Special issue on advances in fractional dynamics in mechanical engineering (original) (raw)
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The Craft of Fractional Modelling in Science and Engineering
MDPI eBooks, 2018
He graduated from the Technical University, Sofia, as an electrical engineer and was awarded his Ph.D. degree in chemical engineering in 1994 with a thesis in the field of magnetic field assisted fluidization. His Doctor of Sciences thesis on nonlinear and anomalous diffusion models was successfully completed in 2018. Prof. Hristov has more than 39 years' experience in the field of chemical engineering with principle research interests in mechanics of particulate materials, fluidization, magnetic field effects of process intensification, mathematical modelling in complex systems, non-linear diffusion and fractional calculus application in modelling with more than 170 articles published in international journals. He is an editorial board member of Thermal Science, Particuology, Progress in Fractional Differentiation and Applications and Fractional and Fractal. v Preface to "The Craft of Fractional Modelling in Science and Engineering" vii Fractional calculus has performed an important role in the fields of mathematics, physics, electronics, mechanics, and engineering in recent years. The modeling methods involving fractional operators have been continuously generalized and enhanced, especially during the last few decades. Many operations in physics and engineering can be defined accurately by using systems of differential equations containing different types of fractional derivatives. This book is a result of the contributions of scientists involved in the special collection of articles organized by the journal Fractal and Fractional (MDPI), most of which have been published at the end of 2017 and the beginning 2018. In accordance with the initial idea of a 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 non-integer differentiation and integration approach in modeling directly related to pertinent problems in science and engineering. It is worth mentioning some principle results from the collected articles, now presented as book chapters, which make this book a contemporary and interesting read for a wide audience: The fractional velocity concept developed by Prodanov [1] is demonstrated as tool to characterize Hölder and in particular, singular functions. Fractional velocities are defined as limits of the difference quotients of a fractional power and they generalize the local notion of a derivative. On the other hand, their properties contrast some of the usual properties of derivatives. One of the most peculiar properties of these operators is that the set of their nontrivial values is disconnected. This can be used, for example, to model instantaneous interactions, such as Langevin dynamics. In this context, the local fractional derivatives and the equivalent fractional velocities have several distinct properties compared to integer-order derivatives. The classical pantograph equation and its generalizations, including fractional order and higher order cases, is developed by Bhalekar and Patade [2]. The special functions are obtained from the series solution of these equations. Different properties of these special functions are established, andtheir relations with other functions are developed. The new direction in fractional calculus involving nonsingular memory kernels, developed in the last three years following the seminar articles of Caputo and Fabrizio in 2015 [3], is hot research topic. Two studies in the collection clearly demonstrate two principle directions: operators with nonsingular exponential kernels, i.e., the so-called Caputo-Fabrizio derivatives [4,5] (Hristov, 2016, Chapter 10), and operators with nonsingular memory kernels based on the Mittag-Leffler function [6,7] (Atangana, Baleanu, 2016; Baleanu, Fennandez, 2018). Yavuz and Ozdemir [8] demonstrate a novel approximate-analytical solution method, called the Laplace homotopy analysis method (LHAM), using the Caputo-Fabrizio (CF) fractional derivative operator based on the exponential kernel. The recommended method is obtained by combining Laplace transform (LT) and the homotopy analysis method (HAM). This study considers the application of LHAM to obtain solutions of the fractional Black-Scholes equations (FBSEs) with the Caputo-Fabrizio (CF) fractional derivative and appropriate initial conditions. The authors demonstrate the efficiencies and accuracies of the suggested method by applying it to the FBS option pricing models with their initial conditions satisfied by the classical European vanilla option. Using real market values from finance literature, it is demonstrated how the option is priced for fractional cases of European call option pricing models. Moreover, the proposed fractional model allows modeling of the price of different financial derivatives such as swaps, warrant, etc., in complete agreement with the corresponding exact solutions.
Fractal and Fractional
Modelling, simulation, and applications of Fractional Calculus have recently become increasingly popular subjects, with impressive growth concerning applications. The founding and limited ideas on fractional derivatives have achieved an incredibly valuable status. The variety of applications in mathematics, physics, engineering, economics, biology, and medicine, have opened new, challenging fields of research. For instance, in soil mechanics, a suitable definition of the fractional operator has shed some light on viscoelasticity, explaining memory effects on materials. Needless to say, these applications require the development of practical mathematical tools in order to extract quantitative information from models, newly reformulated in terms of fractional differential equations. Even confining ourselves to the field of ordinary differential equations, the well-known Bagley-Torvik model showed that fractional derivatives may actually arise naturally within certain physical models, and are not merely fanciful mathematical generalizations. This Special Issue focuses on the most recent advances in fractional calculus, applied to dynamic problems, linear and nonlinear fractional ordinary and partial differential equations, integral fractional differential equations, and stochastic integral problems arising in all fields of science, engineering, and other applied fields. In this issue, we have collected several significant papers devoted to applications of fractional methods with a focus on dynamical aspects. The applications range from theoretical mathematical-numerical aspects [1,2] to bio-medical subjects [3-7]. Applications to complex materials are investigated in [8], aiming at proposing a generalized definition of fractional operators. Special diffusion models are studied in [9-11].
The Craft of Fractional Modeling in Science and Engineering 2017
Fractal and Fractional
Fractional calculus has performed an important role in the fields of mathematics, physics, electronics, mechanics, and engineering in recent years. The modeling methods involving fractional operators have been continuously generalized and enhanced, especially during the last few decades. Many operations in physics and engineering can be defined accurately by using systems of differential equations containing different types of fractional derivatives.
Journal of Advanced Research, 2021
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Fractional Dynamics and Control
2012
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Fractional dynamical systems and applications in mechanics and economics
2007
Using the fractional integration and differentiation on R we build the fractional jet fibre bundle on a differentiable manifold and we emphasize some important geometrical objects. Euler-Lagrange fractional equations are described. Some significant examples from mechanics and economics are presented.
On deterministic fractional models
2010
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Towards the application of Fractional Calculus in Engineering Sciences
This article illustrates several applications of Fractional Calculus (FC) in engineering sciences. This mathematical tool will be used here to model, analyze and control various dynamical systems, such as the Van der Pol oscillator, legged robots, electrical systems, digital circuit synthesis, evolutionary computing, redundant robots and financial modelling. Their effectiveness and utility will be demonstrated.