques based on information system technologies.

fractional calculus --- fractional-order control --- experimental tuning --- smart beam --- vibration suppression --- industrial internet of things --- prediction strategy --- proactive historian --- water industry --- lithium-ion batteries --- fast charging --- battery ageing --- sample and hold --- hierarchical clustering --- multidimensional scaling --- distances --- moving horizon estimation --- Kalman filter --- parameter estimation --- tumor growth estimation --- vibration --- control --- linear quadratic regulator (LQR) --- algebraic Riccati equation --- iteration --- state observer --- viscoelasticity --- mechatronic system --- 6DOF platform --- co-simulation platform --- robust control --- fractional order PID control --- linear parameter varying system --- n/a

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The study of fractional integrals and fractional derivatives has a long history, and they have many real-world applications because of their properties of interpolation between integer-order operators. This field includes classical fractional operators such as Riemann–Liouville, Weyl, Caputo, and Grunwald–Letnikov; nevertheless, especially in the last two decades, many new operators have also appeared that often define using integrals with special functions in the kernel, such as Atangana–Baleanu, Prabhakar, Marichev–Saigo–Maeda, and the tempered fractional equation, as well as their extended or multivariable forms. These have been intensively studied because they can also be useful in modelling and analysing real-world processes, due to their different properties and behaviours from those of the classical cases.Special functions, such as Mittag–Leffler functions, hypergeometric functions, Fox's H-functions, Wright functions, and Bessel and hyper-Bessel functions, also have important connections with fractional calculus. Some of them, such as the Mittag–Leffler function and its generalisations, appear naturally as solutions of fractional differential equations. Furthermore, many interesting relationships between different special functions are found by using the operators of fractional calculus. Certain special functions have also been applied to analyse the qualitative properties of fractional differential equations, e.g., the concept of Mittag–Leffler stability.The aim of this reprint is to explore and highlight the diverse connections between fractional calculus and special functions, and their associated applications.

Caputo-Hadamard fractional derivative --- coupled system --- Hadamard fractional integral --- boundary conditions --- existence --- fixed point theorem --- fractional Langevin equations --- existence and uniqueness solution --- fractional derivatives and integrals --- stochastic processes --- calculus of variations --- Mittag-Leffler functions --- Prabhakar fractional calculus --- Atangana–Baleanu fractional calculus --- complex integrals --- analytic continuation --- k-gamma function --- k-beta function --- Pochhammer symbol --- hypergeometric function --- Appell functions --- integral representation --- reduction and transformation formula --- fractional derivative --- generating function --- physical problems --- fractional derivatives --- fractional modeling --- real-world problems --- electrical circuits --- fractional differential equations --- fixed point theory --- Atangana–Baleanu derivative --- mobile phone worms --- fractional integrals --- Abel equations --- Laplace transforms --- mixed partial derivatives --- second Chebyshev wavelet --- system of Volterra–Fredholm integro-differential equations --- fractional-order Caputo derivative operator --- fractional-order Riemann–Liouville integral operator --- error bound --- n/a --- Atangana-Baleanu fractional calculus --- Atangana-Baleanu derivative --- system of Volterra-Fredholm integro-differential equations --- fractional-order Riemann-Liouville integral operator