Solving Second Order PDE with Dirac Delta I want to find the functional form of the Green function G(x,t) for a parabolic differential equation:
$$ \frac{\partial{}G(x,t)}{\partial{}t}=a\frac{\partial{}^2G(x,t)}{\partial{}x^2}+\delta(t)\delta(x)$$
Then, 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 think I need to use a transform like that of Laplace or Fourier. How can I handle the Dirac delta function in this equation?

you start with a homogeneous equation
$$
\frac{\partial{}G(x,t)}{\partial{}t} - a\frac{\partial{}^2G(x,t)}{\partial{}x^2} =0 
$$
the solution is
$$ G(x,t) = \frac{\mathrm{exp}\left( -\frac{{{x}^{2}}}{  4 \, a \, t}\right) }{\sqrt{t \; 2 \; \pi }}$$
The area below the function graph is 
$$
A_t = \frac{\int_{-\infty }^{\infty }{{e}^{-\frac{{{x}^{2}}}{4\cdot a\cdot t}}}dx}{\sqrt{2 \pi  t}} = \sqrt{2 \pi a}
$$
Therefore,
$$
\lim_{t \rightarrow 0^+} \frac{G(x,t)}{A_t} =\delta(t)
$$
Notice that there is no product of delta's.
Then you look for a solution in a convolution form
$$
T(x,t) = \int_{ -\infty }^\infty \int_{-\infty}^\infty f(x' -x,\tau'-t) G(x',t') dx' \, dt'
$$
and you require that both satisfy the same equation.
That is
$$ \frac{\int_{-\infty }^{\infty }\frac{\left( \frac{d}{d\,t} \mathrm{f}\left( x-z,t-u\right) \right)   {{e}^{-\frac{{{z}^{2}}}{4\cdot a\cdot u}}}}{\sqrt{u}}du}{\sqrt{2 \, \pi }}-\frac{a  \int_{-\infty }^{\infty }\frac{\left( \frac{{{d}^{2}}}{d\,{{x}^{2}}}  \mathrm{f}\left( x-z,t-u\right) \right) \cdot {{e}^{-\frac{{{z}^{2}}}{4\cdot a\cdot u}}}}{\sqrt{u}}du}{\sqrt{2\,\pi }}= \sqrt{2 \pi a} \delta(x- z)
$$
