Fractional order calculus: historical apologia, basic concepts and some applications

Abstract

Fractional order calculus (FOC) deals with integrals and derivatives of arbitrary (i.e., non-integer) order, and shares its origins with classical integral and differential calculus. However, until recently, it has been investigated mainly from a mathematical point of view. Advances in the field of fractals have revealed its subtle relationships with fractional calculus. Nonetheless, fractional calculus is generally excluded from standard courses in mathematics, partly because many mathematicians are unfamiliar with its nature and its applications. This area has emerged as a useful tool among researchers. One of the objectives of this paper is to discuss the usefulness of fractional calculus in applied sciences and engineering. In view of the increasing interest in the development of the new paradigm, another objective is to encourage the use of this mathematical idea in various scientific areas by means of a historical apologia for the development of fractional calculus. Keywords: fractional order calculus, non-integer order systems, dynamical systems.O cálculo de ordem fracionária (COF) lida com derivadas e integrais de ordem arbitrária (ou seja, não-inteiros) e tem as mesmas origens do clássico cálculo diferencial e integral. No entanto, até recentemente, tem sido investigado principalmente do ponto de vista matemático. Progressos nasáreas de fractais revelaram suas sutis relações com o cálculo fracionário. No entanto, o cálculo fracionárioé geralmente excluído dos cursos tradicionais de Matemática, em parte porque muitos matemáticos não estão familiarizados com a sua natureza e suas aplicações. O cálculo fracionário tem se mostrado uma ferramentaútil entre os pesquisadores. Um dos objetivos deste trabalhoé discutir a utilidade do cálculo fracionário nas ciências aplicadas e na engenharia. Além disso, outro propósitoé incentivar o uso dessa idéia matemática como ferramenta para pesquisadores das mais diversaś areas por meio de uma apologia histórica envolvendo o desenvolvimento do cálculo fracionário. Palavras-chave: cálculo fracionário, sistemas de ordem não-inteira, sistemas dinâmicos.

On Hadamard-type inequalities for <i>m</i>-convex functions via Riemann-Liouville fractional integrals

“…Many initiate, using their own notation and methodology, definitions that fit the concept of a non-integer order integral or derivative. The most well-known of these definitions that have been popularized in the subject of fractional calculus are the Riemann-Liouville and the Grunwald-Letnikov definition [4,12]. In [18] Riemann-Liouville fractional integrals are defined as follows: For further details one may see [15,16,17,9,8,13,19].…”

The fractional-nonlinear robotic manipulator: Modeling and dynamic simulations

“…Clearly, numerous mathematicians such as Lacroix, Riemann, Liouville, Cauchy, Laurent, Caputo, among others, have contributed to the history of fractional calculus by attempting to solve a fundamental problem to the best of their understanding [2]. Each researcher sought a definition and therefore different approaches, which has led to various definitions of differentiation and antidifferentiation of noninteger orders that are provenly equivalent.…”

Analytic solutions of time fractional diffusion equations by fractional reduced differential transform method (FRDTM)

“…It is also applicable to problems in polymer science, polymer physics, biophysics, rheology, and thermodynamics (Hilfer, 2000). In addition, it is applicable to problems in: electrochemical process (Millar and Ross, 1993;Oldham and Spanier, 1974;Podlubny, 1999), control theory (David et al, 2011;Podlubny, 1999), physics (Sabatier et al, 2007), science and engineering (Kumar and Saxena, 2016), transport in semi-infinite medium (Oldham and Spanier, 1974), signal processing (Sheng et al, 2011), self-similar protein dynamics (Glockle and Nonnenmacher, 1995), food science (Rahimy, 2010), food gums (David and Katayama, 2013), fractional dynamics (Tarsov, 2011;Zaslavsky, 2005), quantum dynamics (Iomin, 2009), modeling cardiac tissue electrode interface (Magin, 2008), food engineering and econophysics (David et al, 2011), Hamiltonian chaotic systems (Hilfer, 2000;Zaslavsky, 2005), complex dynamics in biological tissues (Margin, 2010), viscoelasticity (Dalir and Bashour, 2010;Mainardi, 2010;Podlubny, 1999;Rahimy, 2010;Sabatier et al, 2007), control science (Shanantu Sabatier et al, 2007), quantum mechanics (Herrmann, 2011), modeling Kenea 15 oscillation systems (Gomez-Aguilar et al, 2015). Some of these mentioned applications were tried to be touched as follows.…”