对称量子演算
Symmetric Quantum Calculus
摘要：
通过研究高阶导数变分问题，推广了Hahn变分理论。研究了对称量子演算，即$α,β$ -对称、$q$ -对称和Hahn对称量子演算.引入了对称量子变分法，证明了$q$ -对称和Hahn对称量子变分法的Euler-Lagrange型方程.定义了时间尺度上的对称导数，并导出了它的一些性质.最后，我们介绍并研究了钻石积分，它是钻石- $α$积分在时间尺度上的一种改进形式。
英文摘要：
We generalize the Hahn variational calculus by studying problems of the calculus of variations with higher-order derivatives. The symmetric quantum calculus is studied, namely the $α,β$-symmetric, the $q$-symmetric, and the Hahn symmetric quantum calculus. We introduce the symmetric quantum variational calculus and an Euler-Lagrange type equation for the $q$-symmetric and Hahn's symmetric quantum calculus is proved. We define a symmetric derivative on time scales and derive some of its properties. Finally, we introduce and study the diamond integral, which is a refined version of the diamond-$α$ integral on time scales.
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