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两端是自由端点的弦的横向振动

Posted by haifeng on 2015-06-04 10:48:54 last update 2015-06-04 15:49:29 | Answers (0)

$\begin{cases} u_{tt}-u_{xx}=0 & \text{in}\ \mathbb{R}\times(0,\ell),\\ u_x(t,0)=u_x(t,\ell)=0 &\text{for}\ t\in\mathbb{R},\\ u(0,x)=u_0(x) &\text{for}\ x\in(0,\ell),\\ u_t(0,x)=u_1(x) &\text{for}\ x\in(0,\ell). \end{cases}\tag{1.1}$

$u(0,x)=u_0(x)$ 是指初始时刻 $t=0$ 时, 弦的方程.

$u_t(0,x)=u_1(x)$ 是弦方程对 $t$ 求偏导在初始时刻的方程, 可以认为是初始速度方程.

$(u_0,u_1)\mapsto u(\cdot,0)|_{(0,T)}\tag{1.2}$

$H:=\Bigl\{v\in L^2(0,\pi)\ :\ \int_0^{\pi}v(x)dx=0\Bigr\},\quad \|v\|_{H}:=\biggl(\int_{0}^{\pi}|v(x)|^2 dx\biggr)^{1/2},$

$V:=\Bigl\{v\in H^{1}(0,\pi)\ :\ \int_{0}^{\pi}v(x)dx=0\Bigr\},\quad \|v\|_{V}:=\biggl(\int_{0}^{\pi}|v'(x)|^2 dx\biggr)^{1/2}.$

$E_0:=\frac{1}{2}(\|u_0\|_V^2+\|u_1\|_H^2),$

$c_1 E_0\leqslant \int_0^T |u_t(t,0)|^2 dt\leqslant c_2 E_0,$

Remark:

Vilmos Komornik, Paola Loreti, Fourier Series in Control Theory. Springer, 2005.