The Role of Salvatore Pincherle in the Development of Fractional Calculus

The present state-of-the-art article is devoted to the analysis of new trends and recent results carried out during the last 10years in the field of fractional calculus application to dynamic problems of solid mechanics. This review involves the papers dealing with study of dynamic behavior of linear and nonlinear 1DOF systems, systems with two and more DOFs, as well as linear and nonlinear systems with an infinite number of degrees of freedom: vibrations of rods, beams, plates, shells, suspension combined systems, and multilayered systems. Impact response of viscoelastic rods and plates is considered as well. The results obtained in the field are critically estimated in the light of the present view of the place and role of the fractional calculus in engineering problems and practice. This articles reviews 337 papers and involves 27 figures.

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The role of fractional calculus in modeling biological phenomena: A review

Communications in Nonlinear Science and Numerical Simulation ◽

10.1016/j.cnsns.2017.04.001 ◽

2017 ◽

Vol 51 ◽

pp. 141-159 ◽

Cited By ~ 169

Author(s):

C. Ionescu ◽

A. Lopes ◽

D. Copot ◽

J.A.T. Machado ◽

J.H.T. Bates

Keyword(s):

Fractional Calculus ◽

Biological Phenomena

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Continuous and discrete Noether's fractional conserved quantities for restricted calculus of variations

Journal of Geometric Mechanics ◽

10.3934/jgm.2021012 ◽

2021 ◽

Vol 0 (0) ◽

pp. 0

Author(s):

Jacky Cresson ◽

Fernando Jiménez ◽

Sina Ober-Blöbaum

Keyword(s):

Fractional Calculus ◽

Calculus Of Variations ◽

Fractional Derivatives ◽

Dissipative Systems ◽

Alternative Formulation ◽

Conserved Quantities ◽

Lagrange Equations ◽

Discrete Counterpart ◽

Fractional Calculus Of Variations

We prove a Noether's theorem of the first kind for the so-called restricted fractional Euler-Lagrange equations and their discrete counterpart, introduced in [<xref ref-type="bibr" rid="b26">26</xref>,<xref ref-type="bibr" rid="b27">27</xref>], based in previous results [<xref ref-type="bibr" rid="b11">11</xref>,<xref ref-type="bibr" rid="b35">35</xref>]. Prior, we compare the restricted fractional calculus of variations to the asymmetric fractional calculus of variations, introduced in [<xref ref-type="bibr" rid="b14">14</xref>], and formulate the restricted calculus of variations using the discrete embedding approach [<xref ref-type="bibr" rid="b12">12</xref>,<xref ref-type="bibr" rid="b18">18</xref>]. The two theories are designed to provide a variational formulation of dissipative systems, and are based on modeling irreversbility by means of fractional derivatives. We explicit the role of time-reversed solutions and causality in the restricted fractional calculus of variations and we propose an alternative formulation. Finally, we implement our results for a particular example and provide simulations, actually showing the constant behaviour in time of the discrete conserved quantities outcoming the Noether's theorems.

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In memory of the honorary founding editors behind the FCAA success story

Fractional Calculus and Applied Analysis ◽

10.1515/fca-2021-0028 ◽

2021 ◽

Vol 24 (3) ◽

pp. 641-666

Author(s):

Virginia Kiryakova ◽

J.A. Tenreiro Machado ◽

Yuri Luchko

Keyword(s):

Fractional Calculus ◽

Main Part ◽

Special Role ◽

Success Story ◽

Biographical Data ◽

Scientific Advance

Abstract In this editorial paper, we start by surveying of the main milestones in the organization, foundation, and development of the journal Fractional Calculus and Applied Analysis (FCAA). The main potential of FCAA is in its readers, authors, and editors who contribute to the scientific advance and promote the progress of the journal. Among the editors, a special role of the honorary editors who contributed significantly to the foundation of the journal should be highlighted. These are prominent scientists, lecturers, and disseminators of FC and related topics. Unfortunately, some of them have already passed away, but they remain our living and lasting memory. In the main part of this survey, we remind the readers some biographical data and achievements related to FCAA topics of these Honorary Founding Editors: Professors Eric Love, Ian Sneddon, Bogoljub Stanković, Rudolf Gorenflo, Danuta Przeworska-Rolewicz, Gary Roach, Anatoly Kilbas, and Wen Chen. In eight sections, each devoted to one of the honorary editors, their ways in science and in FC, as well as their main results and publications are shortly described. Special attention is given to their role in foundation of FCAA and to their contributions to its success story.

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On the role of fractional calculus in electromagnetic theory

IEEE Antennas and Propagation Magazine ◽

10.1109/74.632994 ◽

1997 ◽

Vol 39 (4) ◽

pp. 35-46 ◽

Cited By ~ 143

Author(s):

N. Engheia

Keyword(s):

Fractional Calculus ◽

Electromagnetic Theory

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A Guide to Special Functions in Fractional Calculus

Mathematics ◽

10.3390/math9010106 ◽

2021 ◽

Vol 9 (1) ◽

pp. 106

Author(s):

Virginia Kiryakova

Keyword(s):

Applied Sciences ◽

Fractional Calculus ◽

Hypergeometric Functions ◽

Special Functions ◽

Generalized Hypergeometric Functions ◽

Social Events ◽

Transcendental Functions ◽

Meijer G Function ◽

Wright Generalized Hypergeometric Functions

Dedicated to the memory of Professor Richard Askey (1933–2019) and to pay tribute to the Bateman Project. Harry Bateman planned his “shoe-boxes” project (accomplished after his death as Higher Transcendental Functions, Vols. 1–3, 1953–1955, under the editorship by A. Erdélyi) as a “Guide to the Functions”. This inspired the author to use the modified title of the present survey. Most of the standard (classical) Special Functions are representable in terms of the Meijer G-function and, specially, of the generalized hypergeometric functions pFq. These appeared as solutions of differential equations in mathematical physics and other applied sciences that are of integer order, usually of second order. However, recently, mathematical models of fractional order are preferred because they reflect more adequately the nature and various social events, and these needs attracted attention to “new” classes of special functions as their solutions, the so-called Special Functions of Fractional Calculus (SF of FC). Generally, under this notion, we have in mind the Fox H-functions, their most widely used cases of the Wright generalized hypergeometric functions pΨq and, in particular, the Mittag–Leffler type functions, among them the “Queen function of fractional calculus”, the Mittag–Leffler function. These fractional indices/parameters extensions of the classical special functions became an unavoidable tool when fractalized models of phenomena and events are treated. Here, we try to review some of the basic results on the theory of the SF of FC, obtained in the author’s works for more than 30 years, and support the wide spreading and important role of these functions by several examples.

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Study of Fractional Analytic Functions and Local Fractional Calculus

International Journal of Scientific Research in Science Engineering and Technology ◽

10.32628/ijsrset218482 ◽

2021 ◽

pp. 39-46

Author(s):

Chii-Huei Yu

Keyword(s):

Analytic Function ◽

Fractional Calculus ◽

Analytic Functions ◽

Fundamental Theorem ◽

Product Rule ◽

Integration By Parts ◽

Local Fractional Calculus ◽

Change Of Variable ◽

Variable Integration

In this present paper, the role of fractional analytic function in local fractional calculus is studied. Some important properties and theorems in local fractional calculus are discussed, such as product rule, quotient rule, chain rule, fundamental theorem of local fractional calculus, change of variable, integration by parts and so on. In addition, we propose several examples and formulas of local fractional calculus.

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