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

where

One can use this exact solution to test the results of a computer code.