Toronto Math Forum
APM3462012 => APM346 Math => Home Assignment 1 => Topic started by: Miranda Jarvis on September 22, 2012, 03:55:39 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?)

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

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?

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 :D

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

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.

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.

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 $1$).

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

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!

Ok great  does this mean that I can understand the 2nd auxiliary condition of 5 c) as corresponding physically to a boundary condition on the velocity?

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?

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

I can't really see how part C of problem 5 is any different then a Cauchy problem, except for a change in coordinates and in the notes the initial conditions of the Goursat problem are given with respect to U only and not U_t .
Could someone please explain the differences between Goursat and Cauchy problems? thanks in advance.

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}
instead of that given in the assignment: (this is the IVP)
\begin{equation} u_{x=3t}=t,\quad u_t_{x=3t}=2t. \end{equation}

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.

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.

Professor Ivrii, could you please confirm this?

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 wellposed as well, but it is not a Goursat problem and it is what was intended.

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?

Try using characteristic coordinates, I think it should work.

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).

I used change of variables and found it really helpful :)

Solution to part a and b is attached!

Solution to problem 5

Scanning is barely passable

a) This is a 1D wave PDE general solution of which is discussed in the class.
\begin{equation*}
u(t,x)=\phi(x+3t) + \phi(x3t)
\end{equation*}
b) Using D'Alemblert formula we get:
\begin{equation*}
u(t,x)=\frac{1}{2} \bigl[ (x+3t)^2(x3t)^2 \bigr] + \frac{1}{6}\int_{x3t}^{x+3t} s\,ds \\
=\frac{1}{2}\bigl[ (x+3t)^2(x3t)^2 \bigr]+\frac{1}{12}(x+3t)^2\frac{1}{12}(x3t)^2 \\
=\frac{7}{6}(x+3t)^2+\frac{5}{6}(x3t)^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 $$ u(t,x)=\frac{1}{6}x+\frac{3}{2}t $$

The previous post is definitely an improvement over the preceding one

Professor I used a similar method but got the final answer completely different (see attached).
In the equation Ï•(0)+Ïˆ(âˆ’6t)=2t, shouldn't we have Ïˆ'(âˆ’6t) since the second initial condition gives du/dt?

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?