This paper describes an application of weighted Laguerre polynomial functions to produce an unconditionally stable and accurate finite-difference scheme for the
numerical solution of transient diffusion and convection-diffusion problems. The unconditionally stability of Laguerre-FDM (L-FDM) is guaranteed by expanding the time
dependency of the unknown potential as a series of orthogonal functions in the domain
(0,∞), avoiding thus any time integration scheme. The L-FDM is a marching-on-indegree scheme instead of traditional marching-on-in-time methods. For the two heattransfer problems, we demonstrated the accuracy, numerical stability and computational
efficiency of the proposed L-FDM by comparing its results against closed-form analytical
solutions and numerical results obtained from classical finite-difference schemes as, for
instance, the Alternating Direction Implicit (ADI).