1

$ u_{tt}-u_{t}-u_{xx} =0\;,\;t>0\;,\;x\in(0,\pi) \\ u(0,t)=u(\pi,t)=0,\; t>0 \\ E(t):= \frac{1}{2}\int_0^\pi(u_t^2(x,t)+u_x^2(x,t))dx. \\ $

I solved this equation and I end up with something like that $ E'(t)=\int_0^\pi (u_tu_{tt}+u_xu_{xt})\mathrm dx=\int_0^\pi (u_tu_{xx}+u_xu_{xt})\mathrm dx+\int_0^\pi (u_t)^2\mathrm dx$. Since $\int_0^\pi (u_tu_{xx}+u_xu_{xt})\mathrm dx = \int_0^\pi(u_tu_x)_xdx=0$ we ended with $E'(t)=\int_0^\pi (u_t)^2\mathrm dx$. How can I show that it is decreasing? thanks in advance

Evgeny
  • 5,755
Alpys Rauan
  • 117
  • @Winther I did not forget minus, because I used $u_{tt} = u_{xx} {+} u_t$ and multiplied with $u_t$ and got $u_t u_{tt}=u_t(u_{xx}+u_t) $ and substituted it to integral. – Alpys Rauan May 09 '19 at 21:23
  • 4
    It must be a typo in the problem and it's intended that the equation is $u_{tt} \color{red}{+} u_t - u_{xx} = 0$. Your calculation is correct. As the problem is written the energy will in general increase with time. – Winther May 09 '19 at 22:05
  • 1
    The same problem can be found here: https://math.stackexchange.com/questions/513691/proving-energy-conservation-for-wave-equation Notice that the friction term $u_t$ in that case has the opposite sign as you have here. – Winther May 09 '19 at 22:08

0 Answers0