Heat equation for a finite bar

Let's consider a metal bar of length L, assimilated to the segment [0, L], and let's call u (x, t ) the temperature of the point of abscissa x at the instant t, knowing that the ends of the bar are in contact with the outside (they are cold) and that, at the instant zero, the point of abscissa x is brought to the temperature h (x ).

How will the bar cool down, in other words, how will u (x, t ) evolve? This evolution follows Newton's law of cooling, which, omitting physical constants and posing Ω = ]0, L[ × ]0, ∞[ (Ω is an open in the plane ${ℝ}^2$ ), leads to the following problem:...