Messages

16

Home Assignment 3 / Re: Problem 4

« on: October 08, 2012, 03:53:35 PM »

I don't think that the maximum in the region $0\leq x \leq l$, $0 \leq t \leq T$ must either decreases or increase; I think it also can stay constant (e.g. iron rod that is initially very hot in the middle, then the maximum is found at t=0, x=l/2) and M(T) =const.

Am I wrong?

17

Home Assignment 3 / problem 3

« on: October 07, 2012, 10:15:42 PM »

Could anyone give me a hint on how to integrate problem 3a)? In order to get rid of the absolute value I have to split up the integral and then the -inf to inf formulas don't work any more...

Thanks a lot!

18

Misc Math / proof of proposition 2a) in 9th lecture notes

« on: October 05, 2012, 03:57:42 PM »

we are given
$$
U(x,t)=\int_{-\infty}^xu(x,t)dx
$$
and are to prove that $U$ satisfies $U_t=kU_{xx}$.

The proof given is "one can see easily that as $x\rightarrow -\infty$ and that therefore $U$ and all its derivatives have to be zero". But the integral over any function with the upper bound approaching the lower bound goes to zero!

For instance I take the function $f(x)=x^5$, which obviously does not satisfy the heat equation for $x\neq 0$. Then isn't
$$lim_{x\rightarrow -\infty}\int_{-\infty}^{x}x^5dx=0$$, and according to this "proof" $\int_{-\infty}^{x}x^5dx$ would satisfy the heat equation?

This does not make sense to me...

Thanks!

19

Misc Math / inhomogeneous b.c.

« on: October 05, 2012, 08:59:09 AM »

In the 10th lecture we are asked to consider
$$
0=\int_{II}G(x,y,t-\tau)(-u_{\tau}(y,\tau))+ku_{yy}(y,\tau)d\tau'dy
$$

1. question: What are we integrating over here? Is $\tau'=t-\tau$?

2. Where is this expression coming from? Is it a trial solution that I simply have to take as given, or does it follwow from any other expression?

Thanks!

20

Home Assignment 2 / Re: Problem 2

« on: September 29, 2012, 03:37:34 PM »

should the $\phi$ in eq. 6 read $\phi(r+ct)$ rather than $\phi(x+ct)$?

21

Misc Math / Re: Example 8b

« on: September 28, 2012, 05:15:08 PM »

... and in example 8d): Why should $u$ be given by eq. (9)? Shouldn't that be eq. (9) with all sines exchanged by cosines?

22

Misc Math / Example 8b

« on: September 28, 2012, 05:02:12 PM »

Dear all,

in the 8th lecture notes in the example b) we end up at
$$u(x,t)=\frac{1}{2}(sin(x+tt)+sin(x-t))$$ for $x>t$
and $$u(x,t)=\frac {1}{2}(sin(x+t)-sin(x-t))$$ for 0<x<t. But isn't the latter $u$ valid on $-t<x<t$, and can't we even say that
$$u(x,t)=\frac{1}{2}(-sin(x+t)-sin(x-t))$$ for $x<-t$?

23

Misc Math / Lecture 6 example

« on: September 27, 2012, 04:23:33 PM »

Hello,
I was going through the 6. lecture notes, where in the end an example is brought up that leads to an integral
$$\frac{1}{4}\int_0^tcos(t')(cos(x-ct+ct')-cos(x+ct-ct'))dt'$$
I was trying to do that integral, but the only way that I could do it was to write out the cosines as complex exponentials, which lead me to eight terms in the end... Is there a cleverer way to do this integral?
Thanks!

24

Misc Math / integration constant in wave equation

« on: September 23, 2012, 12:10:16 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!

25

Misc Math / Re: characteristic vs. integral lines

« on: September 21, 2012, 05:59:31 PM »

havin some trouble there... $$x(t)=\frac{b}{a}t+c$$

26

Misc Math / characteristic vs. integral lines

« on: September 21, 2012, 05:57:04 PM »

Heythere,
I am confused by the terms characteristic lines and integral lines. The book introduces characteristic lines as the curves along which a function is constant. Now in the notes integral lines are curves to which the vector field is tangential, i.e. in the case of the gradient vector field the lines along which the function changes most (in abs. value).

So I thought these two should be orthogonal in the case of $au_t+bu_x=0$.

The characteristic lines of u are given by $x=\frac{b}{a}y+c$, but this is also the solution to the ODE
$$\frac{dt}{a}=\frac{dx}{b}$$, which is given for the integral lines in the notes (First order PDEs)...

So are integral lines the same as characteristic lines?