The following IBVP for the diffusion equation in one space variable is an example of a well posed parabolic PDE problem for .
One physical interpretation of this problem is that is the temperature at position x and time t in a one dimensional heat conducting medium (say a metal rod, for example) with thermal diffusivity . The initial condition, , specifies the temperature in the rod at the assigned time t = 0. The boundary conditions, and state that the ends of the rod are held at temperature zero for all time.
Simple problems such as this make excellent validation tools for the computational scientist. Since the exact solution to this IBVP can be shown (by separation of variables and Fourier series methods) to be
One can use this exact solution to test the results of a computer code.