scholarly journals Quasiconvexity in the fractional calculus of variations: Characterization of lower semicontinuity and relaxation

2022 ◽  
Vol 215 ◽  
pp. 112625
Author(s):  
Carolin Kreisbeck ◽  
Hidde Schönberger
Keyword(s):  
Fractional Calculus ◽  
Calculus Of Variations ◽  
Lower Semicontinuity ◽  
Fractional Calculus Of Variations

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

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

<p style='text-indent:20px;'>We prove a Noether's theorem of the first kind for the so-called <i>restricted fractional Euler-Lagrange equations</i> 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 <i>asymmetric fractional calculus of variations</i>, introduced in [<xref ref-type="bibr" rid="b14">14</xref>], and formulate the restricted calculus of variations using the <i>discrete embedding</i> 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.</p>

scholarly journals Fractional calculus of variations: a novel way to look at it

2019 ◽  
Vol 22 (4) ◽  
pp. 1133-1144 ◽  
Author(s):  
Rui A.C. Ferreira
Keyword(s):  
Fractional Calculus ◽  
Calculus Of Variations ◽  
Classical Problem ◽  
Simple Consequence ◽  
Fractional Calculus Of Variations ◽  
Derivative Order

Abstract In this work we look at the original fractional calculus of variations problem in a somewhat different way. As a simple consequence, we show that a fractional generalization of a classical problem has a solution without any restrictions on the derivative-order α.

THE FRACTIONAL CALCULUS OF VARIATIONS FROM EXTENDED ERDÉLYI-KOBER OPERATOR

2009 ◽  
Vol 23 (16) ◽  
pp. 3349-3361 ◽  
Author(s):  
AHMAD RAMI EL-NABULSI
Keyword(s):  
Fractional Calculus ◽  
Harmonic Oscillator ◽  
Calculus Of Variations ◽  
Fractional Derivatives ◽  
Time Dependent ◽  
Dissipative Systems ◽  
Hamilton Equations ◽  
Different Types ◽  
Fractional Calculus Of Variations ◽  
Dependent Mass

Fractional calculus of variations (FCV) has recently attracted considerable attention as it is deeply related to the fractional quantization procedure. In this work, the FCV from extended Erdélyi-Kober fractional integral is constructed. Our main goal is to exhibit a general treatment for dissipative systems, in particular the harmonic oscillator (HO) that has time-dependent mass and time-dependent frequency. The general linear equation of damped Erdélyi-Kober harmonic oscillator is constructed from which a time-dependent mass generalized law was derived exhibiting different types of behavior. This relatively new time-dependent mass law permits us to point out several possible cases simultaneously in contrast to many models discussed in the literature and without making use of any types of fractional derivatives. Some results on Hamiltonian part, namely Hamilton equations for the damped HO were obtained and discussed in detail.

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