The Craft of Fractional Modeling in Science and Engineering 2017

Abstract

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) fracti...

Quenched and annealed disorder mechanisms in comb models with fractional operators

“…where R(α i ) is a normalization constant. This choice corresponds to the Caputo-Fabrizio [59][60][61]. A remarkable feature of this exponential kernel is its connection with resetting processes [61].…”

Solving PDEs of fractional order using the unified transform method

“…In this paper we have considered only fractional differential equations of Riemann-Liouville type. But many real-world processes can be better modelled using other definitions of fractional calculus: for example, the Caputo definition ( 7) is better suited to many initial value problems, and newer definitions such as Caputo-Fabrizio and Atangana-Baleanu have been used to model various types of nonlocal dynamics [6,7,18]. Solving fractional PDEs in these alternative fractional models could be an important result, and it may be possible to do so using the unified transform method.…”

Free convection flow of second grade dusty fluid between two parallel plates using Fick’s and Fourier’s laws: a fractional model

“…In the last few decades, continuous generalization and enhancement of fractional operators have been noticed due to their hereditary properties and material memory effects. Recently, it has been demonstrated that the fractional calculus 33 has an involvement in the modeling of differential equations (DE’s) of non-integer order. Studies reveal that these fractional differential equations can describe more accurately the dynamics of many systems.…”