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### Messages - Victor Ivrii

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2446
##### Home Assignment 2 / Re: Problem 1 -- not done yet!
« on: October 07, 2012, 11:15:56 AM »
Posted by: Rouhollah Ramezani

C) Contains a logical error in the domain $\{âˆ’2t<x<2t\}$ (middle sector) which should be found and fixed. Note that the solution of RR there is not in the form $\phi(x+2t)+\psi(x-2t)$

C) is also amended, but I probably failed to spot the "logical error" and it is still there.

You presume that $u$ should be continuous, which is not the case. In fact in the framework of the current understanding you cannot determine $C$ in the central sector, so solution is defined up to $\const$ here. One needs to dig dipper in the notion of the weak solution.

Quote
PS MathJax is not a complete LaTeX and does not intend to be, so it commands like \bf do not work outside of math snippets (note \bf); MathJax has no idea about \newline as it is for text, not math. For formatting text use either html syntax (in plain html) or forum markup

Yes, I realized that after. And I found out from your other post that we can actually use an html editor+MathJax instead of LaTeX. right?

Not really: you do not use html syntax but a special SMF markdown which translates into html (so you cannot insert a raw html-- but Admin  can if needed.

2447
##### Technical Questions / Re: Printing from the website
« on: October 06, 2012, 03:51:11 PM »
Has any one tried printing from the website (the lec notes)-- the math codes come out very strange when printed. Does any one have a solution?? Thanks
Kanita

The math snippets are really written in LaTeX and parsed through javascript. If I had time (which I don't) I would convert pages (actually their sources)  to pure LaTeX files and then generate PDF as I do with Home Assignments. I can also print the pages directly to PDF (using Web browser--I use Safari 6.0.1 on MacOSX) but this PDF would be good for printing only.

General rule: If you discuss technical difficulties, please provide your OS (including version) and Web Browser (including version).

2448
##### Home Assignment 3 / Re: Problem 2
« on: October 06, 2012, 03:16:06 PM »
In this problem it is stated that we should use formulas (1)-(2). But shouldn't we have an initial condition in order to apply these formulas.

Sure we need: so add $u(x,0)=g(x)$.

2449
##### Misc Math / Re: Method of Continuation
« on: October 06, 2012, 10:16:32 AM »
A method of continuation is a cheap trick to reduce certain BVP to those we already know how to solve. In its easiest form we looked at it in the lectures.

Consider a BVP with one "special" variable $x$ (there could be other variables). This $x$ runs from $0$ to $+\infty$ (there could be other cases). Consider the same problem but with $x$ running from $-\infty$ to $\infty$, thus dropping boundary condition(s) at $x=0$.

Assume that

1) plugging $-x$ instead of $x$ leaves this new boundary problem unchanged. F.e. it happens when we consider equations with the constant coefficients and  containing only even order derivatives by $x$;

Good: $u_{t}+u_{xx}$, $u_{yxx}+ u_{y}-u_{xx}$

Bad: $u_{tx}+u_{xx}$, $u_t+u_{xxx}$

Variable coefficients can affect this situation:

Also good: $u_{tt}- xu_{xxx}$

Bad: $u_t + xu_{xx}$

So far we applied method of continuation to wave and heat equations:
$$u_{tt}-c^2u_{xx}=f, \qquad u|_{t=0}=g, \qquad u_t|_{t=0}=h$$
and
$$u_{t}-ku_{xx}=f, \qquad u|_{t=0}=g.$$

2) Assume that boundary conditions contains only terms with all odd order derivatives with respect to $x$ and are homogeneous:

$u_x|_{x=0}=0$ or $(u_x-u_ {xxx})|_{x=0}=0$ fit the bill.

Note that even functions satisfy these boundary conditions automatically. Then:
We continue all known functions to $x<0$ as even  functions and solve extended problem (ignoring boundary condition(s) at $x=0$.

2*) Alternatively, assume that boundary conditions contains only terms with all even order derivatives with respect to $x$ and are homogeneous:

$u|_{x=0}=0$ or $(u-u_ {xx})|_{x=0}=0$ fit the bill.

Note that odd functions satisfy these boundary conditions automatically. Then:
We continue all known functions to $x<0$ as odd  functions and solve extended problem (ignoring boundary condition(s) at $x=0$.

