Problem 1

1. The time-stepping loop could look something like

     T = pi/2;
     k = T/10;
     time = 0;
   
     for n = 1:10
   
       time = time + k; 
   
     end
   

2. Add U0 = zeros(size(p,2),1) and U0 = zeros(size(p,2),1) at the beginning of the program, and then add U0 = U1 in every step of the loop.
3. The variational formulation that you need to put in the file Heat.m is

       u*v*dx + k*(du'*dv*dx + g(x,d,t)*u*v*ds)

   for the left-hand side and

       k*(f(x,d,t)*v*dx + (g(x,d,t)*gd(x,d,t) - gn(x,d,t))*v*ds) + w*v*dx

   for the right-hand side.