ABSTRACT
We start with a brief derivation of the reaction-diffusion equation. Let a uniform circular tube of cross-sectional area A be held in such a way that that the x-axis is chosen to coincide with its axial direction. We focus on the portion of the tube marked with x = a and x = b (see Figure 5.1). We consider the movement of some substance in this tube. Let the substance be governed by the density function ϕ(x, t), x and t denoting respectively the location and time. The total amount of the substance inside the tube would be Φ ( t ) = ∫ a b ϕ ( x , t ) A d x https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429276477/5e6b97b3-f700-4dc6-82d7-7fdf47b0f880/content/math5_1.jpg"/> Typically, the substance can correspond to a population in which case ϕ(x, t) stands for the population density. But here we will be interested with a flow possessing properties like continuity and differentiability up to sufficient orders.
