Poisson's equation in 1 dim.

''(x)= , u(0)=u(1)=0 .

This is part of the worksheet by HA.

Green's function

> G:=(x,y)->piecewise(y>=0 and y <=x,y*(1-x),x<=y and y<=1,x*(1-y));

> x:=0.1:y:=0.2:simplify(G(x,y)-G(y,x));x:='x':y:='y':

> x:='x':f:='f':sol:=int(G(x,y)*f(y),y=0..1);

> assume(x>0);additionally(x<1);

> f:=x->(x^2);

> u:=factor(eval(sol));

> plot(u,x=0..1);

Check:

> simplify(-diff(u,x,x));subs(x=0,u),subs(x=1,u);

O.K.

> f:=x->exp(x);

> u:=factor(eval(sol));

> plot(u,x=0..1);

> simplify(-diff(u,x,x));subs(x=0,u),subs(x=1,u);

> f:=x->cos(x);

> u:=factor(eval(sol));

> plot(u,x=0..1);

> simplify(-diff(u,x,x));subs(x=0,u),subs(x=1,u);

>