Nuclear fuel materials are those that are capable of undergoing fission reactions along with the liberation of enormous quantity of energy. The fuel cycle costs are 20% of nuclear electricity generation cost. This new book evaluates the use of the damped wave diffusion and relaxation equation to nuclear fuels processes. Simultaneous diffusion and auto-catalytic reactions and the wave diffusion phenomena may have a combined effect of sub-critical damped oscillations in concentration under certain conditions. Closed form analytical solutions are developed from the hyperbolic partial differential equations that govern the phenomena.