${\displaystyle f_{t}-f_{xx}=\pi ^{2}\sin(\pi x)\,}$    on    ${\displaystyle [0,1]\times [0,\infty ]\,}$

with zero initial condition

${\displaystyle f(x,0)=0\,}$

and zero boundary conditions

${\displaystyle f(0,t)=f(1,t)=0\,}$.