粘弹性材料下纳米弦的振动

Vibration of Nano Strings in Viscoelastic Materials

本文给出了一个具有Caputo时间分数阶导数的纳米弦振动的波动方程在不同边界条件下的解析解。根据Mittag-Leffler函数和Sturm-Liouville问题的一组完整的本征函数,采用分离变量法、拉普拉斯变换法,得到在初始条件、边界条件以及外力条件特殊情况下的一些结果,表明整数阶方程的相应解是时间分数阶方程的特殊情况,且该方程可用于描述复杂介质或粘弹性介质中的记忆特性。

In this paper, an analytical solution of the wave equation of nano string vibration with Caputo time-fractional derivative under different boundary conditions is given. According to the Mittag-Leffler function and a complete set of eigenfunctions of Sturm-Liouville problem, some results under the special conditions of initial conditions, boundary conditions and external force conditions are obtained by using the separated variable method and Laplace transform method. It shows that the corresponding solution of the integer-order equation is a special case of the time-fractional order equation, and the equation can be used to describe the memory characteristics in complex media or viscoelastic media.