  A new method for the solution of initial-boundary value problems for
evolution PDEs recently introduced by Fokas is generalised to multidimensions.
Also the relation of this method with the method of images and with the
classical integral transforms is discussed. The new method is easy to
implement, yet it is applicable to problems for which the classical approaches
apparently fail. As illustrative examples, initial-boundary value problems for
the diffusion-convection equation in one and higher dimensions, as well as for
the linearised Korteweg-de Vries equation with the space variables on the
half-line are solved. The suitability of the new method for the analysis of the
the long-time asymptotics is ellucidated.
