Laplace Equation Polar Form Separation of Variables Problem

Question

I need some help on this question:

Solve Laplace's Equation $\frac{1}{r} \frac{\partial}{\partial r} (r \frac{\partial u }{\partial r }) + \frac{1}{r^2} \frac{\partial^{2} u}{\partial \theta^2}$ outside the semi-circle $r>a$ and above the real axis where $u=0$ on Real axis and $u(a,\theta) = f(\theta)$
$(0<\theta<2\pi)$

My attempt:

Assume the PDE is separable and the general solution is in the form $u(r,\theta) = \phi(\theta) G(r) $

Boundary Conditions: $u(r,0) = \phi(0) =0$,
$u(r,\pi) = \phi(\pi) = 0$, $u(0,\theta) = G(\theta) = 0$

By separation of Variables, the PDE would be

$\frac{r}{G} \frac{\partial}{\partial r} (\frac{\partial G}{\partial r}) =\lambda$

$\frac{-1}{\phi} \frac{\partial^{2} \phi}{\partial \theta ^2} = \lambda$ ($\lambda$ is a separation constant)

I have found $\lambda =0, and <0$ would yield trivial solutions

For $\lambda >0$, I have found that $\phi(\theta) = A \cos({\sqrt{\lambda} \theta}) + B \sin({\sqrt{\lambda} \theta}) $

When I initialized the initial conditions, the function would then become a sequence of functions which turns out to be

$\phi_n(\theta) = B_n \sin{n \theta}$ $(n = 1,2,3,...)$

I am not sure how to find G(r), and the Fourier Series of this problem. Also, how would I find the coefficients

Answer 1

The equation of $G$ is $$\frac{r}{G} \frac{\partial}{\partial r} r(\frac{\partial G}{\partial r}) =\lambda \\ r \frac{\partial}{\partial r} r(\frac{\partial G}{\partial r}) =\lambda G \\ r^2G''+r G'=\lambda G \\ r^2G''+r G'-\lambda G=0$$ $G=r^n $ is a solution of this Euler equation and so $$r^2\left(n(n-1)r^{n-2}\right)+r (nr^{n-1})-\lambda r^n=0 \\ \left(n(n-1)+ (n)-\lambda \right)r^n=0 \\ \left(n^2-\lambda \right)r^n=0$$you can take it from here

Answer 2

$$\frac{1}{r} \frac{\partial}{\partial r} (r \frac{\partial u }{\partial r }) + \frac{1}{r^2} \frac{\partial^{2} u}{\partial \theta^2}=0$$ $$u(r,\theta) = \phi(\theta) G(r) $$ $$\frac{r}{G} \frac{\partial}{\partial r} (r \frac{\partial G }{\partial r }) + \frac{1}{\phi} \frac{\partial^{2} \phi}{\partial \theta^2}=0$$ $$\frac{r}{G} \frac{\partial}{\partial r} (r \frac{\partial G }{\partial r }) =\omega^2 \space and \space \frac{1}{\phi} \frac{\partial^{2} \phi}{\partial \theta^2}=-\omega^2$$ $$G=r^\omega \space and \space \phi=A \cos(\omega \theta) + B sin(\omega \theta)$$ Particular solution : $$u=r^\omega \big( A \cos(\omega \theta) + B sin(\omega \theta)\big)$$ General solution, any integrable functions $A(\omega)$ and $B(\omega)$ : $$u=\int r^{\omega} \big( A(\omega) \cos(\omega \theta) + B(\omega) sin(\omega \theta)\big)d\omega$$ Discrete form : $$u=\sum_n r^{\omega_n} \big( A_n \cos(\omega_n \theta) + B_n sin(\omega_n \theta)\big)$$ With condition $u(r,0)=0\space$ then $u=\sum_n r^{\omega_n} B_n sin(\omega_n \theta)$

With condition $u(a,\theta)=f(\theta)$ where $f(\theta)$ is given on the form of Fourier series $f(\theta)=\sum_n c_n sin(2n\pi \theta)$ : $$\omega_n=2n\pi \space and \space B_n=\frac{c_n}{a^{\omega_n}}$$ $$u(r,\theta)=\sum_n r^{2n\pi} \frac{c_n}{a^{2n\pi}} sin(2n\pi \theta)$$