  This paper presents a new numerical method which approximates Neumann type
null controls for the heat equation and is based on the Fokas method. This is a
direct method for solving problems originating from the control theory, which
allows the realisation of an efficient numerical algorithm that requires small
computational effort for determining the null control with exponentially small
error. Furthermore, the unified character of the Fokas method makes the
extension of the numerical algorithm to a wide range of other linear PDEs and
different type of boundary conditions straightforward.
