On Reflection Symmetry and Its Application to the Euler-Lagrange Equations in Fractional Mechanics

2011 ◽  
Author(s):  
Malgorzata Klimek
Keyword(s):  
Fractional Derivatives ◽  
Finite Interval ◽  
Reflection Symmetry ◽  
Symmetric Part ◽  
Fractional Differentiation ◽  
Lagrange Equations ◽  
Integration Formulas ◽  
Lagrange System ◽  
Fractional Mechanics ◽  
Derivatives Of

We study the properties of fractional differentiation with respect to the reflection symmetry in a finite interval. The representation and integration formulas are derived for the symmetric and anti-symmetric fractional derivatives, both of the Riemann -Liouville and Caputo type. The action dependent on the left -sided Caputo derivatives of orders in range (1.2) is considered and we derive the Euler-Lagrange equations for the symmetric and anti-symmetric part of the trajectory. The procedure is illustrated with an example of the action dependent linearly on fractional velocities. For the obtained Euler-Lagrange system we discuss its localization resulting from the subsequent sym-metrization of the action.

scholarly journals On reflection symmetry and its application to the Euler–Lagrange equations in fractional mechanics

2013 ◽  
Vol 371 (1990) ◽  
pp. 20120145 ◽  
Author(s):  
Małgorzata Klimek
Keyword(s):  
Fractional Derivatives ◽  
Finite Interval ◽  
Reflection Symmetry ◽  
Symmetric Part ◽  
Fractional Differentiation ◽  
Lagrange Equations ◽  
Caputo Derivatives ◽  
Lagrange System ◽  
Fractional Mechanics ◽  
Derivatives Of

We study the properties of fractional differentiation with respect to the reflection symmetry in a finite interval. The representation and integration formulae are derived for symmetric and anti-symmetric fractional derivatives, both of the Riemann–Liouville and Caputo type. The action dependent on the left-sided Caputo derivatives of orders in the range (1,2) is considered and we derive the Euler–Lagrange equations for the symmetric and anti-symmetric part of the trajectory. The procedure is illustrated with an example of the action dependent linearly on fractional velocities. For the obtained Euler–Lagrange system, we discuss its localization resulting from the subsequent symmetrization of the action.

Reflection symmetric formulation of generalized fractional variational calculus

2013 ◽  
Vol 16 (1) ◽  
Author(s):  
Małgorzata Klimek ◽  
Maria Lupa
Keyword(s):  
Fractional Derivatives ◽  
Finite Interval ◽  
Equations Of Motion ◽  
Variational Calculus ◽  
Reflection Symmetry ◽  
Lagrange Equations ◽  
Representation Formulas ◽  
Symmetry Properties ◽  
Symmetric Formulation ◽  
Generalized Fractional Derivatives

AbstractWe define generalized fractional derivatives (GFDs) symmetric and anti-symmetric w.r.t. the reflection symmetry in a finite interval. Arbitrary functions are split into parts with well defined reflection symmetry properties in a hierarchy of intervals [0, b/2m], m ∈ ℕ0. For these parts — [J]-projections of function, we derive the representation formulas for generalized fractional operators (GFOs) and examine integration properties. It appears that GFOs can be reduced to operators determined in subintervals [0, b/2m]. The results are applied in the derivation of Euler-Lagrange equations for action dependent on Riemann-Liouville type GFDs. We show that for Lagrangian being a sum (finite or not) of monomials, the obtained equations of motion can be localized in arbitrary short subinterval [0, b/2m].

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.

scholarly journals Time-Domain Analysis of Fractional Electrical Circuit Containing Two Ladder Elements

Electronics ◽  
2021 ◽  
Vol 10 (4) ◽  
pp. 475
Author(s):  
Ewa Piotrowska ◽  
Krzysztof Rogowski
Keyword(s):  
Fractional Derivative ◽  
Fractional Derivatives ◽  
Electric Circuit ◽  
Theoretical Models ◽  
Time Domain Analysis ◽  
Electrical Circuit ◽  
State Variable ◽  
State Space System ◽  
Derivatives Of ◽  
Different Order

The paper is devoted to the theoretical and experimental analysis of an electric circuit consisting of two elements that are described by fractional derivatives of different orders. These elements are designed and performed as RC ladders with properly selected values of resistances and capacitances. Different orders of differentiation lead to the state-space system model, in which each state variable has a different order of fractional derivative. Solutions for such models are presented for three cases of derivative operators: Classical (first-order differentiation), Caputo definition, and Conformable Fractional Derivative (CFD). Using theoretical models, the step responses of the fractional electrical circuit were computed and compared with the measurements of a real electrical system.

scholarly journals Fractional Diffusion Equations for Lattice and Continuum: Grünwald-Letnikov Differences and Derivatives Approach

2014 ◽  
Vol 2014 ◽  
pp. 1-7 ◽  
Author(s):  
Vasily E. Tarasov
Keyword(s):  
Fractional Derivatives ◽  
Three Dimensional ◽  
Lattice Models ◽  
Fractional Diffusion ◽  
Lattice Diffusion ◽  
Diffusion Equations ◽  
Difference Operators ◽  
Fractional Diffusion Equations ◽  
Nonlocal Media ◽  
Derivatives Of

Fractional diffusion equations for three-dimensional lattice models based on fractional-order differences of the Grünwald-Letnikov type are suggested. These lattice fractional diffusion equations contain difference operators that describe long-range jumps from one lattice site to another. In continuum limit, the suggested lattice diffusion equations with noninteger order differences give the diffusion equations with the Grünwald-Letnikov fractional derivatives for continuum. We propose a consistent derivation of the fractional diffusion equation with the fractional derivatives of Grünwald-Letnikov type. The suggested lattice diffusion equations can be considered as a new microstructural basis of space-fractional diffusion in nonlocal media.

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