Fractional Noether’s theorem in the Riesz–Caputo sense

Frederico, Gastão S. F.; Torres, Delfim F. M.

doi:10.1016/j.amc.2010.01.100

Cited by 117 publications

(105 citation statements)

References 42 publications

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“…They all share one common hey are all defined on an interval contrary to integer order differential operators defined following part of this paper the Caputo's type derivative over the interval (a, b) is conall such operator Riesz-Caputo (RC) derivative cf. [58] and its definition includes linear f left and right Caputo's derivatives, namely…”

Section: X1mentioning

confidence: 99%

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Fractional calculus for continuum mechanics – anisotropic non-locality

Sumelka¹

2016

Bulletin of the Polish Academy of Sciences Technical Sciences

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Abstract. In this paper, a generalisation of previous author's formulation of fractional continuum mechanics for the case of anisotropic non-locality is presented. The discussion includes a review of competitive formulations available in literature. The overall concept is based on the fractional deformation gradient which is non-local due to fractional derivative definition. The main advantage of the proposed formulation is its structure, analogous to the general framework of classical continuum mechanics. In this sense, it allows to define similar physical and geometrical meaning of introduced objects. The theoretical discussion is illustrated by numerical examples assuming anisotropy limited to single direction. NotationSection 2and finallywhere] denotes the interval of non-local interaction, and Γ is the Euler gamma functionNext, based on non-standard definition of motion by Eq. (4) the authors define the deformation gradient F α aswhere e a and E A are base vectors in coordinate systems {x a } and {X A }, respectively.

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Section: X1mentioning

confidence: 99%

“…We call such operator Riesz-Caputo (RC) derivative cf. [58] and its definition includes linear combination of left and right Caputo's derivatives, namely…”

Section: Riesz-caputo Fractional Derivativementioning

confidence: 99%

Fractional calculus for continuum mechanics – anisotropic non-locality

Sumelka¹

2016

Bulletin of the Polish Academy of Sciences Technical Sciences

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“…For a continuous function g(t) on the [a, b] interval, left Caputo derivative of order α > 0 is defined as follows [31,32]:…”

Section: Sirv Model With Historical Effectsmentioning

confidence: 99%

Numerical solution and stability analysis of a nonlinear vaccination model with historical ejects

Rostamy¹

2017

HJMS

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In this paper, we extend the classical vaccination epidemic model from a deterministic framework to a model with historical effects by formulating it as a system of fractional-order differential equations (FDEs). The basic reproduction number R0 of the resulting fractional model is computed and it is shown that if R0 is less than one, the disease-free equilibrium is locally asymptotically stable. Particularly, we analytically calculate a certain threshold-value for R0 and present the existence conditions of endemic equilibrium. By using stability analysis, we prove stability and α-stability of the endemic equilibrium points. The proposed model is applied on Pertussis disease and the fractional nonlinear system of the model is solved by applying multi-step generalized differential transform method (MSGDTM). Our results show that historical effects play an important role on the disease spreading.

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“…This is possible through a powerful tool known as the Euler-Lagrange equation [33]. Recently the theory of the calculus of variations has been considered in the fractional context [7,8,10,11,12,13,14,23,25,27,32]. The fractional calculus allows to generalize the ordinary differentiation and integration to an arbitrary (non-integer) order, and provides a powerful tool for modeling and solving various problems in science and engineering [28,29,31].…”

Section: Introductionmentioning

confidence: 99%