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Fractional Euler-Lagrange equations : numerical solutions and applications of reflection operator

Błaszczyk T., Ciesielski M.

Prace Naukowe Instytutu Matematyki i Informatyki Politechniki Częstochowskiej

2010

Vol. 9, nr 2

17-24

In this work numerical solutions of fractional Euler-Lagrange equations describing free motion are considered. This type of equations contains a composition of left and right fractional derivatives. A reflection operator is applied to obtain relations between the Euler-Lagrange equations. In addition we verify the dependence between the respective numerical schemes using the same operator. In the final part of paper the examples of the numerical solutions are shown.

The delta-nabla calculus of variations

Malinowska A. B., Torres D. F. M.

Fasciculi Mathematici

2010

Nr 44

75-83

The discrete-time, the quantum, and the continuous calculus of variations have been recently unified and extended. Two approaches are followed in the literature: one dealing with minimization of delta integrals; the other dealing with minimization of nabla integrals. Here we propose a more general approach to the calculus of variations on time scales that allows to obtain both delta and nabla results as particular cases.

Theoretical foundations of a Dynamic Inertial Bumper (DIB) design

Dobry M. W.

Vibrations in Physical Systems

2006

Vol. 22

107--112

Stability of the Euler-Lagrange-Rassias functional equation

Pietrzyk A.

Demonstratio Mathematica

2006

Vol. 39, nr 3

523-530

Let F be a field, a1, a2 is an element of F, K is an element of {R, C}, s an element of K\{0,1}, X be a linear space over F, S C is contained in X be nonempty, and Y be a Banach space over K. Under some additional assumptions on S we show some stability results for the functional equation Q (a1x + a2y) + Q (a2X - a1y) = s[Q{x) + Q{y)} in the class of function Q : S -> Y.

Field of complex linear frames on real space-time manifold as dynamical variable for generally-covariant models

Godlewski P.

Demonstratio Mathematica

2005

Vol. 38, nr 4

819--835

A model of the self-interacting field of complex linear frames E on an n-dimensional real manifold is investigated. The model is generally-covariant and GL(n, C)- invariant. If n = 4, the components of E can be interpreted as dynamical variables for the gravitational field. A Lagrangian of E is constructed, the Euler-Lagrange equations are derived and a wide class of their solutions is found. The solutions are built of left invariant vector fields on a real semisimple Lie group "deformed" by a complex factor of a natural exponential structure.