2450
##### Home Assignment 2 / Re: Problem 3
« on: October 05, 2012, 08:56:36 PM »
Hi, I understand how to derive the solutions for Problem 3, but now I have a question. For example, in 3.a, the solution is
u=x/2 for 0<x<2t, but u(0,x)=x/2 which does not equal 0, the given initial condition. Also, du/dt(0,x)=0 which does not equal 1, the given initial condition. I know however that these initial conditions are satisfied for u=t as x>2t. Could you please explain this?

Domain $0<x<2t$ does not approach line $t=0$ (except a single point) where initial condition must be satisfied.

2451
##### Misc Math / Re: proof of proposition 2a) in 9th lecture notes
« on: October 05, 2012, 04:46:44 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!

Thanks!

First, a correct citation:
"as $x\to -\infty$ $U$ is fast decaying with all its derivatives".

So, this is true for $R= U_u-kU_{xx}$, right? But we know that $R$ does not depend on $x$ (fact, your 'counterexample' misses) and  therefore $R=0$

2452
##### Misc Math / Re: inhomogeneous b.c.
« on: October 05, 2012, 10:42:42 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!

$'$ is an artefact (removed). BTW domain is $\Pi$, not $II$

$\bigl(-u_{\tau}(y,\tau)+ku_{yy}(y,\tau)\bigr)=0$ due to equation $u_t-ku_{xx}=0$ (if $u_t-ku_{xx}=f$) we would get an extra term (17) in the lecture 10

2453
##### Home Assignment 3 / Re: Problem 6
« on: October 05, 2012, 07:04:07 AM »
so, the only condition for a is Ut = K Uxx?

Obviously

2454
##### Home Assignment 3 / Re: Problem 1
« on: October 05, 2012, 07:03:41 AM »
I think there is still problem 2/ sqrt(pi) instead of sqrt(2/pi), is this true?

$\sqrt{\frac{2}{\pi}}$ as it should be

2455
##### APM346 Misc / Re: Typo in led note?
« on: October 04, 2012, 06:52:25 PM »
Yep...that's what I guess initially....sorry my space doesn't work properly these days

Too much cola spilled on the keyboard?

2456
##### Misc Math / Re: Classification criteria for PDEs
« on: October 04, 2012, 06:51:40 PM »
Is B=7/2?....I thought on one of the book it gives the general form as AUxx+2BUxt+CUtt+...=0?

Ok, but then you need to calculate $B^2-AC$

2457
##### Home Assignment 3 / Re: Problem 6
« on: October 04, 2012, 06:49:38 PM »
In question 6 part a, Are we supposed we have condition like U(l,t)=U(0,t)=0?

No, it was at (a) to be integral on $(-\infty,\infty)$. $u(l,t)=u(0,t)=0$ (or Neumann) would be in (b)

2458
##### Home Assignment 3 / Re: Problem 1
« on: October 04, 2012, 05:52:05 PM »
I noticed the given error function(Erf) in the problem set sheet is different with what's given in WolframAlpha. Which one should I use? Also, are we expected to write our final answers in form of Erf?

It was a misprint in HA3, which I just fixed

2459
##### APM346 Misc / Re: Typo in led note?
« on: October 04, 2012, 04:23:49 PM »
In lec 10 ..1D heat equation,  around the formula 4, there's Aline saying"this solution has the same form as 2." but I feel like the integral should from 0 to infinity, instead of from - infinity to infinity like the formula 2?

Who is Aline?!  OK, it is not a name but "a line"

Not a typo but a bit too casual sentence. I put (and obviously with the different domain of integration $[0,\infty))$:

2460
##### APM346 Misc / Re: When HA marks will be available?
« on: October 04, 2012, 04:16:23 PM »

Does this mean that Problem 2 and 3 are not going to be graded? And also, when and where can we pick up our Home Assignments?

Yes, unfortunately we do not have enough TA office hours and it was explained from the beginning that home assignments will be only partially graded.

I believe that the HA1 for day section will be brought to either Monday or Wednesday class (you wrote your  Section on the paper, right?) Night section' papers will be available during my office hours October 10 Wed http://forum.math.toronto.edu/index.php?topic=46.0

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