I have a question that is bugging me and for which i would appreciate some help or to be pointed out to the litterature on the subject.
Is it possible to solve the heat equation with setting an initial and terminal condition?
that is
$$ \frac{\partial f}{\partial t} = \alpha \frac{\partial^2 f}{\partial x^2}\\ f(0,x)=g(x)\\ f(T,x)=h(x)\\ for \: t \in [0,T], \: x \in \mathbb{R} $$
Thank you for your help!
No. Solutions of the heat equation (if say to limit oneself with bounded functions $u$) is totally defined by the inicial function $g$, so the final function $h$ is determined uniquely by $g$.