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.


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?


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