Summary: The objective of this work is to discuss a more general one-dimensional diffusion equation that accounts for certain aspects such as the variation of a parameter that describes the relaxation time of the mass flux and also the presence of a potential field. The equation will have properties similar to a an hyperbolic equation or parabolic equation according to which values of the relaxation parameter or the potential field we consider. In the hyperbolic case we deal with some discontinuities. We apply a numerical scheme to solve this equation, which consists of using an inverse Laplace transform algorithm. The Laplace method is used to remove the time-dependent terms in the goveming equation and boundary conditions. For a constant potential field general solutions can be determined. On the other hand for a non-constant potential field, a spatial discretisation must be considered. We study the convergence of the numerical scheme based on the inverse of Laplace transform and present some test problems.