Determine the dirichlet problem

Question

Solve the following problems:

1. What would the solution of the following problem look like for various values of time?

    PDE u_tt=u_xx   0    BC's {u(0,t)=0             0           {u(1,t)=sin t
   
    IC's  {u(x,0)=0           0≤x≤1
           {ut(x,0)=0

2. Find the solution to the Dirichlet problem

    PDE  ∇²u=0   0

   BC  u(1,Θ)=sinΘ   0≤ Θ<2π

3. Does the following Neumann problem have a solution inside the circle:

        PDE  ∇²u=0   0
        BC  ∂u/∂r=sin²Θ