Pollutant through a region

[kidia]

Any idea on this please?, Suppose you want to model the spread of pollutant through a region.Suppose D(x,y,t) represents the density of the pollutant per unit area at the point (x,y) in a plane at time t.Suppose D satisfies the diffusion equation

\(\displaystyle \frac{dD}{dx}\)=\(\displaystyle \alpha\)[\(\displaystyle \frac{d^2D}{dx^2}\)+\(\displaystyle \frac{d^2D}{dy^2}]\)

where is some physical constant.If you know that for some particular pollutant D(x,y,t)=\(\displaystyle \exp^{ax+by+ct}\),what can you say about relationship between a,b and c?

 

[Guest]

Presumably this should be:

\(\displaystyle \Large
\frac{\partial D}{\partial t}= \alpha \left[\frac{\partial^2 D}{\partial x^2}+\frac{\partial ^2 D}{\partial y^2}\right]\)

and \(\displaystyle \Large D(x,y,t)=exp^{ax+by+ct}\)

Now:

\(\displaystyle \Large
\frac{\partial D}{\partial t}= cD(x,y,t)\)

and:

\(\displaystyle \Large
\frac{\partial^2 D}{\partial x^2}= a^2 D(x,y,t)\)

\(\displaystyle \Large
\frac{\partial^2 D}{\partial y^2}= b^2 D(x,y,t)\)

So the required relation is between \(\displaystyle \Large \alpha, a, b,\) and \(\displaystyle \Large c\) is:

\(\displaystyle \Large
c=\alpha(a^2+b^2)\)

RonL