Question

See next comment: Proof of the uniquness of the wave equations

Answer 1

Prove the uniquness (of solution) of the wave equation.

\[u_t=u_{xx}\]\[u_x(0,t)=A(t)\]\[u(\pi,t)=B(t)\]\[u(x,0)=f(x)\]\[u_t(x,0)=g(x)\]

Let\[v_t=v_{xx}\]With the same conditions as u_t=u_xx

Let\[w(x,t)=u(x,t)-v(x,t)\]Satisfy w_t=w_xx So that the conditions of w(x,t) all equal zero.

Let\[E(t)=\frac{1}{2} \int\limits w_x^2(x,t)+w_t^2(x,t)dx,\ge 0\]
\[E(0)=\frac{1}{2} \int\limits w_x^2(x,0)+w_t^2(x,t)dx=0\]because both terms in integrad are conditions of w that = 0
\[E'(t)=\int\limits w_xw_{xt}+w_tw_{\tt}dx\]
\[E'(t)=\int\limits_{0}^{\pi} (w_xw_t)_xdx\]
\[E'(t)=(w_xw_t)|_0^\pi=(w_x(\pi,t)w_t(\pi,t)-w_x(0,t)w_t(0,t))=0\]Because those are conditions that = 0

I can't remember the rest of proof?

Answer 2

in the first equation should be u_tt=u_xx
and what is the v(x,t) ?
I know you suppose that w(0,t)=w(pi,t)=0 means zero boundary condition.
also E(t) is an Energy and zero at t=0.
Could you explain all above,then I can give you a solution.

Answer 3

I'm not really sure how to explain it. It's supposed to be a proof showing that every wave equation has one and only one unique solution. E(t) is just a name for an "arbitrary" equation used in the proof. I didn't copy down the last bit that my prof wrote down >.>

Answer 4

ok follow me at least I write them in a right way so you can compare which one has mistake.