### Author Topic: HA1 problem 5  (Read 848 times)

#### Victor Ivrii

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##### HA1 problem 5
« on: January 20, 2015, 06:53:54 AM »
Solutions to be posted as a "Reply" only after January 22, 21:00

a. Find the general solution of

u_{tt}-9u_{xx}=0;
\label{eq-HA1.7}

b. Solve IVP

u|_{t=0}=\sin(x),\quad u_t|_{t=0}=\cos(x)
\label{eq-HA1.8}

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

u|_{x=3t}=t,\quad u|_{x=-3t}=6t.
\label{eq-HA1.9}

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.

#### Yang Liu

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##### Re: HA1 problem 5
« Reply #1 on: January 22, 2015, 10:03:48 PM »
Attached
« Last Edit: January 22, 2015, 10:10:10 PM by Yang Liu »

#### Yang Liu

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##### Re: HA1 problem 5
« Reply #2 on: January 22, 2015, 10:13:19 PM »
By the way Sir, have we talked about the Goursat problem in class? I didn't see it on the notes online.

#### Victor Ivrii

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##### Re: HA1 problem 5
« Reply #3 on: January 23, 2015, 11:14:31 AM »
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)
« Last Edit: January 23, 2015, 11:16:35 AM by Victor Ivrii »