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I am just wondering how would you solve the following PDE if you are not given the "inner circle" boundary condition: enter image description here

So it is given that $u(a, \theta) = \sin(4\pi)$, but nothing is given about $u(1,\theta)$.

I have sketched the graph here, as shown here the inner part is not given any condition. Although I know as $r$ approach $0$, $u(r,\theta)$ is bounded, but it seems this is not relevant as the domain of $r$ won't be less than $1$. So how would I go about solving this? enter image description here

I know this is a Two step problems. I could solve the $\theta$ problem. But not sure how to solve the "R" problem since it seems that nothing is given about the function when $r = 1$. $u(1,0)=?$ $ u(1, \theta)=?$

Mark McClure
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john_w
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    Perhaps it was a typo and they meant as $r\rightarrow1\ \ u(r,\theta)<\infty$, or maybe $r$ is actually bounded between $0$ and $a$? – Eli Bartlett Jun 20 '21 at 22:34
  • @Eli it is really bad that if a person get stuck on the problem for 2, 3 days and end up there is indeed a typo as this question is not very hard to solve if indeed what you said is true. – john_w Jun 20 '21 at 23:05
  • I'd guess there's an implied insulation condition, in which case the solution oughtta look like so - as generated by this web page, where (like a lot of numerical software) the insulation condition is, indeed, implied. – Mark McClure Jun 21 '21 at 00:40
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    If you do think there's an error or omission in the statement of the problem, I recommend considering it to be a learning experience and potentially more valuable to consider ways to fix the problem. It's certainly not realistic to think that sort of thing never happens. – Mark McClure Jun 21 '21 at 00:44

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