As we know, solution to
\begin{align}
&u_t -u_{xx}=0, \label{A}\\
&u|_{t=0}=\delta(x)\label{B}
\end{align}
is
\begin{equation}
\frac{1}{\sqrt{4\pi t}}e^{-x^2/4t}
\label{C}
\end{equation}
where initial condition currently is understood as $u(x,t)\to 0$ as $t\to +0$, $x\ne 0$; $u(0,t)\to \infty$ as $t\to +0$ and $\int_{-\infty}^\infty u(x,t)\,dx=1$.
a. Consider 1D "radioactive cloud" problem:
\begin{align}
&u_t + v u_x-u_{xx}+\beta u=0, \label{D}\\
&u|_{t=0}=\delta(x)\label{E},
\end{align}
where $v$ is a wind velocity, $\beta$ shows the speed of "dropping on the ground".
Hint: Reduce to (\ref{A})--(\ref{B}) by $u= ve^{-\beta t}$ and $x=y+vt$, use (\ref{C}) for $v$ and then write down $u(x,t)$.
b. Find "contamination level" at $x$
\begin{equation}
D(x)=\beta \int _0^\infty u(x,t)\,dt.
\label{F}
\end{equation}
Hint: by change of variables $t= y^2$ with appropriate $z$ reduce to calculation of
\begin{equation}
\int \exp(-ay^2-by^{-2})\,dy
\label{G}
\end{equation}
and calculate it using f.e.
https://www.wolframalpha.com/ with input
int_0^infty exp (-ay^2-b/y^2)dy(you may need to do it few times)
c. Later
