The diffusion process in a transistor is simulated by two partial differential equations: Laplace's and Poisson's. The proposed solution consists in an iterative scheme for a determination of the interface between these two equations.The first one consists in the use of an adjoint potential for the reduction of the domain to the boundary integral. The second method consists in a an analytical reduction of the domain integral and a numerical evaluation of the boundary integral using Simpson's rule and Gauss quadrature scheme. The two methods are applied to a free boundary value problem : A Junction Gate Field Effect Transistor. The results obtained are analysed and compared.