2281

**Home Assignment 1 / Re: Problem 3**

« **on:**September 25, 2012, 10:09:56 AM »

Now it is correct.

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2282

Aida, WTH? In MAT244 your scan was a golden standard of scanning

http://weyl.math.toronto.edu:8888/MAT244-2012S-Forum/index.php?topic=38.msg111#msg111 (see, I remember, so good it was), and now this? Such posts defy the whole purpose of this forum, it is not to submit your papers for grading (you submit it to TA) but even this is difficult as quality of scan is poor (actually you used cellphone without taking care of settings), but to share the solution with your classmates who can comment, find errors or correct them.

From this point the typed solution using MathJax is far the best as I can edit it, just marking the place where I see an error, and anyone can copy-paste code from it. But apart of this your former clean black-white scan with perfect position of the paper was the very best thing.

Here you use colour (not even grayscale) scan and some papers are horizontal and some diagonal ... everyone has a really hard time to read them. I admit, some of your submissions a better than othersbut**from the point of view of this forum I must consider all of them non-existent** . It looks like you just decided to capture the space preventing anyone else from posting the solutions.

So, everyone should feel to post solutions.

------

Also:

some people used paper clips despite our request to staple and some even tried to use "poor man paper clips" just folding several times the corner of the paper and adding a bit of saliva. Sorry, no of these constructions is robust enough to prevent separation of pages. I hope I caught every such attempt and stapled, but I am not sure. I am also not sure if Prof Colliander managed with this. So, if you have not stapled and your pages are separated and lost, you know whom to blame.

Not everyone indicated the section where to bring your papers back (so get such papers from TA who graded it).

Please use the standard paper (letter size). Someone used A4 (a bit more narrow and a longer overseas size). It looks like a little thing but also causes problems when dealing with tens of papers.

And finally, again: please, do not come to submit papers during the class -- only before lecture, during the break or after. Everyone in the night class feels tired (instructor in the least degree because of adrenalin rush) and such distractions are rather disruptive.

http://weyl.math.toronto.edu:8888/MAT244-2012S-Forum/index.php?topic=38.msg111#msg111 (see, I remember, so good it was), and now this? Such posts defy the whole purpose of this forum, it is not to submit your papers for grading (you submit it to TA) but even this is difficult as quality of scan is poor (actually you used cellphone without taking care of settings), but to share the solution with your classmates who can comment, find errors or correct them.

From this point the typed solution using MathJax is far the best as I can edit it, just marking the place where I see an error, and anyone can copy-paste code from it. But apart of this your former clean black-white scan with perfect position of the paper was the very best thing.

Here you use colour (not even grayscale) scan and some papers are horizontal and some diagonal ... everyone has a really hard time to read them. I admit, some of your submissions a better than othersbut

So, everyone should feel to post solutions.

------

Also:

some people used paper clips despite our request to staple and some even tried to use "poor man paper clips" just folding several times the corner of the paper and adding a bit of saliva. Sorry, no of these constructions is robust enough to prevent separation of pages. I hope I caught every such attempt and stapled, but I am not sure. I am also not sure if Prof Colliander managed with this. So, if you have not stapled and your pages are separated and lost, you know whom to blame.

Not everyone indicated the section where to bring your papers back (so get such papers from TA who graded it).

Please use the standard paper (letter size). Someone used A4 (a bit more narrow and a longer overseas size). It looks like a little thing but also causes problems when dealing with tens of papers.

And finally, again: please, do not come to submit papers during the class -- only before lecture, during the break or after. Everyone in the night class feels tired (instructor in the least degree because of adrenalin rush) and such distractions are rather disruptive.

2283

Is it an ancient manuscript? Can anyone read this?

2284

One-way (no way to reverse the direction)

2285

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).

2286

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.

2287

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

$$

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 $\psi=-c_1$.

2288

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

2289

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!

2290

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 $-1$).

2291

$x>b^2$, 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.

2292

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.

2293

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

2294

I will expand this question a little bit: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?"

So, can we use the formula of general solution for wave equation or we should prove/derive it?

There is a subtle difference between

Quote

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

2295

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