Toronto Math Forum

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.

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 $r=1$).

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

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.)

Hopeful solutions to parts a), b), c), and d) attached!

hyperref is probably my favourite package

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

Quote from: Ian Kivlichan on November 19, 2012, 09:41:48 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 ($A_0$).

Calvin: I think you should set your constants to zero, as in Lecture 24.

Hopeful solutions attached!

(part 3)