Derivation of Reaction-Diffusion Equations

Diffusion mechanism models the movement of many individuals in an environment or media. The individuals can be very small such as basic particles in physics, bacteria, molecules, or cells, or very large objects such as animals, plants, or certain kind of events like epidemics, or rumors. The particles reside in a region, which we call Ω, and we assume that Ω is open subset of Rn (the n-th dimensional space with Cartesian coordinate system) with n ≥ 1. In particular, we are interested in the cases of n = 1, 2 and 3, but most material here are true regardless of the dimensions of the space (sometimes n = 1 and n ≥ 2 may be different as we will see.)