This is my first time ever attempting to use Green's Function and I have taken no formal classes on the subject, so my mistake might be rather entry level, so please bare with me. From the Wikipedia article on Green's Function, the solution to a PDE is: $$u(x) = \int_{-\infty}^{\infty}f(s)G(x,s)ds$$. However, I was reading up on an example on the 1-D Diffusion equation whose Green Function according to the table of Green's Functions should be $G(x,t) = \Theta(t)(\frac{1}{4\pi kt})^{1/2}e^{-\frac{x^2}{4kt}}$, and from that example the integral instead had the general form: $$u(x,t) = \int_{-\infty}^{\infty}f(x)G(x-s,t)ds$$.
In the "Definitions and uses" section, it stated that $G(x,s)$ can be written as $G(x-s)$ under certain circumstances which I assume this problem satisfied, so it looks like that's where that change came from. What is still confusing me is why it would be $f(x)$ and not $f(s)$. Is this just another way of writing it or did it have anything to do with changing $G(x,s,t)$ to $G(x-s,t)$?