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Home Assignment 1 / Re: Problem 5

« on: September 23, 2012, 11:15:28 AM »

For the initial conditions for Problem 5 (c) should they both be at x = 3t or is it not a typo?
Thanks

There are no initial conditions in 5(c). Note a special name of the problem (Goursat). It is not IVP!

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Home Assignment 1 / Re: Problem 5

« on: September 23, 2012, 10:14:36 AM »

My reason for the asking the above question is that the examples that I have managed to find regarding Goursat problems and the wave equation all have the x<3t which made more sense to me for the integration - after thinking about it I am guessing #5 has x>3t due to the initial velocity?  The examples I was looking at only included initial positions - I was just trying to get a picture of what was happening.

Both problems have exactly the same properties. In fact if one considers one spatial dimension then $x$ and $t$ could be permuted (and equation multiplied by $-1$) and the type of equation would not change; so in fact $x$ could be a time ant $t$ a spatial coordinate.

Of course, it would not be a case if there were 2 or more spatial variables, in
$u_{tt}-c^2 u_{xx}-c^2u_{yy}=0$  $t$ and $x$ are not on equal footing (as presence $u_{yy}$ prevents from multiplying by $-1$).

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Technical Questions / Re: test for math

« on: September 23, 2012, 05:55:35 AM »

$x>b^2$, is it the same as latex?

Yes -- except MathJax neither is nor intended to be a complete LaTeX. See
http://www.math.toronto.edu/courses/apm346h1/20129/about_mathjax.html

Note: If you have text snippet inside of math you should use \text{Your text} rather than format it as math as you did. You can also use \textrm{Your text}, \textit{Your text}.

Also DoubleDollars are deprecated in LaTeX, you use \[ \] or \begin{equation*} \end{equation*} (without * for a numbered version)

PS I disabled \( \) and \[ \] as they in the forum environment have funny side effects.

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Home Assignment 1 / Re: Problem 4

« on: September 23, 2012, 05:39:08 AM »

But the only condition I have is the original PDE and my general solution just well satisfies that equation. What other conditions should I have? Thanks!

It was a general remark. In Problem 5 there is no other explicit condition. However one needs to take a look on the general solution.

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Home Assignment 1 / Re: Problem 4

« on: September 23, 2012, 04:08:42 AM »

I followed the normal steps and found the general solutions to both equations. I cannot figure out why one of them do not exist. Is it because some function is not defined? Get lost in part (c)...

You need to ask yourself: does a solution you found satisfy all conditions of the problem.

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Home Assignment 1 / Re: Problem 5

« on: September 23, 2012, 04:07:02 AM »

I will expand this question a little bit:
So, can we use the formula of general solution for wave equation or we should prove/derive it?

Yes, you are allowed to use every appropriate (relevant) formula given in the class without rederiving it (unless specifically asked to derive it first). Definitely you need to ask yourself: "can I apply a formula for a general solution?", "Can I apply D'Alembert formula?"

There is a subtle difference between allowed and  can: you can use any appropriate formula but it may happen that you are explicitly asked to use some specific approach in which case you are allowed to use only some specific formulae.

P.S. This is my first post here, so I am not sure if I am asking an appropriate question

Yes, it is a completely appropriate question.

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Home Assignment 1 / Re: Problem 5

« on: September 22, 2012, 10:36:35 PM »

Just wondering if in part c) of problem 5 the question should read: Consider (7) in (x<3t, x>-3t) ?
(instead of x>3t ....)

Just wondering whether you have any reason to prefer $\{x<3t, x>-3t)\}$ to $\{x>3t, x>-3t\}$ or just want to boost the number of posts

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Home Assignment 1 / Re: Problem 5

« on: September 22, 2012, 10:34:31 PM »

I was wondering if for problem 5 if we have to show all the steps to getting answers or if we can simply apply the applicable formulae from lecture notes? (example can we simply use the D'Alembert formula?)

What problem D'Alembert formula solves? Is it the same problem as here?

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Home Assignment 1 / Re: Problem 2

« on: September 22, 2012, 10:31:22 PM »

I am not sure what does it mean by 'explain the difference'. Is it the difference between the solutions or the condition under which they are continuous at (0,0)?

The solutions of (a) and (b) are drastically different. Why?

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Home Assignment 1 / Re: Problem 3

« on: September 22, 2012, 02:33:13 PM »

I did use the method of characteristics, but I parametrized x and y in terms of s and integrated with respect to s . The thing is my solution doesn't depend on Y, is this ok ?

I have no idea what is $s$. TA who will check your paper probably has no idea either. You need to return to the original coordinates $x,y$.

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Home Assignment 1 / Re: Problem 3

« on: September 22, 2012, 01:45:48 PM »

Hello,

For this question I managed to find a solution that satisfies the conditions, but that only depends on X and a constant.

In general, if I find a solution that satisfies all conditions is this solution correct regardless of the method used to find it?

You need to find all solutions and justify that there are no other solutions. The method of characteristics we studied ensures this if correctly applied. Your home-brewed method may not.

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Home Assignment 1 / HA1-pdf

« on: September 22, 2012, 01:40:44 PM »

Here is HA1.pdf - Home assignment 1 printed to pdf (updated Mon 24 Sep 2012 05:03:22 EDT)

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Home Assignment 1 / Re: Problem 2

« on: September 22, 2012, 12:40:30 PM »

Hi, for number 2, say we get $u = f(g(x,y))$ for our general solution when $(x,y) \ne (0,0)$. Then are we just finding a $u(0,0)$ that is equal to the limit of $u = f(g(x,y))$ as $(x,y)$ approaching $(0,0)$ in order to make u continuous at $(0,0)$?

Well, you need to be sure that this limit exists, right?

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Misc Math / Re: characteristic vs. integral lines

« on: September 22, 2012, 02:20:38 AM »

I feel like I understand what is being said here, but I am confused by the second set of lecture notes,  specifically equation (7) which seems to be doing the same thing as equation (4) in the reply to this post except with a + sign instead of a equal. Am I misunderstanding and these are actually different scenarios, or is one a misprint?

Miranda, thanks!  There was a misprint (mistype) in Lecture 2 (equation (7)), should be =.
Now it has been corrected.

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Home Assignment 1 / Re: Problem 4

« on: September 22, 2012, 02:15:54 AM »

In this settings you need to select a solution having certain properties

It may happen that

• Is what one expects
• Unusually broad
• Unusually narrow

Your task is to determine what happens here.