Toronto Math Forum
APM346-2015S => APM346--Home Assignments => HA1 => Topic started by: Victor Ivrii on January 20, 2015, 06:53:54 AM
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Solutions to be posted as a "Reply" only after January 22, 21:00
a. Find the general solution of
\begin{equation}
u_{tt}-9u_{xx}=0;
\label{eq-HA1.7}
\end{equation}
b. Solve IVP
\begin{equation}
u|_{t=0}=\sin(x),\quad u_t|_{t=0}=\cos(x)
\label{eq-HA1.8}
\end{equation}
for (\ref{eq-HA1.7});
c. Consider (\ref{eq-HA1.7}) in $\{t>0, \, 3t> x > -3t\}$ and find a solution to it, satisfying Goursat problem
\begin{equation}
u|_{x=3t}=t,\quad u|_{x=-3t}=6t.
\label{eq-HA1.9}
\end{equation}
Remark.
Goursat problem for wave equation $u_{tt}-c^2u_{xx}=0$ in ${t> 0, -ct<x<ct}$ is $u|_{x=ct, t>0}=\phi(t)$, $u|_{x=-ct, t>0}=\psi(t)$ and one often assumes that compatibility condition $\phi(0)=\psi(0)$ is fulfilled. It is very important that $x=\pm ct$ are characteristics.
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Attached ;)
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By the way Sir, have we talked about the Goursat problem in class? I didn't see it on the notes online. :(
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No, we did not talk about Goursat but I defined it in the assignment and you have all tools to handle it (as you demonstrated) :D