1
$\begingroup$

I've ran across the following diffusion equation:

$$\frac{\partial c_i(r,t)}{\partial t}- a \nabla^2c_i(r,t)=b \delta[x-x_1(t)]$$

where $a$ and $b$ are constants related to the context, $\delta$ is dirac.

The solution in the text is given by $$c_i(r,t)=b\int_0^sds\frac{1}{(4\pi a|t-s|)^\frac{3}{2}}\exp{\left(\frac{-[x-x_1(s)]^2}{4a|x-s|}\right)}$$

And the author only mentions that this is a Green's Function solution.

I can read general examples of Green's Function being exploited to solve diffusion equations, but this one involves a dirac delta which disappears in the solution and I'm a little confused. If anyone can offer a bit of exposition on how this result was arrived at I'd be grateful.

$\endgroup$
0
$\begingroup$

With regards to the 'missing' Detla Function: Recall that the Dirac Delta Function only has a non-zero value when its argument is zero, hence the product of any function and the Dirac Delta Function will only be non-zero when the Delta function's argument is set to be zero in the product (Note that the integral of 0 is 0, hence when the product of these two functions is 0,we get no contribution to the integral). I.e., to use the general property: $\delta(x-a)f(x)=f(a)$. NB: $\delta$ has been'swallowed up' by this properly. Does this help?

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.