 Solution

Write $\tau =t-t^{\prime }$, $y=x-x^{\prime }$ and $G(y,\tau )=(4\pi D\tau {)}^{-1/2}{e}^{-y^2/(4D\tau )}$ for $\tau >0$, with $D>0$. Direct differentiation givesAlso ${\int }_{\mathbb{R}}Gdy=1$ and, for a smooth bounded test function $f$,Thus $G(\cdot ,\tau )$ tends to a Dirac delta distribution as $\tau \downarrow 0$. Defining the retarded kernel $K=\Theta (\tau )G$ supplies the initial impulse:The source term arises from the jump at $\tau =0$ in the distributional sense; differentiating only the smooth positive-time Gaussian would miss it.