91Ó°ÊÓ

(a) Solve the Dirichlet problem $$ \begin{aligned} &\nabla^{2} u(r, \theta)=0, \quad 0

Solve the heat equation $$ \frac{\partial u}{\partial t}=\frac{\partial^{2} u}{\partial r^{2}}+\frac{1}{r} \frac{\partial u}{\partial r}+\frac{1}{r^{2}} \frac{\partial^{2} u}{\partial \theta^{2}}, \quad 00 $$ given the nonzero boundary condition $$ u(1, \theta, t)=\sin 3 \theta $$ and the initial condition \(u(r, \theta, 0)=0\).

Solve the Neumann problem \(\nabla^{2} u=0\) for \(0 \leq r