I need help finding the functional form of the Green function G(x,t) for a parabolic equation (i.e. heat diffusion etc)
$$ \frac{\partial{}G(x,t)}{\partial{}t}=a\frac{\partial{}^2G(x,t)}{\partial{}x^2}+\delta(t)\delta(x)$$
Using this result, I'd like to write the general solution of the heat equation:
$$\frac{\partial{}T(x,t)}{\partial{}t}=a\frac{\partial{}^2T(x,t)}{\partial{}x^2}+f(x,t) $$
where f is a known source function.
I know that I need to solve for G using suitable integral transforms.
I am not sure what transforms that I should be using to solve this problem and would appreciate some help.
