Analytical solutions of the diffusion differential equation

The water diffusivity in silica is affected by swelling stresses in the surface region which are caused by the silica/water reaction. Since the diffusivity is a function of stress, the consequence is a diffusivity that depends on the local water concentration. Then the solution of the diffusion equation is complicated and makes numerical computations necessary. Disadvantage of numerical computations is the fact that the used extend of the depth range must be finite and, consequently, the semiinfiite body can only be approximated. In the following considerations we will give exact and semi-analytical solutions for diffusion problems in the half-space.