I'm trying to solve a Laplace equation in polar coordinates:
\begin{equation} u_{rr}+\dfrac{1}{r}u_{r}+\dfrac{1}{r^{2}}u_{\theta\theta}=0, 0\le r<a\\u_{r}(a,\theta)+\alpha u(a,\theta)=f(\theta), \alpha>0 \end{equation} where $f$ is a $2\pi$ periodic continuous function.
My attempt: Using the separation of variables method $u(r, \theta)=R(r)\Theta(\theta)$ we have $\dfrac{\Theta''}{\Theta}=\dfrac{-r^{2}R''}{R}-\dfrac{rR'}{R}=-\lambda$ is a constant. The boundary condition now is $[R'(a)+\alpha R(a)]\Theta(\theta)=f(\theta)$. If $R'(a)+\alpha R(a)\neq 0$ then $\Theta(0)=\Theta(2\pi)$ and we have a differential equation:
\begin{matrix}\Theta''+\lambda\Theta=0\\\Theta(0)=\Theta(2\pi) \end{matrix}
However, it seems like this boundary condition is not enough to solve this equation. For example, when $\lambda=\beta^{2}>0$ the general solution is $\Theta(\theta)=C\cos{\beta \theta}+D\sin{\beta\theta}$, so how can we find $C, D$ and $\beta$? And what if If $R'(a)+\alpha R(a)=0$? Now I'm stuck, can anyone help?
Moreover, this exercise requires to find reasonable conditions on $f$ so that the formal solution is a solution. I don't understand what does it mean and how can I solve this.
First, the angular function. The periodic condition required $\beta = n$, where $n=0,1,2,\dots$ where $n$ is an integer.
$$ \Theta_n(\theta) = \begin{cases} C_0, & n = 0 \\ C_n\cos(n\theta) + D_n\sin(n\theta), & n > 0 \end{cases} $$
Plugging the eigenvalue into the radial equation, we have $$ r^2R'' + rR' - n^2 R = 0 $$
This has a general solution:
$$ R_n(r) = \begin{cases} A_0 + B_0\ln r, & n = 0 \\ A_nr^n + B_nr^{-n} , & n > 0 \end{cases} $$
Since $R(0)$ needs to be finite, we can reduce some of the constants
$$ R_n(r) = \begin{cases} A_0, & n = 0 \\ A_nr^n, & n > 0 \end{cases} $$
Hence
$$ u_n(r,\theta) = \begin{cases} A_0C_0 , & n = 0 \\ A_nr^n\big(C_n\cos(n\theta) + D_n\sin(n\theta)\big), & n > 0 \end{cases} $$
Using the law of superposition, we find:
$$ u(r,\theta) = \sum_{n=0}^\infty c_nu_n(r,\theta) = c_0 + \sum_{n=1}^\infty r^n\big(c_n\cos(n\theta) + d_n\sin(n\theta)\big) $$
where I've combined several constants into one.
Finally, we can now plug in the boundary condition
$$ u_r(a,\theta) + \alpha u(a,\theta) = f(\theta) = \alpha c_0 + \sum_{n=1}^\infty \big(a^{n-1}+\alpha a^n \big)\big(c_n\cos(n\theta) + d_n\sin(n\theta)\big) $$
To find the remaining constants, compute the Fourier series of $f(\theta)$
\begin{align} \alpha c_0 &= \frac{1}{2\pi}\int_0^{2\pi}f(\theta) d\theta \\ \big(a^{n-1}+\alpha a^n \big)c_n &= \frac{1}{\pi}\int_0^{2\pi} f(\theta)\cos(n\theta)\ d\theta \\ \big(a^{n-1}+\alpha a^n \big)d_n &= \frac{1}{\pi}\int_0^{2\pi} f(\theta)\sin(n\theta)\ d\theta \end{align}