Analytical solutions for transient diffusion problems in infinite media

Abstract

In this paper, analytical solutions for transient diffusion problems in one-dimensional, two-dimensional and three-dimensional infinite/semi-infinite media with the locally prescribed field variable have been derived. Since the locally prescribed field variable forms the boundary condition of the diffusion problem, its distribution is considered to be either linear or quadratic in a local area of the problem domain so that it can exactly match the shape functions of linear and quadratic finite elements. For this reason, the present analytical solutions can be used to make an exact estimation of discretization error of the finite/infinite element method when this method is used to model transient diffusion problems in infinite/semi-infinite media. Moreover, the present solutions can also be employed to study some basic behaviours of transient diffusion problems in infinite media.