a) Solve the Dirichlet problem for the Laplace equation in a disc in $\mathbb{R}^{2}$

$$\begin{aligned} \Delta u &=0 \quad \text { in } \quad G=\left\{x^{2}+y^{2}<R^{2}\right\} \subseteq \mathbb{R}^{2}, R>0 \\ u &=u_{D} \quad \text { on } \quad \partial G \end{aligned}$$

using polar coordinates $(r, \varphi)$ and separation of variables, $u(x, y)=R(r) \Theta(\varphi)$. Then use the ansatz $R(r)=r^{\alpha}$ for the radial function.

b) Solve the Dirichlet problem for the Laplace equation in a square in $\mathbb{R}^{2}$

$$\begin{aligned} &\Delta u=0 \quad \text { in } \quad G=[0, a] \times[0, a] \\ &u(x, 0)=f_{1}(x), \quad u(x, a)=f_{2}(x), \quad u(0, y)=f_{3}(y), \quad u(a, y)=f_{4}(y) \end{aligned}$$