1) Assume a fluid (say, water) occupies a domain D⊂ R^{3} and has velocity field V=V(x, t). A substance (say, a day) is suspended into the fluid and would be transported by the fluid as well as diffused within it; let u= u(x,t) be the concentration of the substance( mass per unit volume). Let Φ= Φ(x,t) be the concentration flux (mass per unit area per unit time, analogous to heat flux). Let B(x) be a ball of radius r > 0 contained in D. Derive the conservation law.
∫_{B(x)}u_{t} dV= -∫_{∂B(x)}Φ.n_{0}dS.
Fick's law for diffusion states that the concentration flux due to diffusion is proportional to the gradient of the concentration flux due to diffusion is proportional to the gradient of the concentration. Deduce that Φ= -k∇+uV.
Apply the divergence theorem to the conservation law and substitute the flux formula to arrive at the diffusion- transport equation:
u_{t}= ∇.(k∇u)+ ∇.(uV).
in the absence of diffusion (k=0), this is the higher- dimensional transport equation. If the fluid is motionless, it is known as diffusion equation.
2) find out the causality principle for the Klein- Gordon equation in one dimension. Deduce that the speed of propagation is at most c.
3) Drive causality principle for weave equation in R^{2} u_{tt}= c^{2} ∇^{2}u.