# Heat Equation, zero BCs # (Thanks to James Herod's website)

#Initializations: restart;

with(plots):

with(PDEtools):

# The PDE u:=(t,x)->X(x)*T(t);

diff(u(t,x),t)=diff(u(t,x),x,x);

#IC:

f:=x->x*(1-x)^3;

plot(f(x),x=0..1);

# Build a partial sum with 7 components:

for n from 1 to 7 do c[n]:=int(f(x)*sin(n*Pi*x),x=0..1)/int(sin(n*Pi*x)^2,x=0..1): od: n:='n';

approx:=x->sum(c[n]*sin(n*Pi*x),n=1..7):

plot([f(x),approx(x)],x=0..1,color=[BLACK,RED]);

u:=(t,x)->sum(c[n]*exp(-n^2*Pi^2*t)*sin(n*Pi*x),n=1..7);

# Animate the curve through time (click on the plot, then press the play key)

animate(u(t,x),x=0..1,t=0..1/3);

# Plot the temperature over position and time in 3-d:

plot3d(u(t,x),x=0..1,t=0..0.3,axes=NORMAL,orientation=[-20,55]);;