Institute of Mathematics, Czestochowa University of Technology, Dabrowskiego 73, 42-200 Czestochowa, Poland.
Philos Trans A Math Phys Eng Sci. 2013 Apr 1;371(1990):20120145. doi: 10.1098/rsta.2012.0145. Print 2013 May 13.
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.
我们研究了有限区间内关于反射对称的分数阶微分的性质。推导出了对称和反对称分数导数的表示和积分公式，包括 Riemann-Liouville 和 Caputo 型。考虑了依赖于左阶 Caputo 导数的作用，并推导出了轨迹对称和反对称部分的欧拉-拉格朗日方程。该过程通过依赖于分数速度的线性作用的实例进行说明。对于所得到的欧拉-拉格朗日系统，我们讨论了由于随后的作用对称化而导致的其定域化。
