# Sept. 26, 2002
# 
# The following lines construct, using Fourier series, an approximate
# solution to the heat equation, zero boundary conditions at  x = 0, and
# at  x = 1, and u(x,0) equals the function f defined below.
# 
> f:=piecewise(x<0.5,x,x<1,1-x,0);
> N:=50;
> for i from 1 to N do b[i]:=evalf(2*Int(f*sin(i*Pi*x),x=0..1));
> od:i:='i':
> u:=sum(b[k]*sin(k*Pi*x)*exp(-k^2*Pi^2*t),k=1..N):
> plot([f,subs(t=0.1,u),subs(t=0.5,u),subs(t=10,u)],x=0..1,color=[red,bl
> ue,green,black]);
>