The non-reduced solution of the fischer model, grain boundary diffusion in R³

We analysed diffusion in a bicrystal formed by two different crystalline phases (grains) having cubic geometry and different transport properties. The contact layer between the two media, the grain boundary, shows different diffusivity also. Through the 'upper' surface of such a bicrystal, the flow of mass occurs and the local concentration of mass on this surface is a known function. The initial distribution of the diffusing element is an arbitrary known function as well. We assume that through the other elements of the external boundary of the bicrystal the transport of mass does not occur. Finally we assume the Fickian diffusion and mass conservation at all interfaces. We have formulated the variational form of such problem. We present an exact analytical formula, a method of its solution, an effective software that allows us to find the distribution of the diffusing element and practical computation of grain boundary diffusion.