% Here's the diary from class.  I started it by typing the command "diary Jan_15"

>> cd ~
>> cd Mat1062
>> cd heat
>> ls
3_30_99			for_class1.ps		heat3.m			heat_eul_dir.m~		neumann_heat_cn.m	tri_diag.m
Jan_15			for_class2.ps		heat3.m~		heat_eul_neu.m		periodic_tridiag.m	up_solve.m
class_jan_15.m		goof.m			heat4.m			heat_eul_neu.m~		test_fft
class_jan_15.m~		goof1.m			heat5.m			heat_eul_robin.m	test_heat1
down_solve.m		heat1.m			heat_cn.m		heat_eul_robin.m~	test_heat2
find_spec.m		heat1.m~		heat_cn.m~		mode.m			test_heat3
find_spec.m~		heat2.m			heat_eul_dir.m		my_LU.m			test_heat4

>> help heat1

 [u,err,x,t] = heat1(t_0,t_f,M,N)
 
 solves the heat equation u_t = D u_xx on [0,2*pi] with initial data u_0 =
 sin(k x) with periodic boundary conditions using finite-differences in
 space and explicit time-stepping.  t_0 is the initial time, t_f is the
 final time, N is the number of intervals in space, and M is the number of
 intervals in time.  err is the error. 

 If I'm going to use the FFT to analyse the solution then I'd like to  take
 N of the form 2^k.  This will also have the advantage of making sure there's
 always a meshpoint at x = pi

>> display('hit any key for more')
>> pause

% set the start time
>> t_0 = 0;
% set the end time
>> t_f = 1;
% set the number of subintervals of [0,2pi]
>> N = 128;
% set the number of subintervals of [t_0,t_f]
>> M = 100;

>> [u1,err,x,t1] = heat1(t_0,t_f,M,N);
>> figure(1)
>> for j=0:20
     plot(x,u1(:,j+1),'.-')
     title(t1(j+1),'FontSize',16)
     xlabel('Explicit Euler','FontSize',16)
     pause(1)
end

>> display('hit any key for more')
>> pause

% Now I want to make the timestep smaller by 1/10.  So I increase the number of time subintervals
>> M = 1000;
% Note that I'm now creating new arrays u2 and t2.  I'm writing over the priorly defined err and x
% (Because I'm not using err and x is the same anyway.)
>> [u2,err,x,t2] = heat1(t_0,t_f,M,N);
>> figure(2)
>> for j=0:20
     plot(x,u2(:,10*j+1),'.-')
     axis([0,2*pi,-1.1,1.1])
     xlabel('Explicit Euler','FontSize',16)
     title(t2(10*j+1),'FontSize',16)
     pause(1)
end

>> display('hit any key for more')
>> pause

>> help heat3
  [u,err,x,t] = heat3(t_0,t_f,M,N)
 
 solves the heat equation u_t = D u_xx on [0,2*pi] with initial data u_0 =
 sin(k x) with periodic boundary conditions using finite-differences in
 space and implicit time-stepping.  t_0 is the initial time, t_f is the
 final time, N is the number of intervals in space, and M is the number of
 intervals in time.  err is the error. 

 If I'm going to use the FFT to analyse the solution then I'd like to  take
 N of the form 2^k.  This will also have the advantage of making sure there's
 always a meshpoint at x = pi

>> display('hit any key for more')
>> pause

% I'm now going back to the original timestep that was bad for explicit Euler
>> M = 100;
>> [u3,err,x,t1] = heat3(t_0,t_f,M,N);
>> figure(1)
>> for j=0:20
     plot(x,u3(:,j+1),'.-')
     axis([0,2*pi,-1.1,1.1])
     xlabel('fully implicit','FontSize',16)
     title(t1(j+1),'FontSize',16)
     pause(1)
end

>> display('hit any key for more')
>> pause

>> help heat_cn
  [u,err,x,t] = heat_cn(t_0,t_f,M,N)
 
 this solves the heat equation u_t = D u_xx with initial data u_0 =
 sin(k x) with periodic boundary conditions using finite-differences in
 space and Crank-Nicolson time-stepping.  t_0 is the initial time, t_f is the
 final time, N is the number of intervals in space, and M is the number of
 intervals in time.  err is the error. 

 If I'm going to use the FFT to analyse the solution then I'd like to  take
 N of the form 2^k.  This will also have the advantage of making sure there's
 always a meshpoint at x = pi

>> display('hit any key for more')
>> pause

>> [u4,err,x,t1] = heat_cn(t_0,t_f,M,N);
>> figure(1)
>> for j=0:20
      plot(x,u4(:,j+1),'.-')
      axis([0,2*pi,-1.1,1.1])
      xlabel('Crank-Nicolson','FontSize',16)
     title(t1(j+1),'FontSize',16)
     pause(1)
end

% to see what objects have been defined and what their dimensions are
>> whos
  Name        Size                Bytes  Class     Attributes

  M           1x1                     8  double              
  N           1x1                     8  double              
  err       128x101              103424  double              
  j           1x1                     8  double              
  t1          1x101                 808  double              
  t2          1x1001               8008  double              
  t_0         1x1                     8  double              
  t_f         1x1                     8  double              
  u1        128x101              103424  double              
  u2        128x1001            1025024  double              
  u3        128x101              103424  double              
  u4        128x101              103424  double              
  x         128x1                  1024  double              

% to see a subset of that information...
>> whos u*
  Name        Size                Bytes  Class     Attributes

  u1        128x101              103424  double              
  u2        128x1001            1025024  double              
  u3        128x101              103424  double              
  u4        128x101              103424  double              

% I now close the diary. 
>> diary off