Author Topic: Problem 5  (Read 8324 times)

James McVittie

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Re: Problem 5
« Reply #15 on: September 23, 2012, 04:00:49 PM »
I think you're right Jinlong. That was what I meant by my previous post. I could not find any other mention of a Goursat problem in the notes except on the first set of notes of the second week.

Laurie Deratnay

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Re: Problem 5
« Reply #16 on: September 23, 2012, 04:03:24 PM »
Thanks Jinlong - I was hoping that was the problem!  I've been trying to figure out where that partial wrt to 't' came from ... now it all makes sense.

James McVittie

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Re: Problem 5
« Reply #17 on: September 23, 2012, 04:37:05 PM »
Professor Ivrii, could you please confirm this?

Victor Ivrii

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Re: Problem 5
« Reply #18 on: September 23, 2012, 06:15:02 PM »
I guess that there is a typo in the assignment for 5.(c) about the Goursat problem:

the formula (9) should be as below to be a Goursat problem
\begin{equation}  u|_{x=3t}=t, \quad u|_{x=-3t}=2t. \end{equation}

Nice spotting! You are right (copy, paste and correct works faster but is more error prone).


PS. Actually problem as stated originally (with $u_t$ instead of $u$) is not IVP problem (as lines don't coincide) and is well-posed as well, but it is not a Goursat problem and it is what was intended. 

James McVittie

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Re: Problem 5
« Reply #19 on: September 23, 2012, 07:42:54 PM »
I was trying to figure out the "typo'ed" equation using material from the other 1D wave equation lectures but to no avail. Would you be able to explain how to approach the old equation or possibly post an outline of the solution?

Djirar

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Re: Problem 5
« Reply #20 on: September 23, 2012, 08:59:09 PM »
Try using characteristic coordinates, I think it should work.

Victor Ivrii

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Re: Problem 5
« Reply #21 on: September 23, 2012, 11:28:50 PM »
The easiest way is to write the general solution and then trying to satisfy boundary conditions. This works for correct settings (and for those with the typo as well).

Kun Guo

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Re: Problem 5
« Reply #22 on: September 23, 2012, 11:33:00 PM »
I used change of variables and found it really helpful :)

Aida Razi

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Re: Problem 5
« Reply #23 on: September 24, 2012, 09:08:46 PM »
Solution to part a and b is attached!

Djirar

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Re: Problem 5
« Reply #24 on: September 25, 2012, 11:04:27 AM »
Solution to problem 5

Victor Ivrii

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Re: Problem 5
« Reply #25 on: September 25, 2012, 03:12:39 PM »
Scanning is barely passable

Rouhollah Ramezani

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Re: Problem 5
« Reply #26 on: September 25, 2012, 05:27:11 PM »
a) This is a 1-D wave PDE general solution of which is discussed in the class. 

\begin{equation*}
u(t,x)=\phi(x+3t) + \phi(x-3t)
\end{equation*}
b) Using D'Alemblert formula we get:
\begin{equation*}
u(t,x)=\frac{1}{2} \bigl[ (x+3t)^2-(x-3t)^2 \bigr] + \frac{1}{6}\int_{x-3t}^{x+3t} s\,ds \\
=\frac{1}{2}\bigl[ (x+3t)^2-(x-3t)^2 \bigr]+\frac{1}{12}(x+3t)^2-\frac{1}{12}(x-3t)^2 \\
=\frac{7}{6}(x+3t)^2+\frac{5}{6}(x-3t)^2
\end{equation*}

c) We impose Goursat problem boundary conditions to general solution and get:
 $$
\phi(6t)+\psi(0)=t \\
\phi(0)+\psi(-6t)=2t
$$
Letting $t=0$ in first equation and subtracting it from the second we get $\phi(0)=\psi(0)=0$. Therefore $$\phi(t)=\frac{t}{6} \\
\psi(t)=\frac{-t}{3}
$$
Final solution for Goursat problem is $$ u(t,x)=\frac{-1}{6}x+\frac{3}{2}t $$

Victor Ivrii

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Re: Problem 5
« Reply #27 on: September 25, 2012, 06:42:11 PM »
The previous post is definitely an improvement over the preceding one

Peishan Wang

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Re: Problem 5
« Reply #28 on: September 26, 2012, 04:12:44 AM »
Professor I used a similar method but got the final answer completely different (see attached).
In the equation ϕ(0)+ψ(−6t)=2t, shouldn't we have ψ'(−6t) since the second initial condition gives du/dt?
« Last Edit: September 26, 2012, 04:15:00 AM by Peishan Wang »

Peishan Wang

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Re: Problem 5
« Reply #29 on: September 26, 2012, 04:22:17 AM »
OMG I didn't realize there was a correction on Sep 23! I finished the assignment on Friday night and didn't expect that there would be any change to the questions just several hours before the assignment is due....

Professor can you give some consideration to the situation this time?
« Last Edit: September 26, 2012, 04:31:54 AM by Peishan Wang »