The Diffusion Poisson Coupled Model describes the evolution of a dense oxide layer appearing at the surface of carbon steel canisters in contact with a claystone formation. This model is a one dimensional free boundary problem involving drift-diffusion equations on the density of species (electrons, ferric cations and oxygen vacancies), coupled with a Poisson equation on the electrostatic potential and with moving boundary equations, which describe the evolution of the position of each unknown interfaces of the spatial domain. Numerical simulations suggest the existence of traveling wave solutions for this model. These solutions are defined by stationary profiles on a fixed size domain with interfaces moving both at the same velocity. In this paper, we present and apply a computer-assisted method in order to prove the existence of these traveling wave solutions. We also establish a precise and certified description of the solutions.