Do Exercise 1.4.12 if α and β are functions of time. Reference Exercise 1.4.12 Sup...

Do Exercise 1.4.12 if α and β are functions of time.

Reference Exercise 1.4.12

Suppose the concentration u(x, t) of a chemical satisfies Fick’s law (1.2.13), and the initial concentration is given as u(x, 0) = f(x). Consider a region 0 < x < L in which the flow is specified at both ends

Assume α and β are constants.

(a) Express the conservation law for the entire region.

(b) Determine the total amount of chemical in the region as a function of time (using the initial condition).

(c) Under what conditions is there an equilibrium chemical concentration, and what is it?