Symmetry and conserved quantities of Hamilton system with comfortable fractional derivatives

2020 ◽  
Author(s):  
Linli Wang
Keyword(s):  
Fractional Derivatives ◽  
Conserved Quantities ◽  
Hamilton System

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 Motion equations and non-Noether symmetries of Lagrangian systems with conformable fractional derivative

2021 ◽  
pp. 35-35
Author(s):  
Jing-Li Fu ◽  
Lijun Zhang ◽  
Chaudry Khalique ◽  
Ma-Li Guo
Keyword(s):  
Fractional Derivative ◽  
Fractional Derivatives ◽  
Lagrangian Systems ◽  
Conserved Quantities ◽  
Conformable Fractional Derivative ◽  
Noether Symmetries ◽  
Hamilton’S Principle ◽  
Hamilton's Principle ◽  
Motion Equations ◽  
Definition Of

In this paper, we present the fractional motion equations and fractional non-Noether symmetries of Lagrangian systems with the conformable fractional derivatives. The exchanging relationship between isochronous variation and fractional derivative, and the fractional Hamilton's principle of the holonomic conservative and non-conservative systems under the conformable fractional derivative are proposed. Then the fractional motion equations of these systems based on the Hamilton's principle are established. The fractional Euler operator, the definition of fractional non-Noether symmetries, non-Noether theorem and Hojman's conserved quantities for the Lagrangian systems are obtained with conformable fractional derivative. An example is given to illustrate the results.

On Fractional Derivatives and Application

2013 ◽  
Vol 1 (2) ◽  
pp. 51-56
Author(s):  
Mark Borres ◽  
◽  
Efren Barabat ◽  
Jocelyn Panduyos ◽  
◽  
...  
Keyword(s):  
Fractional Derivatives

The properties of Bessel-type fractional derivatives and integrals

2016 ◽  
pp. 3973-3982
Author(s):  
V. R. Lakshmi Gorty
Keyword(s):  
Fractional Order ◽  
Real Line ◽  
Computer Algebra System ◽  
Fractional Derivatives ◽  
Graphical Representation ◽  
Fractional Integrals ◽  
The Real ◽  
Fractional Derivatives And Integrals ◽  
Half Axis ◽  
Derivatives Of

The fractional integrals of Bessel-type Fractional Integrals from left-sided and right-sided integrals of fractional order is established on finite and infinite interval of the real-line, half axis and real axis. The Bessel-type fractional derivatives are also established. The properties of Fractional derivatives and integrals are studied. The fractional derivatives of Bessel-type of fractional order on finite of the real-line are studied by graphical representation. Results are direct output of the computer algebra system coded from MATLAB R2011b.

Conserved Quantities for the General Linear Heat Equation

2016 ◽  
pp. 4437-4439
Author(s):  
Adil Jhangeer ◽  
Fahad Al-Mufadi
Keyword(s):  
Evolution Equation ◽  
Heat Equation ◽  
Conservation Laws ◽  
Exact Solutions ◽  
Lagrangian Approach ◽  
General Linear ◽  
Conserved Quantities ◽  
Linear Heat ◽  
The Family ◽  
Partial Lagrangian

In this paper, conserved quantities are computed for a class of evolution equation by using the partial Noether approach [2]. The partial Lagrangian approach is applied to the considered equation, infinite many conservation laws are obtained depending on the coefficients of equation for each n. These results give potential systems for the family of considered equation, which are further helpful to compute the exact solutions.

Solution of inverse coefficient problem for fractional anomalous diffusion equation

2012 ◽  
Vol 9 (2) ◽  
pp. 65-70
Author(s):  
E.V. Karachurina ◽  
S.Yu. Lukashchuk
Keyword(s):  
Anomalous Diffusion ◽  
Fractional Derivatives ◽  
Descent Method ◽  
Extremum Problem ◽  
Steepest Descent Method ◽  
Coefficient Problem ◽  
Caputo Fractional Derivatives ◽  
Inverse Coefficient Problem ◽  
Residual Functional ◽  
Inverse Coefficient

An inverse coefficient problem is considered for time-fractional anomalous diffusion equations with the Riemann-Liouville and Caputo fractional derivatives. A numerical algorithm is proposed for identification of anomalous diffusivity which is considered as a function of concentration. The algorithm is based on transformation of inverse coefficient problem to extremum problem for the residual functional. The steepest descent method is used for numerical solving of this extremum problem. Necessary expressions for calculating gradient of residual functional are presented. The efficiency of the proposed algorithm is illustrated by several test examples.

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