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Topics - aryakim

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Test-2 / Test-2 problem-1 confusion regarding boundary conditions
« on: March 06, 2021, 10:07:36 AM »
I had a question about problem $1$ of my Test$2$. (My version was called alternative-F, night section).There was something confusing about this problem that I realized during the test.
Here is the problem:

\left\{ \begin{aligned}
& u_{tt}-u_{xx}=0, &&0< t < \pi, 0 < x < \pi, &(1.1) \\\
&u|_{t=0}= 2\cos (x),   && 0< x < \pi, &(1.2)\\
&u_t|_{t=0}= 0,  && 0< x < \pi,  &(1.3)\\
&u|_{x=0}= u|_{x= \pi}=0, && 0< t < \pi.  &(1.4)

So, on the $t-x$ diagram, the lines $t = 0$ and $x = \pi$ intersect at ($t=0,x=\pi$), which will be a boundary point of the region where $0<t<x<\pi$.
If this point (i.e. ($t=0,x=\pi$)) on the diagram is approached by the line $t = 0$, equation $(1.2)$ is used to conclude that the value of $u(x,t)$ approaches $-2$.
On the other hand, if the point is approached from the line $x = \pi$,  $u(x,t)$ should become zero, which is in contradiction with the other boundary condition. It seems this would make it impossible to incorporate conditions $(1.2)$ and $(1.4$) at the same time. I was hoping someone could clear my confusion regarding this problem.

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