Mathematical models of carbon diffusion in the solid materials using parabolic partial differential equations

In this paper, the mathematical models of carbon diffusion in some kinds of material, having different structures, by means of parabolic partial differential equations (PDE), are proposed. Ricci flow [1], or parabolic Einstein equation, was introduced by R. Hamilton in 1982 having the prototype the heat equation and in the field of diffusion, the Fick's second low. The diffusion of carbon in various materials is described in a physically manner, based on the Thermodynamics of irreversible processes. The real parameters which arise in the solutions of PDE modeling the diffusion phenomenon, will be determined taking account of initial and boundary conditions or by fitting the appropriate solutions to the experimental data, using numerical methods. The practical goal of these theoretical results is the possibility to improve the Mechanical Properties of the manufactured machinery by powder metallurgy.