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Messages - Ian Kivlichan

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Final Exam / Re: Problem 3
« on: December 20, 2012, 02:26:11 PM »
Chen Ge: I would point out that the solutions to the Euler equation are $R(r) = Ar^n + Br^{-n}$, not $R(r) = Ar^n + \frac{B}{r^{n+1}}$. It makes no difference, but I think it is worth pointing out anyway.

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Final Exam / Re: Problem 1
« on: December 20, 2012, 02:20:06 PM »
I'm adding my solution.. the others do not mention why the solution works for all of $(x, t)$ (though it does not change the final result).

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Final Exam / Problem 3
« on: December 20, 2012, 01:32:15 PM »
Use separation of variables to solve the Dirichlet problem for the Laplacian on the unit disk $\mathbb{D} = \{ (x,y) \in \mathbb{R}^2: x^2 + y^2 < 1\}$ with boundary condition $u(1, \theta) = \cos \theta.$
(The boundary condition is described in polar coordinates $(r, \theta) \rightarrow u(r, \theta)$ along $r=1$).

hopeful solution attached! (since djirar is posting all the solutions right away after 13:30..)

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Misc Math / Re: Lecture Note 23
« on: December 19, 2012, 01:18:00 AM »
Miranda: it is singular insofar as it is not defined at r=0. Though it goes to infinity as r approaches 0, the value is not infinity at r=0.

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Home Assignment 8 / Re: Problem 2
« on: November 28, 2012, 09:53:34 PM »
Peter: I think you should justify how you got to your result, especially since showing how changing $\Delta u$ from positive-definite to negative-definite flips the sign of the inequality in 2.b) is very quick. (As Calvin has now done.)

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Home Assignment 8 / Re: Problem 2
« on: November 28, 2012, 09:32:32 PM »
Peter: You've posted the problem, not the solution. ;P

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Home Assignment 8 / Problem 1
« on: November 28, 2012, 09:30:04 PM »
Hopeful solutions to parts a), b), c), and d) attached!

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Misc Math / Lecture 26 Equation 13 Question
« on: November 28, 2012, 12:34:26 AM »
Hi,

It looks like a mysterious 77 slipped its way into Equation 13 of Lecture 26: http://www.math.toronto.edu/courses/apm346h1/20129/L26.html#mjx-eqn-eq-13 .

I assume this is a typo?

Cheers,

Ian

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Technical Questions / Re: LaTeX advices
« on: November 20, 2012, 12:27:54 PM »
hyperref is probably my favourite package

For graphics (pictures), I would add \usepackage{graphicx}, but it is maybe not so relevant for math.

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Home Assignment 7 / Re: Problem 1
« on: November 20, 2012, 12:20:02 PM »
http://www.math.toronto.edu/courses/apm346h1/20129/HA7.html#problem-7.1

There is an unfinished business: found solutions satisfy $\Delta u=\pm k^2u0$ as $r>0$ but not necessarily in the origin. Which solutions satisfy this equation in the origin?
I would guess that since the solutions depend on $r$ only, at $r=0$ they can only be constants $u=C$. Since they are constant, $\Delta u = 0$, so $C=0$. (Anything with $\frac{1}{r}$ dependence will have a singularity at $r=0$.)

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Home Assignment 7 / Re: Problem 4
« on: November 20, 2012, 12:04:28 PM »
Calvin: I think you should set your constants to zero, as in Lecture 24.

Not really: lecture suggests that we can do it imposing an extra condition $\int _\Omega u \, dS=0$.

A better question: why solution exists? Related: Does solution of the same equation but with b.c. $u_r|_{r=a}=1$ exist?
I don't think we can have solutions with BC $u_r|_{r=a}=1$ since all our coefficients would become 0 (since $\int_{0}^{2\pi}f\left(\theta\right) \sin(n\theta) d \theta = \int_{0}^{2\pi}f\left(\theta\right) \cos(n\theta) d \theta = 0$ for $n \ge 1$). I guess that the condition there should be that $u_r|_{r=a}$ be Fourier-decomposable, without a constant ($A_0$).

12
Home Assignment 7 / Re: Problem 2
« on: November 19, 2012, 09:48:54 PM »
Calvin: I wasn't really sure that for 2.b) you could have a Bessel function of a complex variable, or if you'd need to do something slightly different?

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Home Assignment 7 / Re: Problem 4
« on: November 19, 2012, 09:41:48 PM »
Calvin: I think you should set your constants to zero, as in Lecture 24.

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Home Assignment 7 / Re: Problem 3
« on: November 19, 2012, 09:40:30 PM »
Calvin: it's a bit simpler to write out ((-1)^(n+1) +1) as 0 or 2!

p.s. I am envious of your skills in LaTeX

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Home Assignment 7 / Re: Problem 3
« on: November 19, 2012, 09:31:27 PM »
Hopeful solutions attached!

(part 3)

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