4.8 Show the effect of boundary/initial conditions on 1-D heat PDE

The PDE is \[ \frac {\partial T\left ( x,t\right ) }{\partial t}=k\frac {\partial ^{2}T\left ( x,t\right ) }{\partial x^{2}} \] Problem: given a bar of length \(L\) , solve the above 1-D heat PDE for 4 different boundary/initial condition to show that the solution depends on these.

Mathematica

 
Clear[y,x,t,k]; 
SetDirectory[NotebookDirectory[]]; 
barLength = 4*Pi; 
timeDuration = 10; 
eq = D[y[x, t], t] == k*D[y[x, t], x, x]; 
eq = eq /. k -> 0.5; 
solveHeat[eq_,bc_,ic_,y_,x_,t_,len_,timeDuration_]:=Module[{sol}, 
sol=First@NDSolve[{eq,bc,ic},y[x,t],{x,0,len},{t,0,timeDuration}]; 
Plot3D[y[x,t]/.sol,{x,0,barLength},{t,0,timeDuration}, 
       PlotPoints->30,PlotRange->All,AxesLabel->{"x","time","y[x,t]"}, 
       ImageSize->200,PlotLabel->bc] 
]; 
 
bc={{y[0,t]==1,y[barLength,t]==1}, 
    {y[0,t]==0,y[barLength,t]==0}, 
    {y[0,t]==1,y[barLength,t]==Exp[-t barLength]}, 
    {y[0,t]==0,y[barLength,t]==0} 
}; 
 
ic={y[x,0]== Cos[x], 
y[x,0]==Sin[x], 
y[x,0]==1, 
y[x,0]== Cos[x]Sin[x] 
}; 
 
sol = MapThread[solveHeat[eq,#1,#2,y,x,t,barLength ,timeDuration]&,{bc,ic}]; 
Grid[Partition[sol,2],Frame->All] 
Export["images/mma_e58_1.pdf",%] 
  

Each plot shows the boundary conditions used.