I have some assignments to do and I don't even know where to start. The notes in the course aren't too good, so I didn't understand too much from them.
Given $$ \Omega = \{(x, y) \in \mathbb{R}^2 , \quad 1 < x^2 + y^2 < 4\}$$
and the following problem:
$$u_{xx}(x,y) + u_{yy}(x,y) = 0 \quad \operatorname{for} \; (x,y) \in \Omega, $$
$$u(\cos \theta , \sin \theta ) = f( \theta ), \quad \theta \in [0,2\pi],$$
$$u(2 \cos \theta , 2 \sin \theta ) = g( \theta ). $$
a) Given $v(r, \theta ) = u(r \cos \theta , r \sin \theta ),$ and that $u$ is the solution of the problem, write the equation satisfied by $v$ and the values of $v(1, \theta)$ and $v(2,\theta ).$
b) In order to get the solution $v,$ we use the method of separation of variables. We shall look for particular solutions of the form $$v(r,\theta) = A_k(r)B_k(\theta)$$ with $r \in [1,2]$ and $ \theta \in [0,2\pi].$ Find the form of the particular solutions.
c) Write the general form of the solution $v.$
I don't expect anyone to do this whole assignment for me, but it would be great if anyone gives me some ideas for what I should do and/or some links as a reference.
Thanks!