I am having trouble with this problem. Here is the problem:
I might need some tips on how to go through this problem. I have a sense on solving the cases for the separation constant, but I am having trouble on how to do it in this scenario. Here's the question:
Solve Laplace’s equation $[u_{xx}+u_{yy} = 0.]$ (This equation should be in polar form)
inside a circular annulus $R_1 < r < R_2$ with boundary conditions: $\varphi(R_1,\theta) = 0$ and $\frac{\partial \varphi}{\partial r} (R_2,θ) = f(θ)$
I understand the process of Separation Of Variables and the Laplace Equation. The point I am stuck on at the moment is evaluating the Boundary Condition and the Initial Condition.
Also, this is my first time on this website and I am trying to get familiar with it. I might need some tips on how to input in these figures correctly.
The usual separation of variables in polar coordinates produces
$$\phi \left( r,\theta \right) =\sum _{n=0}^{\infty } \left( A_{{n}}\sin \left( n\theta \right) +B_{{n}}\cos \left( n\theta \right) \right) \left( {r}^{n}C_{{n}}+{r}^{-n}E_{{n}} \right) +F\ln \left( r \right)$$
which is rewritten as
$$\phi \left( r,\theta \right) =\sum _{n=1}^{\infty } \left( A_{{n}}\sin \left( n\theta \right) +B_{{n}}\cos \left( n\theta \right) \right) \left( {r}^{n}C_{{n}}+{r}^{-n}E_{{n}} \right) +G+ F\ln \left( r \right)$$ Applying the boundary condition at $r=R_1$ we obtain
$$\sum _{n=1}^{\infty } \left( A_{{n}}\sin \left( n\theta \right) +B_{{n }}\cos \left( n\theta \right) \right) \left( {R_{{1}}}^{n}C_{{n}}+{R _{{1}}}^{-n}E_{{n}} \right) +F\ln \left( R_{{1}} \right) +G =0 $$
From this last equation we derive that $G=-Fln(R_1)$ and
$${R_{{1}}}^{n}C_{{n}}+{R_{{1}}}^{-n}E_{{n}}= 0$$
Then we have
$$ E_{{n}}=-{R_{{1}}}^{2\,n}C_{{n}} $$
The solution takes the form
$$\phi \left( r,\theta \right) =\sum _{n=1}^{\infty } \left( A_{{n}}\sin \left( n\theta \right) +B_{{n}}\cos \left( n\theta \right) \right) \left( {r}^{n}-{r}^{-n}{R_{{1}}}^{2\,n} \right)-Fln(R_1)+Fln(r) $$
Applying the boundary condition at $r=R_2$ we obtain
$$\sum _{n=1}^{\infty } \left( A_{{n}}\sin \left( n\theta \right) +B_{{n }}\cos \left( n\theta \right) \right) n \left( {R_{{2}}}^{n-1}+{R_{{2 }}}^{-n-1}{R_{{1}}}^{2\,n} \right)+F/R_2 =f \left( \theta \right) $$
Finally the constants $F$, $A_n$ and $B_n$ are determined using the standard expressions of Fourier series.