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I am currently working on solving $$ \begin{cases} \Delta u = 0,\ \ 1 < r < 2\\ u(1,\theta) = 0, \ \ u(2, \theta) = \sin(\theta) \end{cases} $$ After making the ansatz $u(r,\theta) = R(r)\Theta(\theta)$ I have separated the equation into $$ \Theta(\theta) = \mathcal{A}e^{im\theta},\ \ \forall\mathcal{A}\ \in\ \mathbb{C} $$ and solved the Cauchy-Euler equation to find $$ R(r) = Br^m + Cr^{-m} $$ On a circular disk I am used to discarding the constant $C$ as the solution must be bounded at $r = 0$, but in this case however, I am unsure as to how to use the boundary conditions to find my constants $\mathcal{A}, B$ and $C$. To find $\mathcal{A}$ I suppose the second boundary condition should suffice, but the radial equation has got me confused.

Any insights are much appreciated.

braviolli
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  • This might be useful: https://math.stackexchange.com/questions/951755/pde-separation-of-variables-laplace-equation-problem – goblinb Jun 07 '20 at 21:14
  • @goblinb Thank you! Very helpful. A little unsure as to where the F ln(r) term came from, is it a part of the solution to the radial equation? – braviolli Jun 07 '20 at 21:19
  • Yes, $R(r) = c_1 \ln(r) + c_2$ is a solution for $k = 0$, right? See section 7.2.1 in Tveito/Winther, referencing example 7.1. Have you done problem 4 yet? :) – goblinb Jun 07 '20 at 21:23
  • @goblinb I see. I will read the section tomorrow and look at problem 4! – braviolli Jun 07 '20 at 21:58
  • Let me know how it goes! – goblinb Jun 07 '20 at 23:06
  • @goblinb At this point I have arrived at $ u(r,\theta) = E_m\ln r + \sum_{m=1}^{\infty}\mathcal{A}e^{im\theta}(r^{-m} - r^m)$, where I just need to determine $E$ and $\mathcal{A}$ by taking the inner product $\langle u(r,0), sin(m \pi x) \rangle$ I suppose. Still haven't used the second boundary condition, though... – braviolli Jun 08 '20 at 07:58
  • can't you use the second boundary condition to find the Fourier coefficients? – goblinb Jun 08 '20 at 14:40

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