Is the fundamental solution (Green's function) of the 1D advection-diffusion equation
$$\frac{\partial{\phi}}{\partial{t}} = D(t)\frac{\partial{^{2}\phi}}{\partial{x^{2}}} - c(t)\frac{\partial{\phi}}{\partial{x}}-k(t)\phi$$
(where the diffusivity $D(t)$, drift speed $c(t)$ and decay coefficient $k(t)$ are functions of $t$ only and not $x$) given by
$$ \phi(x,t) = \frac{1}{\sqrt{4\pi\int_{t_0}^{t}D\left(t'\right)dt'}}\, \exp\left({-\frac{\left\lbrack x-\int_{t_{0}}^{t}c(t')\,{\rm d}t'\right\rbrack^{2}}{4\int_{t_0}^{t}D\left(t'\right){\rm d}t'}-\int_{t_0}^{t}k\left(t'\right)\,{\rm d}t'}\right) $$
provided that $\int{D dt}$, $\int{c dt}$ and $\int{k dt}$ exist?
I believe that this solution is correct, I can't find an error in my derivation. I'll post more details if this result is incorrect, in order to identify the error. But is it correct?
Thanks!
Yes, it is correct assuming your initial condition is $\phi(x,t_0)=\delta(x)$ and the domain is $x\in\mathbb{R}$, $t>t_0$.
First verify that it solves the equation for $t>t_0$.
input: $$\frac{1}{\sqrt{4\pi \int _0^td[s] ds}}\text{Exp}\left[\frac{-\left(x-\text{x0}-\int _0^tc[s] ds\right){}^2}{4\int _0^td[s] ds}-\int _0^tk[s] ds\right]\text{//}\{-1,d[t],-c[t],-k[t]\}.\{D[\#,t],D[\#,x,x],D[\#,x],\#\}\&\text{//}\text{Simplify}$$ output: $0$
To verify that it satisfies that initial condition, compare it with the solution for constant coefficients, and note that $\int_0^t f(s)\,ds = t f(0) + O(t)$ for continuous functions $f$.
I can't resist making a general note that you prove an expression to be correct by proving it to be correct, rather than comparing with an expression (in a textbook maybe) that you suspect is correct. Otherwise, you seem to have got it right yourself.