Consider a metal rod, with $K=1$ and longitude $\pi$ .
On its extremes, there is heat transmission with the ambient, according to the differential equation: $u_t(x,t)=u_{xx}(x,t)-hu(x,t),$ with $h>0$.
The extremes of the rod have fixed temperatures of $0^\circ C$ on the left, and $1^\circ C$ on the right. The initial temperature of the rod is the function: $f(x)=x(\pi-x).$
Find the temperature distribution $u(x,t)$ on the rod.
Could someone please explain me how to start? I've never seen something like that (with an $hu(x,t)$ added or taken from the equation). I'm really stuck.
Thank you beforehand, and sorry for my english, it's not my natural language.