The backward heat equation.

  You might ask yourself whether we can compute the solution backwards in time. Physically, given the temperature at some time, can we use this information to construct what the temperature was at earlier times? (The change in the direction of time can be simulated by instead merely reversing the sign of $\kappa$ in the heat equation. Such a heat equation with a negative conduction coefficient is called a backward heat equation.) Although you might think solving backwards in time is possible, the difficulty is that small scale features of the temperature distribution in the bar diffuse away very quickly, and are impossible to reconstruct using a computer with a finite accuracy. So generally it is not possible to compute the temperature at earlier times than is given.