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2281

Home Assignment 1 / Re: Problem 5

« 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. 

2282

Misc Math / Re: integration constant in wave equation

« on: September 23, 2012, 02:33:46 PM »

Good morning,
in the notes "Homogeneous 1D Wave equation" we get to
$$
u(x,t)=\phi(x+ct)+\psi(x-ct)
$$
as the general solution, but in the very last paragraph it is mentioned that we could add a constant to $\phi$ if we subtract that same constant from $\psi$, and that this constant would be the only arbitrariness of this solution. But why does this have to be the same constant?
I can add for instance 1434 to $psi$ and subtract -12i from $\phi$ and the sub of the two would still satisfy the PDE, as any derivative of 1434-12i is zero...
Thanks for your help!

It is not an arbitrariness of the solution, but arbitrariness of the representation of the given solution in the given form. Really, consider the same solution
$$
u(x,t)=\phi_1(x+ct)+\psi_1(x-ct)=\phi_2(x+ct)+\psi_2(x-ct) \qquad \forall x,t,
$$
Then $\phi=\phi_1-\phi_2$ and $\psi=\psi_2-\psi_1$ satisfy
\begin{equation}\phi (x+ct)=\psi (x-ct) \qquad \forall x,t
\label{V}
\end{equation}
plugging $t=x/c$ we get$ \phi(2x)=\psi(0)$ and therefore $\phi(x)=C$ for all $x$. Thus $\phi=C$. Then (\ref{V}) implies that $\psi=-c_1$.

2283

Home Assignment 1 / Re: Problem 5

« on: September 23, 2012, 02:22:16 PM »

My question was with the Goursat problem, should it be of the form:

$$u_{tt}-c^{2}u_{xx}=0$$

$$u(t=\tau)=g(x)$$

$$u_{t}(t=\tau)=h(x)$$

like the example from class?

Then it would be not a Goursat, but Cauchy problem

2284

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!

2285

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 $-1$).

2286

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.

2287

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.

2288

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.

2289

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.

2290

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

2291

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?

2292

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?

2293

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 $x,y$.

2294

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.

2295

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)