Our purpose is to build an inverse method which best fits a model of artery flow and experimental measurements (we assume that we are able to measure the displacement of the artery as a function of time at three stations). Having no clinical data, we simulate these measurements with the numerical computations from a "boundary layer" code. First, we revisit the system of Ling and Atabek of boundary layer type for the transmission of a pressure pulse in the arterial system for the case of an elastic wall (but we solve it without any simplification in the $u\partial u/\partial x$ term). Then, using a method analogous to the well known Von Kármán-Pohlhausen method from aeronautics but transposed here for a pulsatile flow, we build a system of three coupled non-linear partial differential equations depending only on time and axial co-ordinate. This system governs the dynamics of internal artery radius, centre velocity and a quantity related to the presence of viscous effects. These two methods give nearly the same numerical results. Second, we construct an inverse method: the aim is to find for the simple integral model, the physical parameters to put in the "boundary layer" code (simulating clinical data). This is done by varying in the integral model the viscosity and elasticity in order to fit best with the data. To achieve this in a rational way, we have to minimise a cost function, which involves the computation of the adjoint system of the integral method. The good set of parameters (i.e. viscosity, and two coefficients of a wall law) is effectively found again. It opens the perspective for application in real clinical cases of this new non-invasive method for evaluating the viscosity of the flow and elasticity of the wall.