Carlo Cattani - Fractional Dynamics

Published by De Gruyter Open Ltd, Warsaw/Berlin
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2015 Carlo Cattani, Hari M. Srivastava, Xiao-Jun Yang and chapters contributors, published by de Gruyter Open

ISBN 978-3-11-029316-6
e-ISBN 978-3-11-029319-7

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This volume is devoted to recent developments in the theory of fractional calculus and its applications. Particular attention is paid to the applicability of this currently popular research field in various branches of pure and applied mathematics. In particular, we have focused on the more recent results in mathematical physics, engineering applications, theoretical and applied physics as quantum mechanics, signal analysis, and in those relevant research fields where nonlinear dynamics occurs and several tools of nonlinear analysis are required. Dynamical processes and dynamical systems of fractional order attract researchers from many areas of sciences and technologies, ranging from mathematics and physics to computer science.

Fractional calculus is an intriguing mathematical theory which dates back to the very foundation of differential calculus when Leibniz (1695) raised some concern about derivatives of fractional order. However, it was Liouville (1832), and nearly contemporarily to him, Riemann (1847) and Fourier (1822), who collected into an independent theory all the basic knowledge about fractional order derivatives and fractional order integrals. Since then and especially in the last century, there has been an increasing interest on fractional calculus and its application in many different fields in Mathematics, Physics, Engineering, Bio-Science, Computer Science, Economics and so on.

In order to explain why fractional calculus is attracting more and more scholars with different scientific and cultural background is due to the following reasons:

1.Although it is almost completely clear what does it mean by fractional derivative and fractional integral and which basic axioms should be taken to define these operators, fractional derivative as operator doesnt fulfill all properties of the ordinary derivative. For this reason, in order to circumvent this problem many additional axioms were assumed, so that, as a consequence, there are different definitions of the same operator, which give rise to different, sometimes competing, fractional derivatives. All definitions are mathematically respectful, but the main problem is that up to now there is not a unique definition of fractional derivative.

2.Since their origins, fractional derivatives were linked to the fractional calculus. There is always a debate on local (for the derivative) and non-local (for derivative) approach to fractional-order operator calculus. For the nonlocal fractional calculus, some boundary conditions are needed and this implies global approach to such problems.

3.The fundamental operators of fractional calculus are based on special functions which, in turn, are strongly depending on efficient computational methods and numerical algorithms. Therefore, the modern advancements on computational and numerical methods have pushed the recent development of fractional calculus and the massive interests of the scientific community.

However, as a consequence of these main concerns, we have reached a situation where there are still some unsolved theoretical problems in fractional calculus whereas there are many different fractional models in applications. In fact, any physical, engineering, and mathematical problems that are being investigated by a fractional calculus might have a variety of fractional models, depending on the local-nonlocal approach and/or on the chosen fractional derivative among the many alternative definitions. Moreover, the many recently developed numerical methods add confusion to the multi-facet fractional models. In fact, different choices of orthogonal functions and bases would lead to different fractional models.

Very recently there have also been several successful attempts to extend local fractional calculus to non differentiable, irregular sets like fractals. The possibility to extend the fractional calculus to self-similar unsmooth objects is opening new frontiers in science, such as, for example, in signal analysis, data sets from complex phenomena, image analysis, nuclear medicine, and so on. Nonlinear analysis of data, collected by modern devices, offer still unsolved analytical problems related not only to complex physics and abstract mathematical theories, but also to nonlinear science. From analytical point of view, these kinds of problems are often leading us to deal with the concepts of scales, fractals, fractional operators, and so on.

In this volume, we have collected some of the main contributions to the fields of local and non-local fractional calculus, aiming to explore the most recent advancement in several branches of science and technology.

This volume consists of a total of 22 chapters which are outlined below.

(by C. Cattani) deals with the fractional calculus applied to Shannon wavelet theory. This chapter introduces a novel definition of local fractional derivative which makes it possible to compute the derivative of any finite energy function, that is, of a localized function which can be represented by wavelet series.

In , V. E. Tarasov studies three-dimensional discrete dynamical systems with long-range properties and gives some applications for different types of dynamical systems.

In , Antnio M. Lopes and J.A. Tenreiro Machado investigate the statistical distributions of earthquakes in Southern California over the time period from the year 1934 up to the year 2013. The reported results reveal relationships and temporal patterns hidden in the data. Also, the investigated methodology and findings can contribute to a comprehensive explanation of these phenomena and to recognize precursory events for earthquake prediction.

(by Akira Asada) deals with an integral transform arising from fractional calculus, which is used for investigating the solution of some differential equations.

In , Jordan Hristov fully shows how to obtain approximate solutions to time-fractional models by integral balance approach. This long chapter completely describes the theory of integral balance approach for finding approximate solution of fractional differential equations.

(by Bashir Ahmad, Ahmed Alsaedi and Hana Al-Hutami) presents a study of sequential fractional q- integro-difference equations with perturbed anti-periodic boundary conditions.