> with(plots):
# input the data functions; try for example:  f=sinx g=0; f=hat g=0,
# f=sinx  g neq 0; f=0 g= small step
# 
> f:=x->(Pi/4-abs(x-Pi/2))*(Heaviside(x-Pi/4)-Heaviside(x-3*Pi/4));
> g:=x->0;
> 

f := x -> (1/4 Pi - | x - 1/2 Pi |)

    (Heaviside(x - 1/4 Pi) - Heaviside(x - 3/4 Pi))


                                g := 0

# input the number of terms and generate the Fourier coeffs.
# 
> N:=10;
> b:=array(1..N);
> 
> B:=array(1..N);
> for k to N do b[k]:=evalf(2/Pi*int(sin(k*x)*f(x),x=0..Pi)) od:
> for k to N do B[k]:=evalf(2/Pi*int(sin(k*x)*g(x),x=0..Pi)) od:

                               N := 10


                       b := array(1 .. 10, [])


                       B := array(1 .. 10, [])

# generate the partial sums of  the solution and the data
# 
> u:=(x,t)->sum('(evalf(b[n])*cos(n*t)+B[n]/n*sin(n*t))*sin(n*x)','n'=1.
> .N);
> sf:=x->sum('evalf(b[n])*sin(n*x)','n'=1..N);
> sg:=x->sum('evalf(B[n])*sin(n*x)','n'=1..N);
> G:=x->int(g(v),v=0..x);
> sG:=x->sum('evalf(B[n])*int(sin(n*w), w=0..x)','n'=1..N);

u := (x, t) ->

       N
     -----
      \      /                       B[n] sin(n t)\
       )    '|evalf(b[n]) cos(n t) + -------------| sin(n x)'
      /      \                             n      /
     -----
    'n' = 1


                            N
                          -----
                           \
              sf := x ->    )    'evalf(b[n]) sin(n x)'
                           /
                          -----
                         'n' = 1


                            N
                          -----
                           \
              sg := x ->    )    'evalf(B[n]) sin(n x)'
                           /
                          -----
                         'n' = 1


                                     x
                                    /
                                   |
                        G := x ->  |   g(v) dv
                                   |
                                  /
                                    0


                        N                    x
                      -----                 /
                       \                   |
          sG := x ->    )    'evalf(B[n])  |   sin(n w) dw'
                       /                   |
                      -----               /
                     'n' = 1                0

# generate the plots of the data, their partial sums, and the solution
# 
> plot3d(u(x,t),x=0..Pi,t=0..10,title=`u(x,t)`);
> plot(sG(x),x=0..Pi,title=`Partial sums of G`,color=red);
> plot(G(x),x=0..Pi,title=`antiderivative G`,color=red);
> plot(sg(x),x=0..Pi,title=`Partial sum of f(x)`,color=orange);
> plot(g(x),x=0..Pi,title=`g(x)`,color=orange);
> plot(sf(x),x=0..Pi,title=`Partial sum of f(x)`,color=blue);
> plot(f(x),x=0..Pi,title=`f(x)`,color=blue);
# 
# animate the traveling waves given by the data
# 
> 
> animate(sf(x-t),x=0..Pi,t=0..10,frames=50,color=blue,title=`f moving
> to the right`);
> animate(sf(x+t),x=0..Pi,t=0..10,frames=50,color=yellow,title=`f moving
> to the left`);
> animate(sG(x-t),x=0..Pi,t=0..10,frames=50,color=blue,title=`G moving
> to the right`);
> animate(sG(x+t),x=0..Pi,t=0..10,frames=50,color=yellow,title=`G moving
> to the left`);
# 
# 
# 
# animation of the solution and the solution and the data
# 
> a1:=animate(u(x,t),x=0..Pi,t=0..10,frames=50,color=green,title=`soluti
> on and data`):
> a2:=animate(1/2*sf(x-t),x=0..Pi,t=0..10,frames=50,color=blue):
> a3:=animate(1/2*sf(x+t),x=0..Pi,t=0..10,frames=50,color=yellow):
> a4:=animate(1/2*sG(x-t),x=0..Pi,t=0..10,frames=50,color=blue):
> a5:=animate(1/2*sG(x+t),x=0..Pi,t=0..10,frames=50,color=yellow): 
> display({a1,a2,a3,a4,a5});
> animate(u(x,t),x=0..Pi,t=0..10,frames=50,color=green,title=`solution`)
> ;
>