First-order problems in time. Heat equation

We consider a metal bar of unit length; we assume that this bar is subjected to a heat input f (x, t ) per unit length and time and that, in addition, the temperature u (x, t ) of the bar is maintained at zero at each of its ends. C is the heat capacity and K the heat diffusion coefficient; remember that the heat diffusion coefficient expresses that temperatures at all points of the bar tend to become uniform. Assuming that the transverse dimension of the bar is negligible compared to its longitudinal dimension, modeling this problem leads to determining u (x, t ) at each point x ∊ [0, 1] and for any instant t ∊ [0, T ], with T representing the integration horizon, the solution to the heat equation :